The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory

Wolfgang Lück^{⋆} and Holger Reich^{⋆⋆}

Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany

Summary. We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C^{∗}-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G. Key words: K- and L-groups of group rings and group C^{∗}-algebras, Baum-Connes Conjecture, Farrell-Jones Conjecture. Mathematics subject classification 2000: 19A31, 19B28, 19D99, 19G24, 19K99, 46L80.

Introduction

This survey article is devoted to the Baum-Connes Conjecture about the topological K-theory of the reduced group C^{∗}-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a discrete group G. We will present a unified approach to these conjectures hoping that it will stimulate further interactions and exchange of methods and ideas between algebraic and geometric topology on the one side and non-commutative geometry on the other.

Each of the above mentioned conjectures has already been proven for astonishingly large classes of groups using a variety of different methods coming from operator theory, controlled topology and homotopy theory. Methods have been developed for this purpose which turned out to be fruitful in other

⋆ email: lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ fax: +49 251 8338370

⋆⋆ email: holger.reich@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/reichh/

contexts. The conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures, the stable Gromov-Lawson-Rosenberg Conjecture about the existence of Riemannian metrics with positive scalar curvature and the Kadison Conjecture about idempotents in the reduced C^{∗}-algebra of a torsionfree discrete group G.

Formulation of the Conjectures

The Baum-Connes and Farrell-Jones Conjectures predict that for every discrete group G the following so called “assembly maps” are isomorphisms.

K_{n}^{G}(E_{FIN} (G)) → K_{n}(C_{r}^{∗}(G));
H_{n}^{G}(E_{VCY}(G); K_{R}) → K_{n}(RG);
H_{n}^{G}(E_{VCY}(G); L_{R}^{⟨−∞⟩}) → L_{n}^{⟨−∞⟩}(RG).

Here the targets are the groups one would like to understand, namely the topological K-groups of the reduced group C^{∗}-algebra in the Baum-Connes case and the algebraic K- or L-groups of the group ring RG for R an associative ring with unit. In each case the source is a G-homology theory evaluated on a certain classifying space. In the Baum-Connes Conjecture the G-homology theory is equivariant topological K-theory and the classifying space E_{FIN} (G) is the classifying space of the family of finite subgroups, which is often called the classifying space for proper G-actions and denoted \underline{E}G in the literature. In the Farrell-Jones Conjecture the G-homology theory is given by a certain K- or L-theory spectrum over the orbit category, and the classifying space E_{VCY}(G) is the one associated to the family of virtually cyclic subgroups. The conjectures say that these assembly maps are isomorphisms.

These conjectures were stated in [28, Conjecture 3.15 on page 254] and [111, 1.6 on page 257]. Our formulations differ from the original ones, but are equivalent. In the case of the Farrell-Jones Conjecture we slightly generalize the original conjecture by allowing arbitrary coefficient rings instead of ℤ. At the time of writing no counterexample to the Baum-Connes Conjecture 2.3 or the Farrell-Jones Conjecture 2.2 is known to the authors.

One can apply methods from algebraic topology such as spectral sequences and Chern characters to the sources of the assembly maps. In this sense the sources are much more accessible than the targets. The conjectures hence lead to very concrete calculations. Probably even more important is the structural insight: to what extent do the target groups show a homological behaviour. These aspects can be treated completely analogously in the Baum-Connes and the Farrell-Jones setting.

However, the conjectures are not merely computational tools. Their importance comes from the fact that the assembly maps have geometric interpretations in terms of indices in the Baum-Connes case and in terms of surgery theory in the Farrell-Jones case. These interpretations are the key ingredient in applications and the reason that the Baum-Connes and Farrell-Jones Conjectures imply so many other conjectures in non-commutative geometry, geometric topology and algebra.

A User’s Guide

A reader who wants to get specific information or focus on a certain topic should consult the detailed table of contents, the index and the index of notation in order to find the right place in the paper. We have tried to write the text in a way such that one can read small units independently from the rest. Moreover, a reader who may only be interested in the Baum-Connes Conjecture or only in the Farrell-Jones Conjecture for K-theory or for L-theory can ignore the other parts. But we emphasize again that one basic idea of this paper is to explain the parallel treatment of these conjectures.

A reader without much prior knowledge about the Baum-Connes Conjecture or the Farrell-Jones Conjecture should begin with Chapter 1. There, the special case of a torsionfree group is treated, since the formulation of the conjectures is less technical in this case and there are already many interesting applications. The applications themselves however, are not needed later. A more experienced reader may pass directly to Chapter 2.

Other (survey) articles on the Farrell-Jones Conjecture and the Baum-Connes Conjecture are [111], [128], [147], [225], [307].

Notations and Conventions

Here is a briefing on our main notational conventions. Details are of course discussed in the text. The columns in the following table contain our notation for: the spectra, their associated homology theory, the right hand side of the corresponding assembly maps, the functor from groupoids to spectra and finally the G-homology theory associated to these spectra valued functors.

BU  K_{n}(X)  K_{n}(C_{r}^{∗}G)K^{top} H_{n}^{G}(X; K^{top})
K(R)  H_{n}(X; K(R)) K_{n}(RG) K_{R} H_{n}^{G}(X; K_{R})
L^{⟨j⟩}(R) H_{n}(X; L^{⟨j⟩}(R))L_{n}^{⟨j⟩}(RG) L_{R}^{⟨j⟩} H_{n}^{G}(X; L_{R}^{⟨j⟩})

We would like to stress that K without any further decoration will always refer to the non-connective K-theory spectrum. L^{⟨j⟩} will always refer to quadratic L-theory with decoration j. For a C^{∗}- or Banach algebra A the symbol K_{n}(A) has two possible interpretations but we will mean the topological K-theory.

A ring is always an associative ring with unit, and ring homomorphisms are always unital. Modules are left modules. We will always work in the category of compactly generated spaces, compare [295] and [330, I.4]. For our conventions concerning spectra see Section 6.2. Spectra are denoted with boldface letters such as E.

Acknowledgements

We would like to thank Arthur Bartels, Indira Chatterji, Tom Farrell, Joachim Grunewald, Nigel Higson, Lowell Jones, Michel Matthey, Guido Mislin, Erik Pedersen, David Rosenthal, Marco Schmidt, Alain Valette, Marco Varisco, Julia Weber, Shmuel Weinberger, Bruce Williams, Guoliang Yu and the referee for useful comments and discussions. The first author thanks the Max-Planck Institute for Mathematics in Bonn for its hospitality during his visit in November and December 2002, and both authors thank the Department of Mathematics of the University of Chicago, in particular Shmuel Weinberger, for the pleasant stay in March 2003, when parts of this paper were written. We particularly enjoyed the “Windy City”-Martini respectively the “Earl Grey” on the 96th floor of the Hancock building, where this introduction was drafted.

Münster, August 2003

1 The Conjectures in the Torsion Free Case . . . . . . . . . . . . . . . . 7

1.1 Algebraic K-Theory - Low Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Applications I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 The s-Cobordism Theorem and the Poincaré Conjecture . . . . . . . . . 9 1.2.2 Finiteness Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.3 Negative K-Groups and Bounded h-Cobordisms . . . . . . . . . . . . . . . . 12 1.3 Algebraic K-Theory - All Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Applications II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.1 The Relation to Pseudo-Isotopy Theory . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.2 Negative K-Groups and Bounded Pseudo-Isotopies . . . . . . . . . . . . . 17 1.5 L-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Applications III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6.1 The Borel Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6.2 Automorphisms of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 The Baum-Connes Conjecture in the Torsion Free Case . . . . . . . . . 23 1.8 Applications IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.8.1 The Trace Conjecture in the Torsion Free Case . . . . . . . . . . . . . . . . . 25 1.8.2 The Kadison Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.8.3 Other Related Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.8.4 L^{2}-Rho-Invariants and L^{2}-Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.9 Applications V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.9.1 Novikov Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.9.2 The Original Novikov Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.9.3 Relations between the Novikov Conjectures . . . . . . . . . . . . . . . . . . . . 31 1.9.4 The Zero-in-the-Spectrum Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 The Conjectures in the General Case . . . . . . . . . . . . . . . . . . . . . 33

2.1 Formulation of the Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.1 G-CW-Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.1.2 Families of Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.1.3 Classifying Spaces for Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.4 G-Homology Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Varying the Family of Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.1 The General Versions Specialize to the Torsion Free Versions . . . . . 41 2.2.2 The Baum-Connes Conjecture and the Family VCY . . . . . . . . . . . . . 42 2.2.3 The Baum-Connes Conjecture and the Family FCY . . . . . . . . . . . . . 42 2.2.4 Algebraic K-Theory for Special Coefficient Rings . . . . . . . . . . . . . . . 42 2.2.5 Splitting off Nil-Terms and Rationalized Algebraic K-Theory …. 44 2.2.6 Inverting 2 in L-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.7 L-theory and Virtually Cyclic Subgroups of the First Kind . . . . . . . 45 2.2.8 Rationally FIN Reduces to FCY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 More Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1 Applications VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.1 Low Dimensional Algebraic K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.2 G-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.3 Bass Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Applications VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Novikov Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.2 Relating Topological K-Theory and L-Theory . . . . . . . . . . . . . . . . . . 55 3.3 Applications VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.1 The Modified Trace Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.2 The Stable Gromov-Lawson-Rosenberg Conjecture . . . . . . . . . . . . . . 58

4 Generalizations and Related Conjectures . . . . . . . . . . . . . . . . . 60

4.1 Variants of the Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . 60 4.1.1 The Real Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.2 The Version for Maximal Group C^{∗}-Algebras . . . . . . . . . . . . . . . . . . 61 4.1.3 The Bost Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1.4 The Baum-Connes Conjecture with Coefficients . . . . . . . . . . . . . . . . 62 4.1.5 The Coarse Baum Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1.6 The Baum-Connes Conjecture for Non-Discrete Groups . . . . . . . . . 65 4.2 Variants of the Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.1 Pseudoisotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.2 Fibered Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2.3 The Isomorphism Conjecture for NK-groups . . . . . . . . . . . . . . . . . . . 69 4.2.4 Algebraic K-Theory of the Hecke Algebra . . . . . . . . . . . . . . . . . . . . . 70

5 Status of the Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Status of the Baum-Connes-Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 70 5.1.1 The Baum-Connes Conjecture with Coefficients . . . . . . . . . . . . . . . . 70 5.1.2 The Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.3 The Injectivity Part of the Baum-Connes Conjecture . . . . . . . . . . . . 73 5.1.4 The Coarse Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Status of the Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.1 The Fibered Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.2 The Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.3 The Farrell-Jones Conjecture for Arbitrary Coefficients . . . . . . . . . . 79 5.2.4 Injectivity Part of the Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . 79 5.3 List of Groups Satisfying the Conjecture . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 Inheritance Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.1 Directed Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.2 Passing to Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.3 Extensions of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.4 Products of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4.5 Subgroups of Finite Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4.6 Groups Acting on Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Equivariant Homology Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1 The Definition of an Equivariant Homology Theory . . . . . . . . . . . . . 89 6.2 Constructing Equivariant Homology Theories . . . . . . . . . . . . . . . . . . 92 6.3 K- and L-Theory Spectra over Groupoids . . . . . . . . . . . . . . . . . . . . . 96 6.4 Assembly Maps in Terms of Homotopy Colimits . . . . . . . . . . . . . . . . 98 6.5 Naturality under Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Methods of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.1 Analytic Equivariant K-Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.2 The Analytic Assembly Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.3 Equivariant KK-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.4 The Dirac-Dual Dirac Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.5 An Example of a Dirac Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.6 Banach KK-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.7 Controlled Topology and Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.8 Assembly as Forget Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.9 Methods to Improve Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.10 The Descent Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.11 Comparing to Other Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.1 K- and L- Groups for Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.1.1 Topological K-Theory for Finite Groups . . . . . . . . . . . . . . . . . . . . . . . 111 8.1.2 Algebraic K-Theory for Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . 112 8.1.3 Algebraic L-Theory for Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.2 Rational Computations for Infinite Groups . . . . . . . . . . . . . . . . . . . . . 114 8.2.1 Rationalized Topological K-Theory for Infinite Groups . . . . . . . . . . 115 8.2.2 Rationalized Algebraic K-Theory for Infinite Groups . . . . . . . . . . . . 115 8.2.3 Rationalized Algebraic L-Theory for Infinite Groups . . . . . . . . . . . . 117 8.3 Integral Computations for Infinite Groups . . . . . . . . . . . . . . . . . . . . . 117 8.4 Techniques for Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.4.1 Equivariant Atiyah-Hirzebruch Spectral Sequence . . . . . . . . . . . . . . . 120 8.4.2 p-Chain Spectral Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.4.3 Equivariant Chern Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

1 The Conjectures in the Torsion Free Case

In this chapter we discuss the Baum-Connes and Farrell-Jones Conjectures in the case of a torsion free group. Their formulation is less technical than in the general case, but already in the torsion free case there are many interesting and illuminating conclusions. In fact some of the most important consequences of the conjectures, like for example the Borel Conjecture (see Conjecture 1.27) or the Kadison Conjecture (see Conjecture 1.39), refer exclusively to the torsion free case. On the other hand in the long run the general case, involving groups with torsion, seems to be unavoidable. The general formulation yields a clearer and more complete picture, and furthermore there are proofs of the conjectures for torsion free groups, where in intermediate steps of the proof it is essential to have the general formulation available (compare Section 7.9).

The statement of the general case and further applications will be presented in the next chapter. The reader may actually skip this chapter and start immediately with Chapter 2.

We have put some effort into dealing with coefficient rings R other than the integers. A topologist may a priori be interested only in the case R = ℤ but other cases are interesting for algebraists and also do occur in computations for integral group rings.

1.1 Algebraic K-Theory - Low Dimensions

A ring R is always understood to be associative with unit. We denote by K_{n}(R) the algebraic K-group of R for n ∈ ℤ. In particular K_{0}(R) is the Grothendieck group of finitely generated projective R-modules and elements in K_{1}(R) can be represented by automorphisms of such modules. In this section we are mostly interested in the K-groups K_{n}(R) with n ≤ 1. For definitions of these groups we refer to [221], [266], [286], [299], [323] for n = 0, 1 and to [22] and [268] for n ≤ 1.

For a ring R and a group G we denote by

A_{0} = K_{0}(i): K_{0}(R) → K_{0}(RG)

the map induced by the natural inclusion i: R → RG. Sending (g, [P]) ∈ G × K_{0}(R) to the class of the RG-automorphism

R[G] ⊗_{R} P → R[G] ⊗_{R} P, u ⊗ x ↦ ug^{−1} ⊗ x

defines a map Φ: G_{ab}⊗_{ℤ}K_{0}(R) → K_{1}(RG), where G_{ab} denotes the abelianized group. We set

A_{1} = Φ ⊕ K_{1}(i): G_{ab} ⊗_{ℤ} K_{0}(R) ⊕ K_{1}(R) → K_{1}(RG).

We recall the notion of a regular ring. We think of modules as left modules unless stated explicitly differently. Recall that R is Noetherian if any submodule of a finitely generated R-module is again finitely generated. It is called regular if it is Noetherian and any R-module has a finite-dimensional projective resolution. Any principal ideal domain such as ℤ or a field is regular.

The Farrell-Jones Conjecture about algebraic K-theory implies for a torsion free group the following conjecture about the low dimensional K-theory groups.

Conjecture 1.1 (The Farrell-Jones Conjecture for Low Dimensional K-Theory and Torsion Free Groups). Let G be a torsion free group and let R be a regular ring. Then

K_{n}(RG) = 0 for n ≤ −1

and the maps

K_{0}(R) ⟶^{A_{0}} K_{0}(RG) and
G_{ab} ⊗_{ℤ} K_{0}(R) ⊕ K_{1}(R) ⟶^{A_{1}} K_{1}(RG)

are both isomorphisms.

Every regular ring satisfies K_{n}(R) = 0 for n ≤ −1 [268, 5.3.30 on page 295] and hence the first statement is equivalent to K_{n}(i): K_{n}(R) → K_{n}(RG) being an isomorphism for n ≤ −1. In Remark 1.15 below we explain why we impose the regularity assumption on the ring R.

For a regular ring R and a group G we define Wh_{1}^{R}(G) as the cokernel of the map A_{1} and Wh_{0}^{R}(G) as the cokernel of the map A_{0}. In the important case where R = ℤ the group Wh_{1}^{ℤ}(G) coincides with the classical Whitehead group Wh(G) which is the quotient of K_{1}(ℤG) by the subgroup consisting of the classes of the units ±g ∈ (ℤG)^{inv} for g ∈ G. Moreover for every ring R we define the reduced algebraic K-groups K̃_{n}(R) as the cokernel of the natural map K_{n}(ℤ) → K_{n}(R). Obviously Wh_{0}^{ℤ}(G) = K̃_{0}(ℤG).

Lemma 1.2. The map A_{0} is always injective. If R is commutative and the natural map ℤ → K_{0}(R), 1 ↦ [R] is an isomorphism, then the map A_{1} is injective.

Proof. The augmentation ϵ: RG → R, which maps each group element g to 1, yields a retraction for the inclusion i: R → RG and hence induces a retraction for A_{0}. If the map ℤ → K_{0}(R), 1 ↦ [R] induces an isomorphism and R is commutative, then we have the commutative diagram

\xymatrix{G_{ab} ⊗_{ℤ} K_{0}(R) ⊕ K_{1}(R) \ar[dd]_{≅} \ar[r]^{A_{1}} & K_{1}(RG) \ar[d] \\  & K_{1}(RG_{ab}) \ar[d]^{(det,K_{1}(ϵ))} \\ G_{ab} ⊕ K_{1}(R) \ar[r] & RG_{ab}^{inv} ⊕ K_{1}(R),}

where the upper vertical arrow on the right is induced from the map G → G_{ab} to the abelianization. Since RG_{ab} is a commutative ring we have the determinant det: K_{1}(RG_{ab}) → (RG_{ab})^{inv}. The lower horizontal arrow is induced from the obvious inclusion of G_{ab} into the invertible elements of the group ring RG_{ab} and in particular injective. ⊓ ⊔

In the special case R = ℤ Conjecture 1.1 above is equivalent to the following conjecture.

Conjecture 1.3 (Vanishing of Low Dimensional K-Theory for Torsionfree Groups and Integral Coefficients). For every torsion free group G we have

K_{n}(ℤG) = 0 for n ≤ −1, K̃_{0}(ℤG) = 0 and Wh(G) = 0.

Remark 1.4 (Torsionfree is Necessary). In general K̃_{0}(ℤG) and Wh(G) do not vanish for finite groups. For example K̃_{0}(ℤ[ℤ/23]) ≅ ℤ/3 [221, page 29, 30] and Wh(ℤ/p) ≅ ℤ^{\frac{p−3}{2}} for p an odd prime [70, 11.5 on page 45]. This shows that the assumption that G is torsion free is crucial in the formulation of Conjecture 1.1 above.

For more information on K̃_{0}(ℤG) and Whitehead groups of finite groups see for instance [22, Chapter XI], [79], [220], [231], [232] and [299].

1.2 Applications I

We will now explain the geometric relevance of the groups whose vanishing is predicted by Conjecture 1.3.

1.2.1 The s-Cobordism Theorem and the Poincaré Conjecture

The Whitehead group Wh(G) plays a key role if one studies manifolds because of the so called s-Cobordism Theorem. In order to state it, we explain the notion of an h-cobordism.

Manifold always means smooth manifold unless it is explicitly stated differently. We say that W or more precisely (W; M^{−}, f^{−}, M^{+}, f^{+}) is an ndimensional cobordism over M^{−} if W is a compact n-dimensional manifold together with the following: a disjoint decomposition of its boundary ∂W into two closed (n − 1)-dimensional manifolds ∂^{−}W and ∂^{+}W, two closed (n − 1)- dimensional manifolds M^{−} and M^{+} and diffeomorphisms f^{−}: M^{−} → ∂^{−}W and f^{+} : M^{+} → ∂^{+}W. The cobordism is called an h-cobordism if the inclusions i^{−}: ∂^{−}W → W and i^{+} : ∂^{+}W → W are both homotopy equivalences. Two cobordisms (W; M^{−}, f^{−}, M^{+}, f^{+}) and (W′; M^{−}, f^{′−}, M^{′+}, f^{′+}) over M^{−} are diffeomorphic relative M^{−} if there is a diffeomorphism F: W → W′ with F ∘ f^{−} = f^{′−}. We call a cobordism over M^{−} trivial, if it is diffeomorphic relative M^{−} to the trivial h-cobordism given by the cylinder M^{−} × [0, 1] together with the obvious inclusions of M^{−} × {0} and M^{−} × {1}. Note that “trivial” implies in particular that M^{−} and M^{+} are diffeomorphic.

The question whether a given h-cobordism is trivial is decided by the Whitehead torsion τ(W; M^{−}) ∈ Wh(G) where G = π_{1}(M^{−}). For the details of the definition of τ(W; M^{−}) the reader should consult [70], [220] or Chapter 2 in [200]. Compare also [266].

Theorem 1.5 (s-Cobordism Theorem). Let M^{−} be a closed connected oriented manifold of dimension n ≥ 5 with fundamental group G = π_{1}(M^{−}). Then

(i) An h-cobordism W over M^{−} is trivial if and only if its Whitehead torsion τ(W, M^{−}) ∈ Wh(G) vanishes.

(ii) Assigning to an h-cobordism over M^{−} its Whitehead torsion yields a bijection from the diffeomorphism classes relative M^{−} of h-cobordisms over M^{−} to the Whitehead group Wh(G).

The s-Cobordism Theorem is due to Barden, Mazur and Stallings. There are also topological and PL-versions. Proofs can be found for instance in [173], [176, Essay III], [200] and [272, page 87-90].

The s-Cobordism Theorem tells us that the vanishing of the Whitehead group (as predicted in Conjecture 1.3 for torsion free groups) has the following geometric interpretation.

Consequence 1.6. For a finitely presented group G the vanishing of the Whitehead group Wh(G) is equivalent to the statement that each h-cobordism over a closed connected manifold M^{−} of dimension dim(M^{−}) ≥ 5 with fundamental group π_{1}(M^{−}) ≅ G is trivial.

Knowing that all h-cobordisms over a given manifold are trivial is a strong and useful statement. In order to illustrate this we would like to discuss the case where the fundamental group is trivial.

Since the ring ℤ has a Gaussian algorithm, the determinant induces an isomorphism K_{1}(ℤ) ⟶^{≅} {±1} (compare [268, Theorem 2.3.2]) and the Whitehead group Wh({1}) of the trivial group vanishes. Hence any h-cobordism over a simply connected closed manifold of dimension ≥ 5 is trivial. As a consequence one obtains the Poincaré Conjecture for high dimensional manifolds.

Theorem 1.7 (Poincaré Conjecture). Suppose n ≥ 5. If the closed manifold M is homotopy equivalent to the sphere S^{n}, then it is homeomorphic to S^{n}.

Proof. We only give the proof for dim(M) ≥ 6. Let f: M → S^{n} be a homotopy equivalence. Let D_{−}^{n} ⊂ M and D_{+}^{n} ⊂ M be two disjoint embedded disks. Let W be the complement of the interior of the two disks in M. Then W turns out to be a simply connected h-cobordism over ∂D_{−}^{n}. Hence we can find a diffeomorphism

F: (∂D_{−}^{n} × [0, 1]; ∂D_{−}^{n} × {0}, ∂D_{−}^{n} × {1}) → (W; ∂D_{−}^{n}, ∂D_{+}^{n})

which is the identity on ∂D_{−}^{n} = ∂D_{−}^{n} × {0} and induces some (unknown) diffeomorphism f^{+} : ∂D_{−}^{n} × {1} → ∂D_{+}^{n}. By the Alexander trick one can extend f^{+} : ∂D_{−}^{n} = ∂D_{−}^{n} × \{1\} → ∂D_{+}^{n} to a homeomorphism \overline{f^{+}} : D_{−}^{n} → D_{+}^{n}. Namely, any homeomorphism f: S^{n−1} → S^{n−1} extends to a homeomorphism \overline{f}: D^{n} → D^{n} by sending t · x for t ∈ [0, 1] and x ∈ S^{n−1} to t · f(x). Now define a homeomorphism h: D_{−}^{n} × {0} ∪_{i_{−}} ∂D_{−}^{n} × [0, 1] ∪_{i_{+}} D_{−}^{n} × {1} → M for the canonical inclusions i_{k}: ∂D_{−}^{n} × {k} → ∂D_{−}^{n} × [0, 1] for k = 0, 1 by h|_{D_{−}^{n}×\{0\}} = id, h|_{∂D_{−}^{n}×[0,1]} = F and h|_{D_{−}^{n}×\{1\}} = \overline{f^{+}}. Since the source of h is obviously homeomorphic to S^{n}, Theorem 1.7 follows. ⊓ ⊔

The Poincaré Conjecture (see Theorem 1.7) is at the time of writing known in all dimensions except dimension 3. It is essential in its formulation that one concludes M to be homeomorphic (as opposed to diffeomorphic) to S^{n}. The Alexander trick does not work differentiably. There are exotic spheres, i.e. smooth manifolds which are homeomorphic but not diffeomorphic to S^{n} [218].

More information about the Poincaré Conjecture, the Whitehead torsion and the s-Cobordism Theorem can be found for instance in [50], [70], [86], [131], [132], [141], [173], [200], [219], [220], [266] and [272].

1.2.2 Finiteness Obstructions

We now discuss the geometric relevance of K̃_{0}(ℤG).

Let X be a CW -complex. It is called finite if it consists of finitely many cells. It is called finitely dominated if there is a finite CW -complex Y together with maps i: X → Y and r: Y → X such that r∘i is homotopic to the identity on X. The fundamental group of a finitely dominated CW -complex is always finitely presented.

While studying existence problems for spaces with prescribed properties (like for example group actions), it happens occasionally that it is relatively easy to construct a finitely dominated CW -complex within a given homotopy type, whereas it is not at all clear whether one can also find a homotopy equivalent finite CW -complex. Wall’s finiteness obstruction, a certain obstruction element õ(X) ∈ K̃_{0}(ℤπ_{1}(X)), decides the question.

Theorem 1.8 (Properties of the Finiteness Obstruction). Let X be a finitely dominated CW -complex with fundamental group π = π_{1}(X).

(i) The space X is homotopy equivalent to a finite CW -complex if and only if õ(X) = 0 ∈ K̃_{0}(ℤπ).

(ii) Every element in K̃_{0}(ℤG) can be realized as the finiteness obstruction õ(X) of a finitely dominated CW -complex X with G = π_{1}(X), provided that G is finitely presented.

(iii) Let Z be a space such that G = π_{1}(Z) is finitely presented. Then there is a bijection between K̃_{0}(ℤG) and the set of equivalence classes of maps f: X → Z with X finitely dominated under the equivalence relation explained below.

The equivalence relation in (iii) is defined as follows: Two maps f: X → Z and f′: X′ → Z with X and X′ finitely dominated are equivalent if there exists a commutative diagram

\xymatrix{X \ar[drr]_{f} \ar[r]^{j} & X_{1} \ar[dr]^{f_{1}} \ar[r]^{h} & X_{2} \ar[d]_{f_{2}} & X_{3} \ar[dl]_{f_{3}} \ar[l]_{h′} & X′ \ar[dll]^{f′} \ar[l]_{j′} \\ & & Z} ,

where h and h′ are homotopy equivalences and j and j′ are inclusions of subcomplexes for which X_{1}, respectively X_{3}, is obtained from X, respectively X′, by attaching a finite number of cells.

The vanishing of K̃_{0}(ℤG) as predicted in Conjecture 1.3 for torsion free groups hence has the following interpretation.

Consequence 1.9. For a finitely presented group G the vanishing of K̃_{0}(ℤG) is equivalent to the statement that any finitely dominated CW -complex X with G ≅ π_{1}(X) is homotopy equivalent to a finite CW -complex.

For more information about the finiteness obstruction we refer for instance to [125], [126], [196], [224], [257], [266], [308], [317] and [318].

1.2.3 Negative K-Groups and Bounded h-Cobordisms

One possible geometric interpretation of negative K-groups is in terms of bounded h-cobordisms. Another interpretation will be explained in Subsection 1.4.2 below.

We consider manifolds W parametrized over ℝ^{k}, i.e. manifolds which are equipped with a surjective proper map p: W → ℝ^{k}. We will always assume that the fundamental group(oid) is bounded, compare [239, Definition 1.3]. A map f: W → W′ between two manifolds parametrized over ℝ^{k} is bounded if {p′ ∘ f(x) − p(x) | x ∈ W } is a bounded subset of ℝ^{k}.

A bounded cobordism (W; M^{−}, f^{−}, M^{+}, f^{+}) is defined just as in Subsection 1.2.1 but compact manifolds are replaced by manifolds parametrized over ℝ^{k} and the parametrization for M^{±} is given by p_{W} ∘ f^{±}. If we assume that the inclusions i^{±}: ∂^{±}W → W are homotopy equivalences, then there exist deformations r^{±}: W × I → W, (x, t) ↦ r_{t}^{±}(x) such that r_{0}^{±} = id_{W} and r_{1}^{±}(W) ⊂ ∂^{±}W.

A bounded cobordism is called a bounded h-cobordism if the inclusions i^{±} are homotopy equivalences and additionally the deformations can be chosen such that the two sets

S^{±} = {p_{W} ∘ r_{t}^{±}(x) − p_{W} ∘ r_{1}^{±}(x) | x ∈ W, t ∈ [0, 1]}

are bounded subsets of ℝ^{k}.

The following theorem (compare [239] and [327, Appendix]) contains the s-Cobordism Theorem 1.5 as a special case, gives another interpretation of elements in K̃_{0}(ℤπ) and explains one aspect of the geometric relevance of negative K-groups.

Theorem 1.10 (Bounded h-Cobordism Theorem). Suppose that M^{−} is parametrized over ℝ^{k} and satisfies dim M^{−} ≥ 5. Let π be its fundamental group(oid). Equivalence classes of bounded h-cobordisms over M^{−} modulo bounded diffeomorphism relative M^{−} correspond bijectively to elements in κ_{1−k}(π), where

κ_{1−k}(π) = \{ \begin{matrix}Wh(π) & if k = 0, \\ K̃_{0}(ℤπ) & if k = 1, \\ K_{1−k}(ℤπ) & if k ≥ 2. \end{matrix}

More information about negative K-groups can be found for instance in [8], [22], [57], [58], [113], [213], [238], [239], [252], [259], [268] and [327, Appendix].

1.3 Algebraic K-Theory - All Dimensions

So far we only considered the K-theory groups in dimensions ≤ 1. We now want to explain how Conjecture 1.1 generalizes to higher algebraic K-theory. For the definition of higher algebraic K-theory groups and the (connective) K-theory spectrum see [35], [52], [158], [249], [268], [292], [315] and [323]. We would like to stress that for us K(R) will always denote the non-connective algebraic K-theory spectrum for which K_{n}(R) = π_{n}(K(R)) holds for all n ∈ ℤ. For its definition see [52], [194] and [237].

The Farrell-Jones Conjecture for algebraic K-theory reduces for a torsion free group to the following conjecture.

Conjecture 1.11 (Farrell-Jones Conjecture for Torsion Free Groups and K-Theory). Let G be a torsion free group. Let R be a regular ring. Then the assembly map

H_{n}(BG; K(R)) → K_{n}(RG)

is an isomorphism for n ∈ ℤ.

Here H_{n}(−; K(R)) denotes the homology theory which is associated to the spectrum K(R). It has the property that H_{n}(pt; K(R)) is K_{n}(R) for n ∈ ℤ, where here and elsewhere pt denotes the space consisting of one point. The space BG is the classifying space of the group G, which up to homotopy is characterized by the property that it is a CW -complex with π_{1}(BG) ≅ G whose universal covering is contractible. The technical details of the construction of H_{n}(−; K(R)) and the assembly map will be explained in a more general setting in Section 2.1.

The point of Conjecture 1.11 is that on the right-hand side of the assembly map we have the group K_{n}(RG) we are interested in, whereas the left-hand side is a homology theory and hence much easier to compute. For every homology theory associated to a spectrum we have the Atiyah-Hirzebruch spectral sequence, which in our case has E_{p,q}^{2} = H_{p}(BG; K_{q}(R)) and converges to H_{p+q}(BG; K(R)).

If R is regular, then the negative K-groups of R vanish and the spectral sequence lives in the first quadrant. Evaluating the spectral sequence for n = p + q ≤ 1 shows that Conjecture 1.11 above implies Conjecture 1.1.

Remark 1.12 (Rational Computation). Rationally an Atiyah-Hirzebruch spectral sequence collapses always and the homological Chern character gives an isomorphism

ch: ⨁_{p+q=n} H_{p}(BG; ℚ) ⊗_{ℚ} (K_{q}(R) ⊗_{ℤ} ℚ) ⟶^{≅} H_{n}(BG; K(R)) ⊗_{ℤ} ℚ.

The Atiyah-Hirzebruch spectral sequence and the Chern character will be discussed in a much more general setting in Chapter 8.

Remark 1.13 (Separation of Variables). We see that the left-hand side of the isomorphism in the previous remark consists of a group homology part and a part which is the rationalized K-theory of R. (Something similar happens before we rationalize at the level of spectra: The left hand side of Conjecture 1.11 can be interpreted as the homotopy groups of the spectrum BG_{+}∧K(R).) So essentially Conjecture 1.11 predicts that the K-theory of RG is built up out of two independent parts: the K-theory of R and the group homology of G. We call this principle separation of variables. This principle also applies to other theories such as algebraic L-theory or topological K-theory. See also Remark 8.11.

Remark 1.14 (K-Theory of the Coefficients). Note that Conjecture 1.11 can only help us to explicitly compute the K-groups of RG in cases where we know enough about the K-groups of R. We obtain no new information about the K-theory of R itself. However, already for very simple rings the computation of their algebraic K-theory groups is an extremely hard problem.

It is known that the groups K_{n}(ℤ) are finitely generated abelian groups [248]. Due to Borel [39] we know that

K_{n}(ℤ) ⊗_{ℤ} ℚ ≅ \{ \begin{matrix}ℚ & if n = 0; \\ ℚ & if n = 4k + 1 with k ≥ 1; \\ 0 & otherwise. \end{matrix}

Since ℤ is regular we know that K_{n}(ℤ) vanishes for n ≤ −1. Moreover, K_{0}(ℤ) ≅ ℤ and K_{1}(ℤ) ≅ {±1}, where the isomorphisms are given by the rank and the determinant. One also knows that K_{2}(ℤ) ≅ ℤ/2, K_{3}(ℤ) ≅ ℤ/48 [189] and K_{4}(ℤ) ≅ 0 [264]. Finite fields belong to the few rings where one has a complete and explicit knowledge of all K-theory groups [247]. We refer the reader for example to [177], [226], [265], [322] and Soulé’s article in [193] for more information about the algebraic K-theory of the integers or more generally of rings of integers in number fields.

Because of Borel’s calculation the left hand side of the isomorphism described in Remark 1.12 specializes for R = ℤ to

H_{n}(BG; ℚ) ⊕ ⨁_{k=1}^{∞} H_{n−(4k+1)}(BG; ℚ)  (1)

and Conjecture 1.11 predicts that this group is isomorphic to K_{n}(ℤG) ⊗_{ℤ} ℚ.

Next we discuss the case where the group G is infinite cyclic.

Remark 1.15 (Bass-Heller-Swan Decomposition). The so called Bass-Heller-Swan-decomposition, also known as the Fundamental Theorem of algebraic K-theory, computes the algebraic K-groups of R[ℤ] in terms of the algebraic K-groups and Nil-groups of R:

K_{n}(R[ℤ]) ≅ K_{n−1}(R) ⊕ K_{n}(R) ⊕ NK_{n}(R) ⊕ NK_{n}(R).

Here the group NK_{n}(R) is defined as the cokernel of the split injection K_{n}(R) → K_{n}(R[t]). It can be identified with the cokernel of the split injection K_{n−1}(R) → K_{n−1}(Nil(R)). Here K_{n}(Nil(R)) denotes the K-theory of the exact category of nilpotent endomorphisms of finitely generated projective R-modules. For negative n it is defined with the help of Bass’ notion of a contracting functor [22] (see also [57]). The groups are known as Nil-groups and often denoted Nil_{n−1}(R).

For proofs of these facts and more information the reader should consult [22, Chapter XII], [25], [135, Theorem on page 236], [249, Corollary in §6 on page 38], [268, Theorems 3.3.3 and 5.3.30], [292, Theorem 9.8] and [300, Theorem 10.1].

If we iterate and use R[ℤ^{n}] = R[ℤ^{n−1}][ℤ] we see that a computation of K_{n}(RG) must in general take into account information about K_{i}(R) for all i ≤ n. In particular we see that it is important to formulate Conjecture 1.11 with the non-connective K-theory spectrum.

Since S^{1} is a model for Bℤ, we get an isomorphism

H_{n}(Bℤ; K(R)) ≅ K_{n−1}(R) ⊕ K_{n}(R)

and hence Conjecture 1.11 predicts

K_{n}(R[ℤ]) ≅ K_{n−1}(R) ⊕ K_{n}(R).

This explains why in the formulation of Conjecture 1.11 the condition that R is regular appears. It guarantees that NK_{n}(R) = 0 [268, Theorem 5.3.30 on page 295]. There are weaker conditions which imply that NK_{n}(R) = 0 but “regular” has the advantage that R regular implies that R[t] and R[ℤ] = R[t^{±1}] are again regular, compare the discussion in Section 2 in [23].

The Nil-terms NK_{n}(R) seem to be hard to compute. For instance NK_{1}(R) either vanishes or is infinitely generated as an abelian group [95]. In Subsection 4.2.3 we will discuss the Isomorphism Conjecture for NK-groups. For more information about Nil-groups see for instance [73], [74], [146], [324] and [325].

1.4 Applications II

1.4.1 The Relation to Pseudo-Isotopy Theory

Let I denote the unit interval [0, 1]. A topological pseudoisotopy of a compact manifold M is a homeomorphism h: M × I → M × I, which restricted to M × {0} ∪ ∂M × I is the obvious inclusion. The space P (M) of pseudoisotopies is the (simplicial) group of all such homeomorphisms. Pseudoisotopies play an important role if one tries to understand the homotopy type of the space Top(M) of self-homeomorphisms of a manifold M. We will see below in Subsection 1.6.2 how the results about pseudoisotopies discussed in this section combined with surgery theory lead to quite explicit results about the homotopy groups of Top(M).

There is a stabilization map P (M) → P (M × I) given by crossing a pseudoisotopy with the identity on the interval I and the stable pseudoisotopy space is defined as P(M) = colim_{k} P (M × I^{k}). In fact P(−) can be extended to a functor on all spaces [144]. The natural inclusion P (M) → P(M) induces an isomorphism on the i-th homotopy group if the dimension of M is large compared to i, see [43] and [157].

Waldhausen [314], [315] defines the algebraic K-theory of spaces functor A(X) and the functor Wh^{P L}(X) from spaces to spectra (or infinite loop spaces) and a fibration sequence

X_{+}∧A(pt) → A(X) → Wh^{P L}(X).

Here X_{+}∧A(pt) → A(X) is an assembly map, which can be compared to the algebraic K-theory assembly map that appears in Conjecture 1.11 via a commutative diagram

\xymatrix{H_{n}(X; A(pt)) \ar[d] \ar[r] \ar[d] & π_{n}(A(X)) \ar[d] \\ H_{n}(Bπ_{1}(X); K(ℤ)) \ar[r] & K_{n}(ℤπ_{1}(X)).}

In the case where X ≃ BG is aspherical the vertical maps induce isomorphisms after rationalization for n ≥ 1, compare [314, Proposition 2.2]. Since Ω^{2} Wh^{P L}(X) ≃ P(X) (a guided tour through the literature concerning this and related results can be found in [90, Section 9]), Conjecture 1.11 implies rational vanishing results for the groups π_{n}(P(M)) if M is an aspherical manifold. Compare also Remark 4.9.

Consequence 1.16. Suppose M is a closed aspherical manifold and Conjecture 1.11 holds for R = ℤ and G = π_{1}(M), then for all n ≥ 0

π_{n}(P(M)) ⊗_{ℤ} ℚ = 0.

Similarly as above one defines smooth pseudoisotopies and the space of stable smooth pseudoisotopies P^{Diff}(M). There is also a smooth version of the Whitehead space Wh^{Diff}(X) and Ω^{2} Wh^{Diff}(M) ≃ P^{Diff}(M). Again there is a close relation to A-theory via the natural splitting A(X) ≃ Σ^{∞}(X_{+}) ∨ Wh^{Diff}(X), see [316]. Here Σ^{∞}(X_{+}) denotes the suspension spectrum associated to X_{+}. Using this one can split off an assembly map H_{n}(X; Wh^{Diff}(pt)) → π_{n}(Wh^{Diff}(X)) from the A-theory assembly map. Since for every space π_{n}(Σ^{∞}(X_{+})) ⊗_{ℤ} ℚ ≅ H_{n}(X; ℚ) Conjecture 1.11 combined with the rational computation in (1) yields the following result.

Consequence 1.17. Suppose M is a closed aspherical manifold and Conjecture 1.11 holds for R = ℤ and G = π_{1}(M). Then for n ≥ 0 we have

π_{n}(P^{Diff}(M)) ⊗_{ℤ} ℚ = ⨁_{k=1}^{∞} H_{n−4k+1}(M; ℚ).

Observe that the fundamental difference between the smooth and the topological case occurs already when G is the trivial group.

1.4.2 Negative K-Groups and Bounded Pseudo-Isotopies

We briefly explain a further geometric interpretation of negative K-groups, which parallels the discussion of bounded h-cobordisms in Subsection 1.2.3.

Let p: M × ℝ^{k} → ℝ^{k} denote the natural projection. The space P_{b}(M; ℝ^{k}) of bounded pseudoisotopies is the space of all self-homeomorphisms h: M × ℝ^{k} × I → M × ℝ^{k} × I such that restricted to M × ℝ^{k} × {0} the map h is the inclusion and such that h is bounded, i.e. the set {p ∘ h(y) − p(y) | y ∈ M × ℝ^{k} × I} is a bounded subset of ℝ^{k}. There is again a stabilization map P_{b}(M; ℝ^{k}) → P_{b}(M × I; ℝ^{k}) and a stable bounded pseudoisotopy space P_{b}(M; ℝ^{k}) = colim_{j} P_{b}(M × I^{j}; ℝ^{k}). There is a homotopy equivalence P_{b}(M; ℝ^{k}) → ΩP_{b}(M; ℝ^{k+1}) [144, Appendix II] and hence the sequence of spaces P_{b}(M; ℝ^{k}) for k = 0, 1, … is an Ω-spectrum P(M). Analogously one defines the differentiable bounded pseudoisotopies P_{b}^{diff}(M; ℝ^{k}) and an Ωspectrum P^{diff}(M). The negative homotopy groups of these spectra have an interpretation in terms of low and negative dimensional K-groups. In terms of unstable homotopy groups this is explained in the following theorem which is closely related to Theorem 1.10 about bounded h-cobordisms.

Theorem 1.18 (Negative Homotopy Groups of Pseudoisotopies). Let G = π_{1}(M). Suppose n and k are such that n + k ≥ 0, then for k ≥ 1 there are isomorphisms

π_{n+k}(P_{b}(M; ℝ^{k})) = \{ \begin{matrix}Wh(G) & if n = −1, \\ K̃_{0}(ℤG) & if n = −2, \\ K_{n+2}(ℤG) & if n < −2 \end{matrix}

The same result holds in the differentiable case.

Note that Conjecture 1.11 predicts that these groups vanish if G is torsionfree. The result above is due to Anderson and Hsiang [8] and is also discussed in [327, Appendix].

1.5 L-Theory

We now move on to the L-theoretic version of the Farrell-Jones Conjecture. We will still stick to the case where the group is torsion free. The conjecture is obtained by replacing K-theory and the K-theory spectrum in Conjecture 1.11 by 4-periodic L-theory and the L-theory spectrum L^{⟨−∞⟩}(R). Explanations will follow below.

Conjecture 1.19 (Farrell-Jones Conjecture for Torsion Free Groups and L-Theory). Let G be a torsion free group and let R be a ring with involution. Then the assembly map

H_{n}(BG; L^{⟨−∞⟩}(R)) → L_{n}^{⟨−∞⟩}(RG)

is an isomorphism for n ∈ ℤ.

To a ring with involution one can associate (decorated) symmetric or quadratic algebraic L-groups, compare [44], [45], [256], [259] and [332]. We will exclusively deal with the quadratic algebraic L-groups and denote them by L_{n}^{⟨j⟩}(R). Here n ∈ ℤ and j ∈ {−∞} ⨿ {j ∈ ℤ | j ≤ 2} is the so called decoration. The decorations j = 0, 1 correspond to the decorations p, h and j = 2 is related to the decoration s appearing in the literature. Decorations will be discussed in Remark 1.21 below. The L-groups L_{n}^{⟨j⟩}(R) are 4-periodic, i.e. L_{n}^{⟨j⟩}(R) ≅ L_{n+4}^{⟨j⟩}(R) for n ∈ ℤ.

If we are given an involution r ↦ \overline{r} on a ring R, we will always equip RG with the involution that extends the given one and satisfies \overline{g} = g^{−1}. On ℤ, ℚ and ℝ we always use the trivial involution and on ℂ the complex conjugation.

One can construct an L-theory spectrum L^{⟨j⟩}(R) such that π_{n}(L^{⟨j⟩}(R)) = L_{n}^{⟨j⟩}(R), compare [258, § 13]. Above and in the sequel H_{n}(−; L^{⟨j⟩}(R)) denotes the homology theory which is associated to this spectrum. In particular we have H_{n}(pt; L^{⟨j⟩}(R)) = L_{n}^{⟨j⟩}(R). We postpone the discussion of the assembly map to Section 2.1 where we will construct it in greater generality.

Remark 1.20 (The Coefficients in the L-Theory Case). In contrast to K-theory (compare Remark 1.14) the L-theory of the most interesting coefficient ring R = ℤ is well known. The groups L_{n}^{⟨j⟩}(ℤ) for fixed n and varying j ∈ {−∞} ⨿ {j ∈ ℤ | j ≤ 2} are all naturally isomorphic (compare Proposition 1.23 below) and we have L_{0}^{⟨j⟩}(ℤ) ≅ ℤ and L_{2}^{⟨j⟩}(ℤ) ≅ ℤ/2, where the isomorphisms are given by the signature divided by 8 and the Arf invariant, and L_{1}^{⟨j⟩}(ℤ) = L_{3}^{⟨j⟩}(ℤ) = 0, see [41, Chapter III], [256, Proposition 4.3.1 on page 419].

Remark 1.21 (Decorations). L-groups are designed as obstruction groups for surgery problems. The decoration reflects what kind of surgery problem one is interested in. All L-groups can be described as cobordism classes of suitable quadratic Poincaré chain complexes. If one works with chain complexes of finitely generated free based R-modules and requires that the torsion of the Poincaré chain homotopy equivalence vanishes in K̃_{1}(R), then the corresponding L-groups are denoted L_{n}^{⟨2⟩}(R). If one drops the torsion condition, one obtains L_{n}^{⟨1⟩}(R), which is usually denoted L^{h}(R). If one works with finitely generated projective modules, one obtains L^{⟨0⟩}(R), which is also known as L^{p}(R).

The L-groups with negative decorations can be defined inductively via the Shaneson splitting, compare Remark 1.26 below. Assuming that the L-groups with decorations j have already been defined one sets

L_{n−1}^{<j−1>}(R) = coker(L_{n}^{<j>}(R) → L_{n}^{<j>}(R[ℤ])).

Compare [259, Definition 17.1 on page 145]. Alternatively these groups can be obtained via a process which is in the spirit of Subsection 1.2.3 and Subsection 1.4.2. One can define them as L-theory groups of suitable categories of modules parametrized over ℝ^{k}. For details the reader could consult [55, Section 4]. There are forgetful maps L_{n}^{⟨j+1⟩}(R) → L_{n}^{⟨j⟩}(R). The group L_{n}^{⟨−∞⟩}(R) is defined as the colimit over these maps. For more information see [254], [259].

For group rings we also define L_{n}^{s}(RG) similar to L_{n}^{⟨2⟩}(RG) but we require the torsion to lie in im A_{1} ⊂ K̃_{1}(RG), where A_{1} is the map defined in Section 1.1. Observe that L_{n}^{s}(RG) really depends on the pair (R, G) and differs in general from L_{n}^{⟨2⟩}(RG).

Remark 1.22 (The Interplay of K- and L-Theory). For j ≤ 1 there are forgetful maps L_{n}^{⟨j+1⟩}(R) → L_{n}^{⟨j⟩}(R) which sit inside the following sequence, which is known as the Rothenberg sequence [256, Proposition 1.10.1 on page 104], [259, 17.2].

… → L_{n}^{⟨j+1⟩}(R) → L_{n}^{⟨j⟩}(R) → Ĥ^{n}(ℤ/2; K̃_{j}(R))

→ L_{n−1}^{⟨j+1⟩}(R) → L_{n−1}^{⟨j⟩}(R) → …. (2)

Here Ĥ^{n}(ℤ/2; K̃_{j}(R)) is the Tate-cohomology of the group ℤ/2 with coefficients in the ℤ[ℤ/2]-module K̃_{j}(R). The involution on K̃_{j}(R) comes from the involution on R. There is a similar sequence relating L_{n}^{s}(RG) and L_{n}^{h}(RG), where the third term is the ℤ/2-Tate-cohomology of Wh_{1}^{R}(G). Note that Tatecohomology groups of the group ℤ/2 are always annihilated by multiplication with 2. In particular L_{n}^{⟨j⟩}(R)[\frac{1}{2}] = L_{n}^{⟨j⟩}(R) ⊗_{ℤ} ℤ[\frac{1}{2}] is always independent of j.

Let us formulate explicitly what we obtain from the above sequences for R = ℤG.

Proposition 1.23. Let G be a torsion free group, then Conjecture 1.3 about the vanishing of Wh(G), K̃_{0}(ℤG) and K_{−i}(ℤG) for i ≥ 1 implies that for fixed n and varying j ∈ {−∞} ⨿ {j ∈ ℤ | j ≤ 1} the L-groups L_{n}^{⟨j⟩}(ℤG) are all naturally isomorphic and moreover L_{n}^{⟨1⟩}(ℤG) = L_{n}^{h}(ℤG) ≅ L_{n}^{s}(ℤG).

Remark 1.24 (Rational Computation). As in the K-theory case we have an Atiyah-Hirzebruch spectral sequence:

E_{p,q}^{2} = H_{p}(BG; L_{q}^{⟨−∞⟩}(R)) ⇒ H_{p+q}(BG; L^{⟨−∞⟩}(R)).

Rationally this spectral sequence collapses and the homological Chern character gives for n ∈ ℤ an isomorphism

ch: ⨁_{p+q=n} H_{p}(BG; ℚ) ⊗_{ℚ} (L_{q}^{⟨−∞⟩}(R) ⊗_{ℤ} ℚ)

⟶^{≅} H_{n}(BG; L^{⟨−∞⟩}(R)) ⊗_{ℤ} ℚ. (3)

In particular we obtain in the case R = ℤ from Remark 1.20 for all n ∈ ℤ and all decorations j an isomorphism

\xymatrix{ch: ⨁_{k=0}^{∞} H_{n−4k}(BG; ℚ) \ar[r]^{≅} & H_{n}(BG; L^{⟨j⟩}(ℤ)) ⊗_{ℤ} ℚ.}  (4)

This spectral sequence and the Chern character above will be discussed in a much more general setting in Chapter 8.

Remark 1.25 (Torsion Free is Necessary). If G is finite, R = ℤ and n = 0, then the rationalized left hand side of the assembly equals ℚ, whereas the right hand side is isomorphic to the rationalization of the real representation ring. Since the group homology of a finite group vanishes rationally except in dimension 0, the previous remark shows that we need to assume the group to be torsion free in Conjecture 1.19

Remark 1.26 (Shaneson splitting). The Bass-Heller-Swan decomposition in K-theory (see Remark 1.15) has the following analogue for the algebraic L-groups, which is known as the Shaneson splitting [284]

L_{n}^{⟨j⟩}(R[ℤ]) ≅ L_{n−1}^{⟨j−1⟩}(R) ⊕ L_{n}^{⟨j⟩}(R).  (5)

Here for the decoration j = −∞ one has to interpret j − 1 as −∞. Since S^{1} is a model for Bℤ, we get an isomorphisms

H_{n}(Bℤ; L^{⟨j⟩}(R)) ≅ L_{n−1}^{⟨j⟩}(R) ⊕ L_{n}^{⟨j⟩}(R).

This explains why in the formulation of the L-theoretic Farrell-Jones Conjecture for torsion free groups (see Conjecture 1.19) we use the decoration j = −∞.

As long as one deals with torsion free groups and one believes in the low dimensional part of the K-theoretic Farrell-Jones Conjecture (predicting the vanishing of Wh(G), K̃_{0}(ℤG) and of the negative K-groups, see Conjecture 1.3) there is no difference between the various decorations j, compare Proposition 1.23. But as soon as one allows torsion in G, the decorations make a difference and it indeed turns out that if one replaces the decoration j = −∞ by j = s, h or p there are counterexamples for the L-theoretic version of Conjecture 2.2 even for R = ℤ [123].

Even though in the above Shaneson splitting (5) there are no terms analogous to the Nil-terms in Remark 1.15 such Nil-phenomena do also occur in L-theory, as soon as one considers amalgamated free products. The corresponding groups are the UNil-groups. They vanish if one inverts 2 [49]. For more information about the UNil-groups we refer to [15] [46], [47], [74], [77], [96], [260].

1.6 Applications III

1.6.1 The Borel Conjecture

One of the driving forces for the development of the Farrell-Jones Conjectures is still the following topological rigidity conjecture about closed aspherical manifolds, compare [107]. Recall that a manifold, or more generally a CW - complex, is called aspherical if its universal covering is contractible. An aspherical CW -complex X with π_{1}(X) = G is a model for the classifying space BG. If X is an aspherical manifold and hence finite dimensional, then G is necessarily torsionfree.

Conjecture 1.27 (Borel Conjecture). Let f: M → N be a homotopy equivalence of aspherical closed topological manifolds. Then f is homotopic to a homeomorphism. In particular two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic.

Closely related to the Borel Conjecture is the conjecture that each aspherical finitely dominated Poincaré complex is homotopy equivalent to a closed topological manifold. The Borel Conjecture 1.27 is false in the smooth category, i.e. if one replaces topological manifold by smooth manifold and homeomorphism by diffeomorphism [106].

Using surgery theory one can show that in dimensions ≥ 5 the Borel Conjecture is implied by the K-theoretic vanishing Conjecture 1.3 combined with the L-theoretic Farrell-Jones Conjecture.

Theorem 1.28 (The Farrell-Jones Conjecture Implies the Borel Conjecture). Let G be a torsion free group. If Wh(G), K̃_{0}(ℤG) and all the groups K_{−i}(ℤG) with i ≥ 1 vanish and if the assembly map

H_{n}(BG; L^{⟨−∞⟩}(ℤ)) → L_{n}^{⟨−∞⟩}(ℤG)

is an isomorphism for all n, then the Borel Conjecture holds for all orientable manifolds of dimension ≥ 5 whose fundamental group is G.

The Borel Conjecture 1.27 can be reformulated in the language of surgery theory to the statement that the topological structure set S^{top}(M) of an aspherical closed topological manifold M consists of a single point. This set is the set of equivalence classes of homotopy equivalences f: M′ → M with a topological closed manifold as source and M as target under the equivalence relation, for which f_{0} : M_{0} → M and f_{1} : M_{1} → M are equivalent if there is a homeomorphism g: M_{0} → M_{1} such that f_{1} ∘ g and f_{0} are homotopic.

The surgery sequence of a closed orientable topological manifold M of dimension n ≥ 5 is the exact sequence

… → N_{n+1}(M × [0, 1], M × {0, 1}) ⟶^{σ} L_{n+1}^{s}(ℤπ_{1}(M)) ⟶^{∂} S^{top}(M)

⟶^{η} N_{n}(M) ⟶^{σ} L_{n}^{s}(ℤπ_{1}(M)),

which extends infinitely to the left. It is the basic tool for the classification of topological manifolds. (There is also a smooth version of it.) The map σ appearing in the sequence sends a normal map of degree one to its surgery obstruction. This map can be identified with the version of the L-theory assembly map where one works with the 1-connected cover L^{s}(ℤ)⟨1⟩ of L^{s}(ℤ). The map H_{k}(M; L^{s}(ℤ)⟨1⟩) → H_{k}(M; L^{s}(ℤ)) is injective for k = n and an isomorphism for k > n. Because of the K-theoretic assumptions we can replace the s-decoration with the ⟨−∞⟩-decoration, compare Proposition 1.23. Therefore the Farrell-Jones Conjecture 1.19 implies that the maps σ: N_{n}(M) →

L_{n}^{s}(ℤπ_{1}(M)) and N_{n+1}(M × [0, 1], M × {0, 1}) ⟶^{σ} L_{n+1}^{s}(ℤπ_{1}(M)) are injec-

tive respectively bijective and thus by the surgery sequence that S^{top}(M) is a point and hence the Borel Conjecture 1.27 holds for M. More details can be found e.g. in [127, pages 17,18,28], [258, Chapter 18].

For more information about surgery theory we refer for instance to [41], [44], [45], [121], [122], [167], [178], [253], [294], [293], and [320].

1.6.2 Automorphisms of Manifolds

If one additionally also assumes the Farrell-Jones Conjectures for higher Ktheory, one can combine the surgery theoretic results with the results about pseudoisotopies from Subsection 1.4.1 to obtain the following results.

Theorem 1.29 (Homotopy Groups of Top(M)). Let M be an orientable closed aspherical manifold of dimension > 10 with fundamental group G. Suppose the L-theory assembly map

H_{n}(BG; L^{⟨−∞⟩}(ℤ)) → L_{n}^{⟨−∞⟩}(ℤG)

is an isomorphism for all n and suppose the K-theory assembly map

H_{n}(BG; K(ℤ)) → K_{n}(ℤG)

is an isomorphism for n ≤ 1 and a rational isomorphism for n ≥ 2. Then for 1 ≤ i ≤ (dim M − 7)/3 one has

π_{i}(Top(M)) ⊗_{ℤ} ℚ = \{ \begin{matrix}center(G) ⊗_{ℤ} ℚ & if i = 1, \\ 0 & if i > 1 \end{matrix}

In the differentiable case one additionally needs to study involutions on the higher K-theory groups. The corresponding result reads:

Theorem 1.30 (Homotopy Groups of Diff(M)). Let M be an orientable closed aspherical differentiable manifold of dimension > 10 with fundamental group G. Then under the same assumptions as in Theorem 1.29 we have for 1 ≤ i ≤ (dim M − 7)/3

π_{i}(Diff(M)) ⊗_{ℤ} ℚ = \{ \begin{matrix}center(G) ⊗_{ℤ} ℚ & if i = 1; \\ ⨁_{j=1}^{∞} H_{(i+1)−4j}(M; ℚ) & if i > 1 and dim M odd ; \\ 0 & if i > 1 and dim M even . \end{matrix}

See for instance [97], [109, Section 2] and [120, Lecture 5]. For a modern survey on automorphisms of manifolds we refer to [329].

1.7 The Baum-Connes Conjecture in the Torsion Free Case

We denote by K_{∗}(Y) the complex K-homology of a topological space Y and by K_{∗}(C_{r}^{∗}(G)) the (topological) K-theory of the reduced group C^{∗}-algebra. More explanations will follow below.

Conjecture 1.31 (Baum-Connes Conjecture for Torsion Free Groups). Let G be a torsion free group. Then the Baum-Connes assembly map

K_{n}(BG) → K_{n}(C_{r}^{∗}(G))

is bijective for all n ∈ ℤ.

Complex K-homology K_{∗}(Y) is the homology theory associated to the topological (complex) K-theory spectrum K^{top} (which is is often denoted BU) and could also be written as K_{∗}(Y) = H_{∗}(Y; K^{top}). The cohomology theory associated to the spectrum K^{top} is the well known complex K-theory defined in terms of complex vector bundles. Complex K-homology is a 2periodic theory, i.e. K_{n}(Y) ≅ K_{n+2}(Y).

Also the topological K-groups K_{n}(A) of a C^{∗}-algebra A are 2-periodic. Whereas K_{0}(A) coincides with the algebraically defined K_{0}-group, the other groups K_{n}(A) take the topology of the C^{∗}-algebra A into account, for instance K_{n}(A) = π_{n−1}(GL(A)) for n ≥ 1.

Let B(l^{2}(G)) denote the bounded linear operators on the Hilbert space l^{2}(G) whose orthonormal basis is G. The reduced complex group C^{∗}-algebra C_{r}^{∗}(G) is the closure in the norm topology of the image of the regular representation ℂG → B(l^{2}(G)), which sends an element u ∈ ℂG to the (left) G-equivariant bounded operator l^{2}(G) → l^{2}(G) given by right multiplication with u. In particular one has natural inclusions

ℂG ⊆ C_{r}^{∗}(G) ⊆ B(l^{2}(G))^{G} ⊆ B(l^{2}(G)).

It is essential to use the reduced group C^{∗}-algebra in the Baum-Connes Conjecture, there are counterexamples for the version with the maximal group C^{∗}-algebra, compare Subsection 4.1.2. For information about C^{∗}-algebras and their topological K-theory we refer for instance to [37], [71], [80], [154], [188], [228], [279] and [321].

Remark 1.32 (The Coefficients in the Case of Topological K-Theory). If we specialize to the trivial group G = {1}, then the complex reduced group C^{∗}-algebra reduces to C_{r}^{∗}(G) = ℂ and the topological K-theory is well known: by periodicity it suffices to know that K_{0}(ℂ) ≅ ℤ, where the homomorphism is given by the dimension, and K_{1}(ℂ) = 0. Correspondingly we have K_{q}(pt) = ℤ for q even and K_{q}(pt) = 0 for odd q.

Remark 1.33 (Rational Computation). There is an Atiyah-Hirzebruch spectral sequence which converges to K_{p+q}(BG) and whose E^{2}-term is E_{p,q}^{2} = H_{p}(BG; K_{q}(pt)). Rationally this spectral sequence collapses and the homological Chern character gives an isomorphism for n ∈ ℤ

ch: ⨁_{k∈ℤ} H_{n−2k}(BG; ℚ) = ⨁_{p+q=n} H_{p}(BG; ℚ) ⊗_{ℚ} (K_{q}(ℂ) ⊗_{ℤ} ℚ)

⟶^{≅} K_{n}(BG) ⊗_{ℤ} ℚ. (6)

Remark 1.34 (Torsionfree is Necessary). In the case where G is a finite group the reduced group C^{∗}-algebra C_{r}^{∗}(G) coincides with the complex group ring ℂG and K_{0}(C_{r}^{∗}(G)) coincides with the complex representation ring of G. Since the group homology of a finite group vanishes rationally except in dimension 0, the previous remark shows that we need to assume the group to be torsion free in Conjecture 1.31.

Remark 1.35. (Bass-Heller-Swan-Decomposition for Topological K-Theory) There is an analogue of the Bass-Heller-Swan decomposition in algebraic K-theory (see Remark 1.15) or of the Shaneson splitting in L-theory (see Remark 1.26) for topological K-theory. Namely we have

K_{n}(C_{r}^{∗}(G × ℤ)) ≅ K_{n}(C_{r}^{∗}(G)) ⊕ K_{n−1}(C_{r}^{∗}(G)),

see [243, Theorem 3.1 on page 151] or more generally [244, Theorem 18 on page 632]. This is consistent with the obvious isomorphism

K_{n}(B(G × ℤ)) = K_{n}(BG × S^{1}) ≅ K_{n−1}(BG) ⊕ K_{n}(BG).

Notice that here in contrast to the algebraic K-theory no Nil-terms occur (see Remark 1.15) and that there is no analogue of the complications in algebraic L-theory coming from the varying decorations (see Remark 1.26). This absence of Nil-terms or decorations is the reason why in the final formulation of the Baum-Connes Conjecture it suffices to deal with the family of finite subgroups, whereas in the algebraic K- and L-theory case one must consider the larger and harder to handle family of virtually cyclic subgroups. This in some sense makes the computation of topological K-theory of reduced group C^{∗}-algebras easier than the computation of K_{n}(ℤG) or L_{n}(ℤG).

Remark 1.36 (Real Version of the Baum-Connes Conjecture). There is an obvious real version of the Baum-Connes Conjecture. It says that for a torsion free group the real assembly map

KO_{n}(BG) → KO_{n}(C_{r}^{∗}(G; ℝ))

is bijective for n ∈ ℤ. We will discuss in Subsection 4.1.1 below that this real version of the Baum-Connes Conjecture is implied by the complex version Conjecture 1.31.

Here KO_{n}(C_{r}^{∗}(G; ℝ)) is the topological K-theory of the real reduced group C^{∗}-algebra C_{r}^{∗}(G; ℝ). We use KO instead of K as a reminder that we work with real C^{∗}-algebras. The topological real K-theory KO_{∗}(Y) is the homology theory associated to the spectrum BO, whose associated cohomology theory is given in terms of real vector bundles. Both, topological K-theory of a real C^{∗}- algebra and KO-homology of a space are 8-periodic and KO_{n}(pt) = K_{n}(ℝ) is ℤ, if n = 0, 4 (8), is ℤ/2 if n = 1, 2 (8) and is 0 if n = 3, 5, 6, 7 (8).

More information about the K-theory of real C^{∗}-algebras can be found in [281].

1.8 Applications IV

We now discuss some consequences of the Baum-Connes Conjecture for Torsion Free Groups 1.31.

1.8.1 The Trace Conjecture in the Torsion Free Case

The assembly map appearing in the Baum-Connes Conjecture has an interpretation in terms of index theory. This is important for geometric applications.

It is of the same significance as the interpretation of the L-theoretic assembly map as the map σ appearing in the exact surgery sequence discussed in Section 1.5. We proceed to explain this.

An element η ∈ K_{0}(BG) can be represented by a pair (M, P^{∗}) consisting of a cocompact free proper smooth G-manifold M with Riemannian metric together with an elliptic G-complex P^{∗} of differential operators of order 1 on M [29]. To such a pair one can assign an index ind_{C_{r}^{∗}(G)}(M, P^{∗}) in K_{0}(C_{r}^{∗}(G))

[223] which is the image of η under the assembly map K_{0}(BG) → K_{0}(C_{r}^{∗}(G)) appearing in Conjecture 1.31. With this interpretation the surjectivity of the assembly map for a torsion free group says that any element in K_{0}(C_{r}^{∗}(G)) can be realized as an index. This allows to apply index theorems to get interesting information.

Here is a prototype of such an argument. The standard trace

tr_{C_{r}^{∗}(G)} : C_{r}^{∗}(G) → ℂ  (7)

sends an element f ∈ C_{r}^{∗}(G) ⊆ B(l^{2}(G)) to ⟨f(1), 1⟩_{l^{2}(G)}. Applying the trace to idempotent matrices yields a homomorphism

tr_{C_{r}^{∗}(G)} : K_{0}(C_{r}^{∗}(G)) → ℝ.

Let pr: BG → pt be the projection. For a group G the following diagram commutes

\xymatrix{K_{0}(BG) \ar[d]_{K_{0}(pr)} \ar[r]^{A} & K_{0}(C_{r}^{∗}(G)) \ar[r]^{^{tr^{C^{∗}_{r}(G)}}} & ℝ \\ K_{0}(pt) \ar[r]^{≅} & K_{0}(ℂ) \ar[r]_{_{tr_{ℂ}}}^{≅} & ℤ. \ar[u]_{i}}  (8)

Here i: ℤ → ℝ is the inclusion and A is the assembly map. This non-trivial statement follows from Atiyah’s L^{2}-index theorem [12]. Atiyah’s theorem says that the L^{2}-index tr_{C_{r}^{∗}(G)} ∘A(η) of an element η ∈ K_{0}(BG), which is represented by a pair (M, P^{∗}), agrees with the ordinary index of (G\M; G\P^{∗}), which is tr_{ℂ} ∘K_{0}(pr)(η) ∈ ℤ.

The following conjecture is taken from [27, page 21].

Conjecture 1.37 (Trace Conjecture for Torsion Free Groups). For a torsion free group G the image of

tr_{C_{r}^{∗}(G)} : K_{0}(C_{r}^{∗}(G)) → ℝ

consists of the integers.

The commutativity of diagram (8) above implies

Consequence 1.38. The surjectivity of the Baum-Connes assembly map

K_{0}(BG) → K_{0}(C_{r}^{∗}(G))

implies Conjecture 1.37, the Trace Conjecture for Torsion Free Groups.

1.8.2 The Kadison Conjecture

Conjecture 1.39 (Kadison Conjecture). If G is a torsion free group, then the only idempotent elements in C_{r}^{∗}(G) are 0 and 1.

Lemma 1.40. The Trace Conjecture for Torsion Free Groups 1.37 implies the Kadison Conjecture 1.39.

Proof. Assume that p ∈ C_{r}^{∗}(G) is an idempotent different from 0 or 1. From p one can construct a non-trivial projection q ∈ C_{r}^{∗}(G), i.e. q^{2} = q, q^{∗} = q, with im(p) = im(q) and hence with 0 < q < 1. Since the standard trace tr_{C_{r}^{∗}(G)} is faithful, we conclude tr_{C_{r}^{∗}(G)}(q) ∈ ℝ with 0 < tr_{C_{r}^{∗}(G)}(q) < 1. Since tr_{C_{r}^{∗}(G)}(q) is by definition the image of the element [im(q)] ∈ K_{0}(C_{r}^{∗}(G)) under tr_{C_{r}^{∗}(G)} : K_{0}(C_{r}^{∗}(G)) → ℝ, we get a contradiction to the assumption im(tr_{C_{r}^{∗}(G)}) ⊆ ℤ. ⊓ ⊔

Recall that a ring R is called an integral domain if it has no non-trivial zero-divisors, i.e. if r, s ∈ R satisfy rs = 0, then r or s is 0. Obviously the Kadison Conjecture 1.39 implies for R ⊆ ℂ the following.

Conjecture 1.41 (Idempotent Conjecture). Let R be an integral domain and let G be a torsion free group. Then the only idempotents in RG are 0 and 1.

The statement in the conjecture above is a purely algebraic statement. If R = ℂ, it is by the arguments above related to questions about operator algebras, and thus methods from operator algebras can be used to attack it.

1.8.3 Other Related Conjectures

We would now like to mention several conjectures which are not directly implied by the Baum-Connes or Farrell-Jones Conjectures, but which are closely related to the Kadison Conjecture and the Idempotent Conjecture mentioned above.

The next conjecture is also called the Kaplansky Conjecture.

Conjecture 1.42 (Zero-Divisor-Conjecture). Let R be an integral domain and G be a torsion free group. Then RG is an integral domain.

Obviously the Zero-Divisor-Conjecture 1.42 implies the Idempotent Conjecture 1.41. The Zero-Divisor-Conjecture for R = ℚ is implied by the following version of the Atiyah Conjecture (see [202, Lemma 10.5 and Lemma 10.15]).

Conjecture 1.43 (Atiyah-Conjecture for Torsion Free Groups). Let G be a torsion free group and let M be a closed Riemannian manifold. Let \overline{M} → M be a regular covering with G as group of deck transformations. Then all L^{2}-Betti numbers b_{p}^{(2)}(\overline{M}; N (G)) are integers.

For the precise definition and more information about L^{2}-Betti numbers and the group von Neumann algebra N (G) we refer for instance to [202], [205].

This more geometric formulation of the Atiyah Conjecture is in fact implied by the following more operator theoretic version. (The two would be equivalent if one would work with rational instead of complex coefficients below.)

Conjecture 1.44 (Strong Atiyah-Conjecture for Torsion Free Groups). Let G be a torsion free group. Then for all (m, n)-matrices A over ℂG the von Neumann dimension of the kernel of the induced G-equivariant bounded operator

r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}

is an integer.

The Strong Atiyah-Conjecture for Torsion Free Groups implies both the Atiyah-Conjecture for Torsion Free Groups 1.43 [202, Lemma 10.5 on page 371] and the Zero-Divisor-Conjecture 1.42 for R = ℂ [202, Lemma 10.15 on page 376].

Conjecture 1.45 (Embedding Conjecture). Let G be a torsion free group. Then ℂG admits an embedding into a skewfield.

Obviously the Embedding Conjecture implies the Zero-Divisor-Conjecture 1.42 for R = ℂ. If G is a torsion free amenable group, then the Strong Atiyah-Conjecture for Torsion Free Groups 1.44 and the Zero-Divisor-Conjecture 1.42 for R = ℂ are equivalent [202, Lemma 10.16 on page 376]. For more information about the Atiyah Conjecture we refer for instance to [192], [202, Chapter 10] and [261].

Finally we would like to mention the Unit Conjecture.

Conjecture 1.46 (Unit-Conjecture). Let R be an integral domain and G be a torsion free group. Then every unit in RG is trivial, i.e. of the form r · g for some unit r ∈ R^{inv} and g ∈ G.

The Unit Conjecture 1.46 implies the Zero-Divisor-Conjecture 1.42. For a proof of this fact and for more information we refer to [187, Proposition 6.21 on page 95].

1.8.4 L^{2}-Rho-Invariants and L^{2}-Signatures

Let M be a closed connected orientable Riemannian manifold. Denote by η(M) ∈ ℝ the eta-invariant of M and by η^{(2)}(M̃) ∈ ℝ the L^{2}-eta-invariant of the π_{1}(M)-covering given by the universal covering M̃ → M. Let ρ^{(2)}(M) ∈ ℝ be the L^{2}-rho-invariant which is defined to be the difference η^{(2)}(M̃) − η(M). These invariants were studied by Cheeger and Gromov [64], [65]. They show that ρ^{(2)}(M) depends only on the diffeomorphism type of M and is in contrast to η(M) and η^{(2)}(M̃) independent of the choice of Riemannian metric on M. The following conjecture is taken from Mathai [214].

Conjecture 1.47 (Homotopy Invariance of the L^{2}-Rho-Invariant for Torsionfree Groups). If π_{1}(M) is torsionfree, then ρ^{(2)}(M) is a homotopy invariant.

Chang-Weinberger [62] assign to a closed connected oriented (4k − 1)- dimensional manifold M a Hirzebruch-type invariant τ^{(2)}(M) ∈ ℝ as follows. By a result of Hausmann [145] there is a closed connected oriented 4k-dimensional manifold W with M = ∂W such that the inclusion ∂W → W induces an injection on the fundamental groups. Define τ^{(2)}(M) as the difference sign^{(2)}(W̃)−sign(W) of the L^{2}-signature of the π_{1}(W)-covering given by the universal covering W̃ → W and the signature of W. This is indeed independent of the choice of W. It is reasonable to believe that ρ^{(2)}(M) = τ^{(2)}(M) is always true. Chang-Weinberger [62] use τ^{(2)} to prove that if π_{1}(M) is not torsionfree there are infinitely many diffeomorphically distinct manifolds of dimension 4k+3 with k ≥ 1, which are tangentially simple homotopy equivalent to M.

Theorem 1.48 (Homotopy Invariance of τ^{(2)}(M) and ρ^{(2)}(M)). Let M be a closed connected oriented (4k − 1)-dimensional manifold M such that G = π_{1}(M) is torsionfree.

(i) If the assembly map K_{0}(BG) → K_{0}(C_{max}^{∗}(G)) for the maximal group C^{∗}- algebra is surjective (see Subsection 4.1.2), then ρ^{(2)}(M) is a homotopy invariant.

(ii) Suppose that the Farrell-Jones Conjecture for L-theory 1.19 is rationally true for R = ℤ, i.e. the rationalized assembly map

H_{n}(BG; L^{⟨−∞⟩}(ℤ)) ⊗_{ℤ} ℚ → L_{n}^{⟨−∞⟩}(ℤG) ⊗_{ℤ} ℚ

is an isomorphism for n ∈ ℤ. Then τ^{(2)}(M) is a homotopy invariant. If furthermore G is residually finite, then ρ^{(2)}(M) is a homotopy invariant.

Proof. (i) This is proved by Keswani [174], [175].

(ii) This is proved by Chang [61] and Chang-Weinberger [62] using [210]. ⊓ ⊔

Remark 1.49. Let X be a 4n-dimensional Poincaré space over ℚ. Let \overline{X} → X be a normal covering with torsion-free covering group G. Suppose that the assembly map K_{0}(BG) → K_{0}(C_{max}^{∗}(G)) for the maximal group C^{∗}-algebra is surjective (see Subsection 4.1.2) or suppose that the rationalized assembly map for L-theory

H_{4n}(BG; L^{⟨−∞⟩}(ℤ)) ⊗_{ℤ} ℚ → L_{4n}^{⟨−∞⟩}(ℤG) ⊗_{ℤ} ℚ

is an isomorphism. Then the following L^{2}-signature theorem is proved in Lück-Schick [211, Theorem 2.3]

sign^{(2)}(\overline{X}) = sign(X).

If one drops the condition that G is torsionfree this equality becomes false. Namely, Wall has constructed a finite Poincaré space X with a finite G covering \overline{X} → X for which sign(\overline{X}) ≠ |G| · sign(X) holds (see [258, Example 22.28], [319, Corollary 5.4.1]).

Remark 1.50. Cochran-Orr-Teichner give in [69] new obstructions for a knot to be slice which are sharper than the Casson-Gordon invariants. They use L^{2}- signatures and the Baum-Connes Conjecture 2.3. We also refer to the survey article [68] about non-commutative geometry and knot theory.

1.9 Applications V

1.9.1 Novikov Conjectures

The Baum-Connes and Farrell-Jones Conjectures discussed so far imply obviously that for torsion free groups the rationalized assembly maps

H_{∗}(BG; K(ℤ)) ⊗_{ℤ} ℚ → K_{∗}(ℤG) ⊗_{ℤ} ℚ
H_{∗}(BG; L^{⟨−∞⟩}(ℤ)) ⊗_{ℤ} ℚ → L_{∗}^{⟨−∞⟩}(ℤG) ⊗_{ℤ} ℚ
K_{∗}(BG) ⊗_{ℤ} ℚ → K_{∗}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ

are injective. For reasons that will be explained below these “rational injectivity conjectures” are known as “Novikov Conjectures”. In fact one expects these injectivity results also when the groups contain torsion. So there are the following conjectures.

Conjecture 1.51 (K- and L-theoretic Novikov Conjectures). For every group G the assembly maps

H_{∗}(BG; K(ℤ)) ⊗_{ℤ} ℚ → K_{∗}(ℤG) ⊗_{ℤ} ℚ
H_{∗}(BG; L^{p}(ℤ)) ⊗_{ℤ} ℚ → L_{∗}^{p}(ℤG) ⊗_{ℤ} ℚ
K_{∗}(BG) ⊗_{ℤ} ℚ → K_{∗}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ

are injective.

Observe that, since the ℤ/2-Tate cohomology groups vanish rationally, there is no difference between the various decorations in L-theory because of the Rothenberg sequence. We have chosen the p-decoration above.

1.9.2 The Original Novikov Conjecture

We now explain the Novikov Conjecture in its original formulation.

Let G be a (not necessarily torsion free) group and u: M → BG be a map from a closed oriented smooth manifold M to BG. Let L(M) ∈ ∏_{k≥0} H^{k}(M; ℚ) be the L-class of M, which is a certain polynomial in the Pontrjagin classes and hence depends a priori on the tangent bundle and hence on the differentiable structure of M. For x ∈ ∏_{k≥0} H^{k}(BG; ℚ) define the higher signature of M associated to x and u to be

sign_{x}(M, u) := ⟨L(M) ∪ u^{∗}x, [M]⟩ ∈ ℚ.  (9)

The Hirzebruch signature formula says that for x = 1 the signature sign_{1}(M, u) coincides with the ordinary signature sign(M) of M, if dim(M) = 4n, and is zero, if dim(M) is not divisible by four. Recall that for dim(M) = 4n the signature sign(M) of M is the signature of the non-degenerate bilinear symmetric pairing on the middle cohomology H^{2n}(M; ℝ) given by the intersection pairing (a, b) ↦ ⟨a ∪ b, [M]⟩. Obviously sign(M) depends only on the oriented homotopy type of M. We say that sign_{x} for x ∈ H^{∗}(BG; ℚ) is homotopy invariant if for two closed oriented smooth manifolds M and N with reference maps u: M → BG and v: N → BG we have

sign_{x}(M, u) = sign_{x}(N, v)

if there is an orientation preserving homotopy equivalence f: M → N such that v ∘ f and u are homotopic.

Conjecture 1.52 (Novikov Conjecture). Let G be a group. Then sign_{x} is homotopy invariant for all x ∈ ∏_{k≥0} H^{k}(BG; ℚ).

By Hirzebruch’s signature formula the Novikov Conjecture 1.52 is true for x = 1.

1.9.3 Relations between the Novikov Conjectures

Using surgery theory one can show [260, Proposition 6.6 on page 300] the following.

Proposition 1.53. For a group G the original Novikov Conjecture 1.52 is equivalent to the L-theoretic Novikov Conjecture, i.e. the injectivity of the assembly map

H_{∗}(BG; L^{p}(ℤ)) ⊗_{ℤ} ℚ → L_{∗}^{p}(ℤG) ⊗_{ℤ} ℚ.

In particular for torsion free groups the L-theoretic Farrell-Jones Conjecture 1.19 implies the Novikov Conjecture 1.52. Later in Proposition 3.19 we will prove in particular the following statement.

Proposition 1.54. The Novikov Conjecture for topological K-theory, i.e. the injectivity of the assembly map

K_{∗}(BG) ⊗_{ℤ} ℚ → K_{∗}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ

implies the L-theoretic Novikov Conjecture and hence the original Novikov Conjecture.

For more information about the Novikov Conjectures we refer for instance to [38], [52], [55], [81], [120], [127], [179], [258] and [269].

1.9.4 The Zero-in-the-Spectrum Conjecture

The following Conjecture is due to Gromov [136, page 120].

Conjecture 1.55 (Zero-in-the-spectrum Conjecture). Suppose that M̃ is the universal covering of an aspherical closed Riemannian manifold M (equipped with the lifted Riemannian metric). Then zero is in the spectrum of the minimal closure

(Δ_{p})_{min} : L^{2}Ω^{p}(M̃) ⊃ dom(Δ_{p})_{min} → L^{2}Ω^{p}(M̃),

for some p ∈ {0, 1, …, dim M}, where Δ_{p} denotes the Laplacian acting on smooth p-forms on M̃.

Proposition 1.56. Suppose that M is an aspherical closed Riemannian manifold with fundamental group G, then the injectivity of the assembly map

K_{∗}(BG) ⊗_{ℤ} ℚ → K_{∗}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ

implies the Zero-in-the-spectrum Conjecture for M̃.

Proof. We give a sketch of the proof. More details can be found in [195, Corollary 4]. We only explain that the assumption that in every dimension zero is not in the spectrum of the Laplacian on M̃, yields a contradiction in the case that n = dim(M) is even. Namely, this assumption implies that the C_{r}^{∗}(G)-valued index of the signature operator twisted with the flat bundle M̃ ×_{G} C_{r}^{∗}(G) → M in K_{0}(C_{r}^{∗}(G)) is zero, where G = π_{1}(M). This index is the image of the class [S] defined by the signature operator in K_{0}(BG) under the assembly map K_{0}(BG) → K_{0}(C_{r}^{∗}(G)). Since by assumption the assembly map is rationally injective, this implies [S] = 0 in K_{0}(BG) ⊗_{ℤ} ℚ. Notice that M is aspherical by assumption and hence M = BG. The homological Chern character defines an isomorphism

K_{0}(BG) ⊗_{ℤ} ℚ = K_{0}(M) ⊗_{ℤ} ℚ ⟶^{≅} ⨁_{p≥0} H_{2p}(M; ℚ)

which sends [S] to the Poincaré dual L(M) ∩ [M] of the Hirzebruch L-class L(M) ∈ ⨁_{p≥0} H^{2p}(M; ℚ). This implies that L(M) ∩ [M] = 0 and hence L(M) = 0. This contradicts the fact that the component of L(M) in H^{0}(M; ℚ) is 1. ⊓ ⊔

More information about the Zero-in-the-spectrum Conjecture 1.55 can be found for instance in [195] and [202, Section 12].

2 The Conjectures in the General Case

In this chapter we will formulate the Baum-Connes and Farrell-Jones Conjectures. We try to emphasize the unifying principle that underlies these conjectures. The point of view taken in this chapter is that all three conjectures are conjectures about specific equivariant homology theories. Some of the technical details concerning the actual construction of these homology theories are deferred to Chapter 6.

2.1 Formulation of the Conjectures

Suppose we are given

• A discrete group G;

• A family F of subgroups of G, i.e. a set of subgroups which is closed under conjugation with elements of G and under taking finite intersections;

• A G-homology theory H_{∗}^{G}(−).

Then one can formulate the following Meta-Conjecture.

Meta-Conjecture 2.1. The assembly map

A_{F}: H_{n}^{G}(E_{F}(G)) → H_{n}^{G}(pt)

which is the map induced by the projection E_{F}(G) → pt, is an isomorphism for n ∈ ℤ.

Here E_{F}(G) is the classifying space of the family F, a certain G-space which specializes to the universal free G-space EG if the family contains only the trivial subgroup. A G-homology theory is the “obvious” G-equivariant generalization of the concept of a homology theory to a suitable category of G-spaces, in particular it is a functor on such spaces and the map A_{F} is simply the map induced by the projection E_{F}(G) → pt. We devote the Subsections 2.1.1 to 2.1.4 below to a discussion of G-homology theories, classifying spaces for families of subgroups and related things. The reader who never encountered these concepts should maybe first take a look at these subsections.

Of course the conjecture above is not true for arbitrary G, F and H_{∗}^{G}(−), but the Farrell-Jones and Baum-Connes Conjectures state that for specific Ghomology theories there is a natural choice of a family F = F(G) of subgroups for every group G such that A_{F(G)} becomes an isomorphism for all groups G.

Let R be a ring (with involution). In Proposition 6.7 we will describe the construction of G-homology theories which will be denoted

H_{n}^{G}(−; K_{R}), H_{n}^{G}(−; L_{R}^{⟨−∞⟩}) and H_{n}^{G}(−; K^{top}).

If G is the trivial group, these homology theories specialize to the (nonequivariant) homology theories with similar names that appeared in Chapter 1, namely to

H_{n}(−; K(R)), H_{n}(−; L^{⟨−∞⟩}(R)) and K_{n}(−).

Another main feature of these G-homology theories is that evaluated on the one point space pt (considered as a trivial G-space) we obtain the K- and L-theory of the group ring RG, respectively the topological K-theory of the reduced C^{∗}-algebra (see Proposition 6.7 and Theorem 6.9 (iii))

K_{n}(RG) ≅ H_{n}^{G}(pt; K_{R}),
L_{n}^{⟨−∞⟩}(RG) ≅ H_{n}^{G}(pt; L_{R}^{⟨−∞⟩}) and
K_{n}(C_{r}^{∗}(G)) ≅ H_{n}^{G}(pt; K^{top}).

We are now prepared to formulate the conjectures around which this article is centered. Let FIN be the family of finite subgroups and let VCY be the family of virtually cyclic subgroups.

Conjecture 2.2 (Farrell-Jones Conjecture for K- and L-theory). Let R be a ring (with involution) and let G be a group. Then for all n ∈ ℤ the maps

A_{VCY}: H_{n}^{G}(E_{VCY}(G); K_{R}) → H_{n}^{G}(pt; K_{R}) ≅ K_{n}(RG);
A_{VCY}: H_{n}^{G}(E_{VCY}(G); L_{R}^{⟨−∞⟩}) → H_{n}^{G}(pt; L_{R}^{⟨−∞⟩}) ≅ L_{n}^{⟨−∞⟩}(RG),

which are induced by the projection E_{VCY}(G) → pt, are isomorphisms.

The conjecture for the topological K-theory of C^{∗}-algebras is known as the Baum-Connes Conjecture and reads as follows.

Conjecture 2.3 (Baum-Connes Conjecture). Let G be a group. Then for all n ∈ ℤ the map

A_{FIN} : H_{n}^{G}(E_{FIN} (G); K^{top}) → H_{n}^{G}(pt; K^{top}) ≅ K_{n}(C_{r}^{∗}(G))

induced by the projection E_{FIN} (G) → pt is an isomorphism.

We will explain the analytic assembly map ind_{G}: K_{n}^{G}(X) → K_{n}(C_{r}^{∗}(G)), which can be identified with the assembly map appearing in the Baum-Connes Conjecture 2.3 in Section 7.2.

Remark 2.4. Of course the conjectures really come to life only if the abstract point of view taken in this chapter is connected up with more concrete descriptions of the assembly maps. We have already discussed a surgery theoretic description in Theorem 1.28 and an interpretation in terms index theory in Subsection 1.8.1. More information about alternative interpretations of assembly maps can be found in Section 7.2 and 7.8. These concrete interpretations of the assembly maps lead to applications. We already discussed many such applications in Chapter 1 and encourage the reader to go ahead and browse through Chapter 3 in order to get further ideas about these more concrete aspects.

Remark 2.5 (Relation to the “classical” assembly maps). The value of an equivariant homology theory H_{∗}^{G}(−) on the universal free G-space EG = E_{{1}}(G) (a free G-CW -complex whose quotient EG/G is a model for BG) can be identified with the corresponding non-equivariant homology theory evaluated on BG, if we assume that H_{∗}^{G} is the special value of an equivariant homology theory H_{∗}^{?} at ? = G. This means that there exists an induction structure (a mild condition satisfied in our examples, compare Section 6.1), which yields an identification

H_{n}^{G}(EG) ≅ H_{n}^{{1}}(BG) = H_{n}(BG).

Using these identifications the “classical” assembly maps, which appeared in Chapter 1 in the versions of the Farrell-Jones and Baum-Connes Conjectures for torsion free groups (see Conjecture 1.11, 1.19 and 1.31),

H_{n}(BG; K(R)) ≅ H_{n}^{G}(EG; K_{R}) → H_{n}^{G}(pt; K_{R}) ≅ K_{n}(RG);
H_{n}(BG; L^{⟨−∞⟩}(R)) ≅ H_{n}^{G}(EG; L_{R}^{⟨−∞⟩}) → H_{n}^{G}(pt; L_{R}^{⟨−∞⟩}) ≅ L_{n}^{⟨−∞⟩}(RG);
and K_{n}(BG) ≅ H_{n}^{G}(EG; K^{top}) → H_{n}^{G}(pt; K^{top}) ≅ K_{n}(C_{r}^{∗}(G)),

correspond to the assembly maps for the family F = {1} consisting only of the trivial group and are simply the maps induced by the projection EG → pt.

Remark 2.6 (The choice of the right family). As explained above the Farrell-Jones and Baum-Connes Conjectures 2.2 and 2.3 can be considered as special cases of the Meta-Conjecture 2.1. In all three cases we are interested in a computation of the right hand side H_{n}^{G}(pt) of the assembly map, which can be identified with K_{n}(RG), L_{n}^{⟨−∞⟩}(RG) or K_{n}(C_{r}^{∗}(G)). The left hand side H_{n}^{G}(E_{F}(G)) of such an assembly map is much more accessible and the smaller F is, the easier it is to compute H_{n}^{G}(E_{F}(G)) using homological methods like spectral sequences, Mayer-Vietoris arguments and Chern characters.

In the extreme case where F = ALL is the family of all subgroups the assembly map A_{ALL}: H_{n}^{G}(E_{ALL}(G)) → H_{n}^{G}(pt) is always an isomorphism for the trivial reason that the one point space pt is a model for E_{ALL}(G) (compare Subsection 2.1.3) and hence the assembly map is the identity. The goal however is to have an isomorphism for a family which is as small as possible.

We have already seen in Remark 1.4, Remark 1.25 and Remark 1.34 that in all three cases the classical assembly map, which corresponds to the trivial family, is not surjective for finite groups. This forces one to include at least the family FIN of finite groups. The K- or L-theory of the finite subgroups of the given group G will then enter in a computation of the left hand side of the assembly map similar as the K- and L-theory of the trivial subgroup appeared on the left hand side in the classical case, compare e.g. Remark 1.14. In the Baum-Connes case the family FIN seems to suffice. However in the case of algebraic K-theory we saw in Remark 1.15 that already the simplest torsion free group, the infinite cyclic group, causes problems because of the Nil-terms that appear in the Bass-Heller-Swan formula. The infinite dihedral group is a “minimal counterexample” which shows that the family FIN is not sufficient in the L_{ℤ}^{⟨−∞⟩}-case. There are non-vanishing UNil-terms, compare 2.2.6 and 2.2.7. Also the version of the L-theoretic Farrell-Jones Conjecture with the decoration s, h = ⟨1⟩ or p = ⟨0⟩ instead of ⟨−∞⟩ is definitely false. Counterexamples are given in [123]. Recall that there were no Nil-terms in the topological K-theory context, compare Remark 1.35.

The choice of the family VCY of virtually cyclic subgroups in the Farrell-Jones conjectures pushes all the Nil-problems appearing in algebraic K- and L-theory into the source of the assembly map so that they do not occur if one tries to prove the Farrell-Jones Conjecture 2.2. Of course they do come up again when one wants to compute the source of the assembly map.

We now take up the promised detailed discussion of some notions like equivariant homology theories and classifying spaces for families we used above. The reader who is familiar with these concepts may of course skip the following subsections.

2.1.1 G-CW-Complexes

A G-CW -complex X is a G-space X together with a filtration X_{−1} = ∅ ⊆ X_{0} ⊆ X_{1} ⊆ … ⊆ X such that X = colim_{n→∞} X_{n} and for each n there is a G-pushout

\begin{matrix}∐_{i∈I_{n}} G/H_{i} × S^{n−1} & \underrightarrow{∐_{i∈In} q_{i}^{n}} & X_{n−1} \\ ↓ & & ↓ \\ ∐_{i∈I_{n}} G/H_{i} × D^{n} & \overrightarrow{∐_{i∈In} Q_{i}^{n}} & X_{n} \end{matrix}.

This definition makes also sense for topological groups. The following alternative definition only applies to discrete groups. A G-CW -complex is a CW - complex with a G-action by cellular maps such that for each open cell e and each g ∈ G with ge ∩ e ≠ ∅ we have gx = x for all x ∈ e. There is an obvious notion of a G-CW -pair.

A G-CW -complex X is called finite if it is built out of finitely many G-cells G/H_{i}×D^{n}. This is the case if and only if it is cocompact, i.e. the quotient space G\X is compact. More information about G-CW -complexes can be found for instance in [197, Sections 1 and 2], [304, Sections II.1 and II.2].

2.1.2 Families of Subgroups

A family F of subgroups of G is a set of subgroups of G closed under conjugation, i.e. H ∈ F, g ∈ G implies g^{−1}Hg ∈ F, and finite intersections, i.e. H, K ∈ F implies H ∩ K ∈ F. Throughout the text we will use the notations

{1}, FCY, FIN , CYC, VCY_{I}, VCY and ALL

for the families consisting of the trivial, all finite cyclic, all finite, all (possibly infinite) cyclic, all virtually cyclic of the first kind, all virtually cyclic, respectively all subgroups of a given group G. Recall that a group is called virtually cyclic if it is finite or contains an infinite cyclic subgroup of finite index. A group is virtually cyclic of the first kind if it admits a surjection onto an infinite cyclic group with finite kernel, compare Lemma 2.15.

2.1.3 Classifying Spaces for Families

Let F be a family of subgroups of G. A G-CW-complex, all whose isotropy groups belong to F and whose H-fixed point sets are contractible for all H ∈ F, is called a classifying space for the family F and will be denoted E_{F}(G). Such a space is unique up to G-homotopy because it is characterized by the property that for any G-CW -complex X, all whose isotropy groups belong to F, there is up to G-homotopy precisely one G-map from X to E_{F}(G). These spaces were introduced by tom Dieck [303], [304, I.6].

A functorial “bar-type” construction is given in [82, section 7].

If F ⊂ G are families of subgroups for G, then by the universal property there is up to G-homotopy precisely one G-map E_{F}(G) → E_{G}(G).

The space E_{{1}}(G) is the same as the space EG which is by definition the total space of the universal G-principal bundle G → EG → BG, or, equivalently, the universal covering of BG. A model for E_{ALL}(G) is given by the space G/G = pt consisting of one point.

The space E_{FIN} (G) is also known as the classifying space for proper Gactions and denoted by \underline{E}G in the literature. Recall that a G-CW -complex X is proper if and only if all its isotropy groups are finite (see for instance [197, Theorem 1.23 on page 18]). There are often nice models for E_{FIN} (G). If G is word hyperbolic in the sense of Gromov, then the Rips-complex is a finite model [216], [217].

If G is a discrete subgroup of a Lie group L with finitely many path components, then for any maximal compact subgroup K ⊆ L the space L/K with its left G-action is a model for E_{FIN} (G) [2, Corollary 4.14]. More information about E_{FIN} (G) can be found for instance in [28, section 2], [180], [199], [206],

[207] and [282].

2.1.4 G-Homology Theories

Fix a group G and an associative commutative ring Λ with unit. A G-homology theory H_{∗}^{G} with values in Λ-modules is a collection of covariant functors H_{n}^{G} from the category of G-CW -pairs to the category of Λ-modules indexed by n ∈ ℤ together with natural transformations

∂_{n}^{G}(X, A): H_{n}^{G}(X, A) → H_{n−1}^{G}(A) := H_{n−1}^{G}(A, ∅)

for n ∈ ℤ such that the following axioms are satisfied:

(i) G-homotopy invariance If f_{0} and f_{1} are G-homotopic maps (X, A) → (Y, B) of G-CW -pairs, then H_{n}^{G}(f_{0}) = H_{n}^{G}(f_{1}) for n ∈ ℤ.

(ii) Long exact sequence of a pair Given a pair (X, A) of G-CW -complexes, there is a long exact sequence

… ⟶^{H_{n+1}^{G}(j)} H_{n+1}^{G}(X, A) ⟶^{∂_{n+1}^{G}} H_{n}^{G}(A) ⟶^{H_{n}^{G}(i)} H_{n}^{G}(X)

⟶^{H_{n}^{G}(j)} H_{n}^{G}(X, A) ⟶^{∂_{n}^{G}} H_{n−1}^{G}(A) ⟶^{H_{n−1}^{G}(i)} …,

where i: A → X and j: X → (X, A) are the inclusions. (iii) Excision

Let (X, A) be a G-CW -pair and let f: A → B be a cellular G-map of G-CW -complexes. Equip (X ∪_{f} B, B) with the induced structure of a G-CW -pair. Then the canonical map (F, f): (X, A) → (X ∪_{f} B, B) induces for each n ∈ ℤ an isomorphism

H_{n}^{G}(F, f): H_{n}^{G}(X, A) ⟶^{≅} H_{n}^{G}(X ∪_{f} B, B).

(iv) Disjoint union axiom Let {X_{i} | i ∈ I} be a family of G-CW -complexes. Denote by j_{i}: X_{i} → ∐_{i∈I} X_{i} the canonical inclusion. Then the map

⨁_{i∈I} H_{n}^{G}(j_{i}): ⨁_{i∈I} H_{n}^{G}(X_{i}) ⟶^{≅} H_{n}^{G} (∐_{i∈I} X_{i})

is bijective for each n ∈ ℤ.

Of course a G-homology theory for the trivial group G = {1} is a homology theory (satisfying the disjoint union axiom) in the classical non-equivariant sense.

The disjoint union axiom ensures that we can pass from finite G-CW-complexes to arbitrary ones using the following lemma.

Lemma 2.7. Let H_{∗}^{G} be a G-homology theory. Let X be a G-CW -complex and {X_{i} | i ∈ I} be a directed system of G-CW -subcomplexes directed by inclusion such that X = ∪_{i∈I}X_{i}. Then for all n ∈ ℤ the natural map

colim_{i∈I} H_{n}^{G}(X_{i}) ⟶^{≅} H_{n}^{G}(X)

is bijective.

Proof. Compare for example with [301, Proposition 7.53 on page 121], where the non-equivariant case for I = ℕ is treated. ⊓ ⊔ Example 2.8 (Bredon Homology). The most basic G-homology theory is Bredon homology. The orbit category Or(G) has as objects the homogeneous spaces G/H and as morphisms G-maps. Let X be a G-CW -complex. It defines a contravariant functor from the orbit category Or(G) to the category of CW - complexes by sending G/H to map_{G}(G/H, X) = X^{H}. Composing it with the functor cellular chain complex yields a contravariant functor

C_{∗}^{c}(X): Or(G) → ℤ-CHCOM

into the category of ℤ-chain complexes. Let Λ be a commutative ring and let

M: Or(G) → Λ-MODULES

be a covariant functor. Then one can form the tensor product over the orbit category (see for instance [197, 9.12 on page 166]) and obtains the Λ-chain complex C_{∗}^{c}(X) ⊗_{ℤOr (G)} M. Its homology is the Bredon homology of X with coefficients in M

H_{∗}^{G}(X; M) = H_{∗}(C_{∗}^{c}(X) ⊗_{ℤOr (G)} M).

Thus we get a G-homology theory H_{∗}^{G} with values in Λ-modules. For a trivial group G this reduces to the cellular homology of X with coefficients in the Λ-module M.

More information about equivariant homology theories will be given in Section 6.1.

2.2 Varying the Family of Subgroups

Suppose we are given a family of subgroups F′ and a subfamily F ⊂ F′. Since all isotropy groups of E_{F}(G) lie in F′ we know from the universal property of E_{F′}(G) (compare Subsection 2.1.3) that there is a G-map E_{F}(G) → E_{F′}(G) which is unique up to G-homotopy. For every G-homology theory H_{∗}^{G} we hence obtain a relative assembly map

A_{F→F′}: H_{n}^{G}(E_{F}(G)) → H_{n}^{G}(E_{F′}(G)).

If F′ = ALL, then E_{F′}(G) = pt and A_{F→F′} specializes to the assembly map A_{F} we discussed in the previous section. If we now gradually increase the family, we obtain a factorization of the classical assembly A = A_{{1}→ALL} into several relative assembly maps. We obtain for example from the inclusions

{1} ⊂ FCY ⊂ FIN ⊂ VCY ⊂ ALL

for every G-homology theory H_{n}^{G}(−) the following commutative diagram.

\xymatrix{H_{n}^{G}(EG) \ar[rr]^{A} \ar[d] &  & H_{n}^{G}(pt) \\ H_{n}^{G}(E_{FCY}(G)) \ar[r] & H_{n}^{G}(E_{FIN} (G)) \ar[r] \ar[ur]^{A_{FIN}} & H_{n}^{G}(E_{VCY}(G)). \ar[u]_{A_{VCY}}}  (10)

Here A is the “classical” assembly map and A_{FIN} and A_{VCY} are the assembly maps that for specific G-homology theories appear in the Baum-Connes and Farrell-Jones Conjectures.

Such a factorization is extremely useful because one can study the relative assembly map A_{F→F′} in terms of absolute assembly maps corresponding to groups in the bigger family. For example the relative assembly map

A_{FIN →VCY}: H_{n}^{G}(E_{FIN} (G)) → H_{n}^{G}(E_{VCY}(G))

is an isomorphism if for all virtually cyclic subgroups V of G the assembly map

A_{FIN} = A_{FIN →ALL}: H_{n}^{V} (E_{FIN} (V)) → H_{n}^{V} (pt)

is an isomorphism. Of course here we need to assume that the G-homology theory H_{∗}^{G} and the V -homology theory H_{∗}^{V} are somehow related. In fact all the G-homology theories H_{∗}^{G} we care about are defined simultaneously for all groups G and for varying G these G-homology theories are related via a so called “induction structure”. Induction structures will be discussed in detail in Section 6.1.

For a family F of subgroups of G and a subgroup H ⊂ G we define a family of subgroups of H

F ∩ H = {K ∩ H | K ∈ F}.

The general statement about relative assembly maps reads now as follows.

Theorem 2.9 (Transitivity Principle). Let H_{∗}^{?}(−) be an equivariant homology theory in the sense of Section 6.1. Suppose F ⊂ F′ are two families of subgroups of G. Suppose that K ∩ H ∈ F for each K ∈ F and H ∈ F′ (this is automatic if F is closed under taking subgroups). Let N be an integer. If for every H ∈ F′ and every n ≤ N the assembly map

A_{F∩H→ALL}: H_{n}^{H}(E_{F∩H}(H)) → H_{n}^{H}(pt)

is an isomorphism, then for every n ≤ N the relative assembly map

A_{F→F′}: H_{n}^{G}(E_{F}(G)) → H_{n}^{G}(E_{F′}(G))

is an isomorphism.

Proof. If we equip E_{F}(G) × E_{F′}(G) with the diagonal G-action, it is a model for E_{F}(G). Now apply Lemma 6.4 in the special case Z = E_{F′}(G). ⊓ ⊔

This principle can be used in many ways. For example we will derive from it that the general versions of the Baum-Connes and Farrell-Jones Conjectures specialize to the conjectures we discussed in Chapter 1 in the case where the group is torsion free. If we are willing to make compromises, e.g. to invert 2, to rationalize the theories or to restrict ourselves to small dimensions or special classes of groups, then it is often possible to get away with a smaller family, i.e. to conclude from the Baum-Connes or Farrell-Jones Conjectures that an assembly map with respect to a family smaller than the family of finite or virtually cyclic subgroups is an isomorphism. The left hand side becomes more computable and this leads to new corollaries of the Baum-Connes and Farrell-Jones Conjectures.

2.2.1 The General Versions Specialize to the Torsion Free Versions

If G is a torsion free group, then the family FIN obviously coincides with the trivial family {1}. Since a nontrivial torsion free virtually cyclic group is infinite cyclic we also know that the family VCY reduces to the family of all cyclic subgroups, denoted CYC.

Proposition 2.10. Let G be a torsion free group.

(i) If R is a regular ring, then the relative assembly map

A_{{1}→CYC}: H_{n}^{G}(E_{{1}}(G); K_{R}) → H_{n}^{G}(E_{CYC}(G); K_{R})

is an isomorphism. (ii) For every ring R the relative assembly map

A_{{1}→CYC}: H_{n}^{G}(E_{{1}}(G); L_{R}^{⟨−∞⟩}) → H_{n}^{G}(E_{CYC}(G); L_{R}^{⟨−∞⟩})

is an isomorphism.

Proof. Because of the Transitivity Principle 2.9 it suffices in both cases to prove that the classical assembly map A = A_{{1}→ALL} is an isomorphism in the case where G is an infinite cyclic group. For regular rings in the K-theory case and with the −∞-decoration in the L-theory case this is true as we discussed in Remark 1.15 respectively Remark 1.26. ⊓ ⊔

As an immediate consequence we obtain.

Corollary 2.11. (i) For a torsion free group the Baum-Connes Conjec-

ture 2.3 is equivalent to its “torsion free version” Conjecture 1.31. (ii) For a torsion free group the Farrell-Jones Conjecture 2.2 for algebraic K-

is equivalent to the “torsion free version” Conjecture 1.11, provided R is regular.

(iii) For a torsion free group the Farrell-Jones Conjecture 2.2 for algebraic L-theory is equivalent to the “torsion free version” Conjecture 1.19.

2.2.2 The Baum-Connes Conjecture and the Family VCY

Replacing the family FIN of finite subgroups by the family VCY of virtually cyclic subgroups would not make any difference in the Baum-Connes Conjecture 2.3. The Transitivity Principle 2.9 and the fact that the Baum-Connes Conjecture 2.3 is known for virtually cyclic groups implies the following.

Proposition 2.12. For every group G and every n ∈ ℤ the relative assembly map for topological K-theory

A_{FIN →VCY}: H_{n}^{G}(E_{FIN} (G); K^{top}) → H_{n}^{G}(E_{VCY}(G); K^{top})

is an isomorphism.

2.2.3 The Baum-Connes Conjecture and the Family FCY

The following result is proven in [215].

Proposition 2.13. For every group G and every n ∈ ℤ the relative assembly map for topological K-theory

A_{FCY→FIN} : H_{n}^{G}(E_{FCY}(G); K^{top}) → H_{n}^{G}(E_{FIN} (G); K^{top})

is an isomorphism.

In particular the Baum-Connes Conjecture predicts that the FCY-assembly map

A_{FCY}: H_{n}^{G}(E_{FCY}(G); K^{top}) → K_{n}(C_{r}^{∗}(G))

is always an isomorphism.

2.2.4 Algebraic K-Theory for Special Coefficient Rings

In the algebraic K-theory case we can reduce to the family of finite subgroups if we assume special coefficient rings.

Proposition 2.14. Suppose R is a regular ring in which the orders of all finite subgroups of G are invertible. Then for every n ∈ ℤ the relative assembly map for algebraic K-theory

A_{FIN →VCY}: H_{n}^{G}(E_{FIN} (G); K_{R}) → H_{n}^{G}(E_{VCY}(G); K_{R})

is an isomorphism. In particular if R is a regular ring which is a ℚ-algebra (for example a field of characteristic 0) the above applies to all groups G.

Proof. We first show that RH is regular for a finite group H. Since R is Noetherian and H is finite, RH is Noetherian. It remains to show that every RH-module M has a finite dimensional projective resolution. By assumption M considered as an R-module has a finite dimensional projective resolution.

If one applies RH ⊗_{R} − this yields a finite dimensional RH-resolution of RH ⊗_{R} res M. Since |H| is invertible, the RH-module M is a direct summand of RH ⊗_{R} res M and hence has a finite dimensional projective resolution.

Because of the Transitivity Principle 2.9 we need to prove that the FIN - assembly map A_{FIN} is an isomorphism for virtually cyclic groups V. Because of Lemma 2.15 we can assume that either V ≅ H ⋊ ℤ or V ≅ K_{1} ∗_{H} K_{2} with finite groups H, K_{1} and K_{2}. From [313] we obtain in both cases long exact sequences involving the algebraic K-theory of the constituents, the algebraic K-theory of V and also additional Nil-terms. However, in both cases the Nilterms vanish if RH is a regular ring (compare Theorem 4 on page 138 and the Remark on page 216 in [313]). Thus we get long exact sequences

… → K_{n}(RH) → K_{n}(RH) → K_{n}(RV) → K_{n−1}(RH) → K_{n−1}(RH) → …

and

… → K_{n}(RH) → K_{n}(RK_{1}) ⊕ K_{n}(RK_{2}) → K_{n}(RV) → K_{n−1}(RH) → K_{n−1}(RK_{1}) ⊕ K_{n−1}(RK_{2}) → …

One obtains analogous exact sequences for the sources of the various assembly maps from the fact that the sources are equivariant homology theories and one can find specific models for E_{FIN} (V). These sequences are compatible with the assembly maps. The assembly maps for the finite groups H, K_{1} and K_{2} are bijective. Now a Five-Lemma argument shows that also the one for V is bijective. ⊓ ⊔

In particular for regular coefficient rings R which are ℚ-algebras the Ktheoretic Farrell-Jones Conjecture specializes to the conjecture that the assembly map

A_{FIN} : H_{n}^{G}(E_{FIN} (G); K_{R}) → H_{n}^{G}(pt; K_{R}) ≅ K_{n}(RG)

is an isomorphism.

In the proof above we used the following important fact about virtually cyclic groups.

Lemma 2.15. If G is an infinite virtually cyclic group then we have the following dichotomy.

(I) Either G admits a surjection with finite kernel onto the infinite cyclic

group ℤ, or (II) G admits a surjection with finite kernel onto the infinite dihedral group

ℤ/2 ∗ ℤ/2.

Proof. This is not difficult and proven as Lemma 2.5 in [113]. ⊓ ⊔ 2.2.5 Splitting off Nil-Terms and Rationalized Algebraic K-Theory

Recall that the Nil-terms, which prohibit the classical assembly map from being an isomorphism, are direct summands of the K-theory of the infinite cyclic group (see Remark 1.15). Something similar remains true in general [16].

Proposition 2.16. (i) For every group G, every ring R and every n ∈ ℤ the relative assembly map

A_{FIN →VCY}: H_{n}^{G}(E_{FIN} (G); K_{R}) → H_{n}^{G}(E_{VCY}(G); K_{R})

is split-injective. (ii) Suppose R is such that K_{−i}(RV) = 0 for all virtually cyclic subgroups

V of G and for sufficiently large i (for example R = ℤ will do, compare Proposition 3.2). Then the relative assembly map

A_{FIN →VCY}: H_{n}^{G}(E_{FIN} (G); L_{R}^{⟨−∞⟩}) → H_{n}^{G}(E_{VCY}(G); L_{R}^{⟨−∞⟩})

is split-injective.

Combined with the Farrell-Jones Conjectures we obtain that the homology group H_{n}^{G}(E_{FIN} (G); K_{R}) is a direct summand in K_{n}(RG). It is much better understood (compare Chapter 8) than the remaining summand which is isomorphic to H_{n}^{G}(E_{VCY}(G), E_{FIN} (G); K_{R}). This remaining summand is the one which plays the role of the Nil-terms for a general group. It is known that for R = ℤ the negative dimensional Nil-groups which are responsible for virtually cyclic groups vanish [113]. They vanish rationally, in dimension 0 by [76] and in higher dimensions by [182]. For more information see also [75]. Analogously to the proof of Proposition 2.14 we obtain the following proposition.

Proposition 2.17. We have

H_{n}^{G}(E_{VCY}(G), E_{FIN} (G); K_{ℤ}) = 0 for n < 0 and
H_{n}^{G}(E_{VCY}(G), E_{FIN} (G); K_{ℤ}) ⊗_{ℤ} ℚ = 0 for all n ∈ ℤ.

In particular the Farrell-Jones Conjecture for the algebraic K-theory of the integral group ring predicts that the map

A_{FIN} : H_{n}^{G}(E_{FIN} (G); K_{ℤ}) ⊗_{ℤ} ℚ → K_{n}(ℤG) ⊗_{ℤ} ℚ

is always an isomorphism.

2.2.6 Inverting 2 in L-Theory

Proposition 2.18. For every group G, every ring R with involution, every decoration j and all n ∈ ℤ the relative assembly map

A_{FIN →VCY}: H_{n}^{G}(E_{FIN} (G); L_{R}^{⟨j⟩})[\frac{1}{2}] → H_{n}^{G}(E_{VCY}(G); L_{R}^{⟨j⟩})[\frac{1}{2}]

is an isomorphism.

Proof. According to the Transitivity Principle it suffices to prove the claim for a virtually cyclic group. Now argue analogously to the proof of Proposition 2.14 using the exact sequences in [48] and the fact that the UNil-terms appearing there vanish after inverting two [48]. Also recall from Remark 1.22 that after inverting 2 there are no differences between the decorations. ⊓ ⊔

In particular the L-theoretic Farrell-Jones Conjecture implies that for every decoration j the assembly map

A_{FIN} : H_{n}^{G}(E_{FIN} (G); L_{R}^{⟨j⟩})[\frac{1}{2}] → L_{n}^{⟨j⟩}(RG)[\frac{1}{2}]

is an isomorphism.

2.2.7 L-theory and Virtually Cyclic Subgroups of the First Kind

Recall that a group is virtually cyclic of the first kind if it admits a surjection with finite kernel onto the infinite cyclic group. The family of these groups is denoted VCY_{I}.

Proposition 2.19. For all groups G, all rings R and all n ∈ ℤ the relative assembly map

A_{FIN →VCY_{I}}: H_{n}^{G}(E_{FIN} (G); L_{R}^{⟨−∞⟩}) → H_{n}^{G}(E_{VCY_{I}} (G); L_{R}^{⟨−∞⟩})

is an isomorphism.

Proof. The point is that there are no UNil-terms for infinite virtually cyclic groups of the first kind. This follows essentially from [254] and [255] as carried out in [204]. ⊓ ⊔

2.2.8 Rationally FIN Reduces to FCY

We will see later (compare Theorem 8.4, 8.5 and 8.10) that in all three cases, topological K-theory, algebraic K-theory and L-theory, the rationalized left hand side of the FIN -assembly map can be computed very explicitly using the equivariant Chern-Character. As a by-product these computations yield that after rationalizing the family FIN can be reduced to the family FCY of finite cyclic groups and that the rationalized relative assembly maps A_{{1}→FCY} are injective.

Proposition 2.20. For every ring R, every group G and all n ∈ ℤ the relative assembly maps

A_{FCY→FIN} : H_{n}^{G}(E_{FCY}(G); K_{R}) ⊗_{ℤ} ℚ → H_{n}^{G}(E_{FIN} (G); K_{R}) ⊗_{ℤ} ℚ
A_{FCY→FIN} : H_{n}^{G}(E_{FCY}(G); L_{R}^{⟨−∞⟩}) ⊗_{ℤ} ℚ → H_{n}^{G}(E_{FIN} (G); L_{R}^{⟨−∞⟩}) ⊗_{ℤ} ℚ
A_{FCY→FIN} : H_{n}^{G}(E_{FCY}(G); K^{top}) ⊗_{ℤ} ℚ → H_{n}^{G}(E_{FIN} (G); K^{top}) ⊗_{ℤ} ℚ

are isomorphisms and the corresponding relative assembly maps A_{{1}→FCY} are all rationally injective.

Recall that the statement for topological K-theory is even known integrally, compare Proposition 2.13. Combining the above with Proposition 2.17 and Proposition 2.18 we see that the Farrell-Jones Conjecture predicts in particular that the FCY-assembly maps

A_{FCY}: H_{n}^{G}(E_{FCY}(G); L_{R}^{⟨−∞⟩}) ⊗_{ℤ} ℚ → L_{n}^{⟨−∞⟩}(RG) ⊗_{ℤ} ℚ
A_{FCY}: H_{n}^{G}(E_{FCY}(G); K_{ℤ}) ⊗_{ℤ} ℚ → K_{n}(ℤG) ⊗_{ℤ} ℚ

are always isomorphisms.

3 More Applications

3.1 Applications VI

3.1.1 Low Dimensional Algebraic K-Theory

As opposed to topological K-theory and L-theory, which are periodic, the algebraic K-theory groups of coefficient rings such as ℤ, ℚ or ℂ are known to be bounded below. Using the spectral sequences for the left hand side of an assembly map that will be discussed in Subsection 8.4.1, this leads to vanishing results in negative dimensions and a concrete description of the groups in the first non-vanishing dimension.

The following conjecture is a consequence of the K-theoretic Farrell-Jones Conjecture in the case R = ℤ. Note that by the results discussed in Subsection 2.2.5 we know that in negative dimensions we can reduce to the family of finite subgroups. Explanations about the colimit that appears follow below.

Conjecture 3.1 (The Farrell-Jones Conjecture for K_{n}(ℤG) for n ≤ −1). For every group G we have

K_{−n}(ℤG) = 0 for n ≥ 2,

and the map

\xymatrix{colim_{H∈Sub _{FIN} (G)} K_{−1}(ℤH) \ar[r]^{≅} & K_{−1}(ℤG)}

is an isomorphism.

We can consider a family F of subgroups of G as a category Sub_{F}(G) as follows. The objects are the subgroups H with H ∈ F. For H, K ∈ F let conhom_{G}(H, K) be the set of all group homomorphisms f: H → K, for which there exists a group element g ∈ G such that f is given by conjugation with g. The group of inner automorphism inn(K) consists of those automorphisms K → K, which are given by conjugation with an element k ∈ K. It acts on conhom(H, K) from the left by composition. Define the set of morphisms in Sub_{F}(G) from H to K to be inn(K)\ conhom(H, K). Composition of group homomorphisms defines the composition of morphisms in Sub_{F}(G). We mention that Sub_{F}(G) is a quotient category of the orbit category Or_{F}(G) which we will introduce in Section 6.4. Note that there is a morphism from H to K only if H is conjugate to a subgroup of K. Clearly K_{n}(R(−)) yields a functor from Sub_{F}(G) to abelian groups since inner automorphisms on a group G induce the identity on K_{n}(RG). Using the inclusions into G, one obtains a map

colim_{H∈Sub _{F} (G)} K_{n}(RH) → K_{n}(RG).

The colimit can be interpreted as the 0-th Bredon homology group

H_{0}^{G}(E_{F}(G); K_{n}(R(?)))

(compare Example 2.8) and the map is the edge homomorphism in the equivariant Atiyah-Hirzebruch spectral sequence discussed in Subsection 8.4.1. In Conjecture 3.1 we consider the first non-vanishing entry in the lower left hand corner of the E_{2}-term because of the following vanishing result [113, Theorem 2.1] which generalizes vanishing results for finite groups from [57].

Proposition 3.2. If V is a virtually cyclic group, then K_{−n}(ℤV) = 0 for n ≥ 2.

If our coefficient ring R is a regular ring in which the orders of all finite subgroups of G are invertible, then we know already from Subsection 2.2.4 that we can reduce to the family of finite subgroups. In the proof of Proposition 2.14 we have seen that then RH is again regular if H ⊂ G is finite. Since negative K-groups vanish for regular rings [268, 5.3.30 on page 295], the following is implied by the Farrell-Jones Conjecture 2.2.

Conjecture 3.3 (Farrell-Jones Conjecture for K_{0}(ℚG)). Suppose R is a regular ring in which the orders of all finite subgroups of G are invertible (for example a field of characteristic 0), then

K_{−n}(RG) = 0 for n ≥ 1

and the map

\xymatrix{colim_{H∈Sub _{FIN} (G)} K_{0}(RH) \ar[r]^{≅} & K_{0}(RG)}

is an isomorphism.

The conjecture above holds if G is virtually poly-cyclic. Surjectivity is proven in [227] (see also [67] and Chapter 8 in [235]), injectivity in [271]. We will show in Lemma 3.11 (i) that the map appearing in the conjecture is always rationally injective for R = ℂ.

The conjectures above describe the first non-vanishing term in the equivariant Atiyah-Hirzebruch spectral sequence. Already the next step is much harder to analyze in general because there are potentially non-vanishing differentials. We know however that after rationalizing the equivariant Atiyah-Hirzebruch spectral sequence for the left hand side of the FIN -assembly map collapses. As a consequence we obtain that the following conjecture follows from the K-theoretic Farrell-Jones Conjecture 2.2.

Conjecture 3.4. For every group G, every ring R and every n ∈ ℤ the map

colim_{H∈Sub _{FIN} (G)} K_{n}(RH) ⊗_{ℤ} ℚ → K_{n}(RG) ⊗_{ℤ} ℚ

is injective.

Note that for K_{0}(ℤG)⊗_{ℤ}ℚ the conjecture above is always true but not very interesting, because for a finite group H it is known that K̃_{0}(ℤH) ⊗_{ℤ} ℚ = 0, compare [298, Proposition 9.1], and hence the left hand side reduces to K_{0}(ℤ) ⊗_{ℤ} ℚ. However, the full answer for K_{0}(ℤG) should involve the negative K-groups, compare Example 8.8.

Analogously to Conjecture 3.4 the following can be derived from the Ktheoretic Farrell-Jones Conjecture 2.2, compare [208].

Conjecture 3.5. The map

colim_{H∈Sub _{FIN} (G)} Wh(H) ⊗_{ℤ} ℚ → Wh(G) ⊗_{ℤ} ℚ

is always injective.

In general one does not expect this map to be an isomorphism. There should be additional contributions coming from negative K-groups. Conjecture 3.5 is true for groups satisfying a mild homological finiteness condition, see Theorem 5.26.

Remark 3.6 (The Conjectures as Generalized Induction Theorems). The above discussion shows that one may think of the Farrell-Jones Conjectures 2.2 and the Baum-Connes Conjecture 2.3 as “generalized induction theorems”. The prototype of an induction theorem is Artin’s theorem about the complex representation ring R_{ℂ}(G) of a finite group G. Let us recall Artin’s theorem.

For finite groups H the complex representation ring R_{ℂ}(H) coincides with K_{0}(ℂH). Artin’s Theorem [283, Theorem 17 in 9.2 on page 70] implies that the obvious induction homomorphism

colim_{H∈Sub _{CYC}(G)} R_{ℂ}(H) ⊗_{ℤ} ℚ ⟶^{≅} R_{ℂ}(G) ⊗_{ℤ} ℚ

is an isomorphism. Note that this is a very special case of Theorem 8.4 or 8.5, compare Remark 8.9.

Artin’s theorem says that rationally one can compute R_{ℂ}(G) if one knows all the values R_{ℂ}(C) (including all maps coming from induction with group homomorphisms induced by conjugation with elements in G) for all cyclic subgroups C ⊆ G. The idea behind the Farrell-Jones Conjectures 2.2 and the Baum-Connes Conjecture 2.3 is analogous. We want to compute the functors K_{n}(RG), L_{n}(RG) and K_{n}(C_{r}^{∗}(G)) from their values (including their functorial properties under induction) on elements of the family FIN or VCY.

The situation in the Farrell Jones and Baum-Connes Conjectures is more complicated than in Artin’s Theorem, since we have already seen in Remarks 1.15, 1.26 and 1.35 that a computation of K_{n}(RG), L_{n}^{⟨−∞⟩}(RG) and K_{n}(C_{r}^{∗}(G)) does involve also the values K_{p}(RH), L_{p}^{⟨−∞⟩}(RH) and K_{p}(C_{r}^{∗}(H)) for p ≤ n. A degree mixing occurs.

3.1.2 G-Theory

Instead of considering finitely generated projective modules one may apply the standard K-theory machinery to the category of finitely generated modules. This leads to the definition of the groups G_{n}(R) for n ≥ 0. For instance G_{0}(R) is the abelian group whose generators are isomorphism classes [M] of finitely generated R-modules and whose relations are given by [M_{0}] − [M_{1}] + [M_{2}] for any exact sequence 0 → M_{0} → M_{1} → M_{2} → 0 of finitely generated modules. One may ask whether versions of the Farrell-Jones Conjectures for G-theory instead of K-theory might be true. The answer is negative as the following discussion explains.

For a finite group H the ring ℂH is semisimple. Hence any finitely generated ℂH-module is automatically projective and K_{0}(ℂH) = G_{0}(ℂH). Recall that a group G is called virtually poly-cyclic if there exists a subgroup of finite index H ⊆ G together with a filtration {1} = H_{0} ⊆ H_{1} ⊆ H_{2} ⊆ … ⊆ H_{r} = H such that H_{i−1} is normal in H_{i} and the quotient H_{i}/H_{i−1} is cyclic. More generally for all n ∈ ℤ the forgetful map

f: K_{n}(ℂG) → G_{n}(ℂG)

is an isomorphism if G is virtually poly-cyclic, since then ℂG is regular [273, Theorem 8.2.2 and Theorem 8.2.20] and the forgetful map f is an isomorphism for regular rings, compare [268, Corollary 53.26 on page 293]. In particular this applies to virtually cyclic groups and so the left hand side of the Farrell-Jones assembly map does not see the difference between K- and G-theory if we work with complex coefficients. We obtain a commutative diagram

\xymatrix{colim_{H∈Sub _{FIN} (G)} K_{0}(ℂH) \ar[d]_{≅} \ar[r] & K_{0}(ℂG) \ar[d]^{f} \\ colim_{H∈Sub _{FIN} (G)} G_{0}(ℂH) \ar[r] & G_{0}(ℂG)}  (11)

where, as indicated, the left hand vertical map is an isomorphism. Conjecture 3.3, which is implied by the Farrell-Jones Conjecture, says that the upper horizontal arrow is an isomorphism. A G-theoretic analogue of the Farrell-Jones Conjecture would say that the lower horizontal map is an isomorphism. There are however cases where the upper horizontal arrow is known to be an isomorphism, but the forgetful map f on the right is not injective or not surjective:

If G contains a non-abelian free subgroup, then the class [ℂG] ∈ G_{0}(ℂG) vanishes [202, Theorem 9.66 on page 364] and hence the map f: K_{0}(ℂG) → G_{0}(ℂG) has an infinite kernel ([ℂG] generates an infinite cyclic subgroup in K_{0}(ℂG)). The Farrell-Jones Conjecture for K_{0}(ℂG) is known for non-abelian free groups.

The Farrell-Jones Conjecture is also known for A = ⨁_{n∈ℤ} ℤ/2 and hence K_{0}(ℂA) is countable, whereas G_{0}(ℂA) is not countable [202, Example 10.13 on page 375]. Hence the map f cannot be surjective.

At the time of writing we do not know a counterexample to the statement that for an amenable group G, for which there is an upper bound on the orders of its finite subgroups, the forgetful map f: K_{0}(ℂG) → G_{0}(ℂG) is an isomorphism. We do not know a counterexample to the statement that for a group G, which is not amenable, G_{0}(ℂG) = {0}. We also do not know whether G_{0}(ℂG) = {0} is true for G = ℤ ∗ ℤ.

For more information about G_{0}(ℂG) we refer for instance to [202, Subsection 9.5.3].

3.1.3 Bass Conjectures

Complex representations of a finite group can be studied using characters. We now want to define the Hattori-Stallings rank of a finitely generated projective ℂG-module which should be seen as a generalization of characters to infinite groups.

Let con(G) be the set of conjugacy classes (g) of elements g ∈ G. Denote by con(G)_{f} the subset of con(G) consisting of those conjugacy classes (g) for which each representative g has finite order. Let class_{0}(G) and class_{0}(G)_{f} be the ℂ-vector space with the set con(G) and con(G)_{f} as basis. This is the same as the ℂ-vector space of ℂ-valued functions on con(G) and con(G)_{f} with finite support. Define the universal ℂ-trace as

tr_{ℂG}^{u}: ℂG → class_{0}(G), ∑_{g∈G} λ_{g} · g ↦ ∑_{g∈G} λ_{g} · (g).  (12)

It extends to a function tr_{ℂG}^{u}: M_{n}(ℂG) → class_{0}(G) on (n, n)-matrices over ℂG by taking the sum of the traces of the diagonal entries. Let P be a finitely generated projective ℂG-module. Choose a matrix A ∈ M_{n}(ℂG) such that A^{2} = A and the image of the ℂG-map r_{A}: ℂG^{n} → ℂG^{n} given by right multiplication with A is ℂG-isomorphic to P. Define the Hattori-Stallings rank of P as

HS_{ℂG}(P) = tr_{ℂG}^{u}(A) ∈ class_{0}(G).  (13)

The Hattori-Stallings rank depends only on the isomorphism class of the ℂGmodule P and induces a homomorphism HS_{ℂG}: K_{0}(ℂG) → class_{0}(G).

Conjecture 3.7 (Strong Bass Conjecture for K_{0}(ℂG)). The ℂ-vector space spanned by the image of the map

HS_{ℂG}: K_{0}(ℂG) → class_{0}(G)

is class_{0}(G)_{f}.

This conjecture is implied by the surjectivity of the map

colim_{H∈Sub _{FIN} (G)} K_{0}(ℂH) ⊗_{ℤ} ℂ → K_{0}(ℂG) ⊗_{ℤ} ℂ,  (14)

(compare Conjecture 3.3) and hence by the K-theoretic Farrell-Jones Conjecture for K_{0}(ℂG). We will see below that the surjectivity of the map (14) also implies that the map K_{0}(ℂG) ⊗_{ℤ} ℂ → class_{0}(G), which is induced by the Hattori-Stallings rank, is injective. Hence we do expect that the Hattori-Stallings rank induces an isomorphism

K_{0}(ℂG) ⊗_{ℤ} ℂ ≅ class_{0}(G)_{f}.

There are also versions of the Bass conjecture for other coefficients than ℂ. It follows from results of Linnell [191, Theorem 4.1 on page 96] that the following version is implied by the Strong Bass Conjecture for K_{0}(ℂG).

Conjecture 3.8 (The Strong Bass Conjecture for K_{0}(ℤG)). The image of the composition

K_{0}(ℤG) → K_{0}(ℂG) ⟶^{HS_{ℂG}} class_{0}(G)

is contained in the ℂ-vector space of those functions f: con(G) → ℂ which vanish for (g) ∈ con(g) with g ≠ 1.

The conjecture says that for every finitely generated projective ℤG-module P the Hattori-Stallings rank of ℂG⊗_{ℤG} P looks like the Hattori-Stallings rank of a free ℂG-module. A natural explanation for this behaviour is the following conjecture which clearly implies the Strong Bass Conjecture for K_{0}(ℤG).

Conjecture 3.9 (Rational K̃_{0}(ℤG)-to-K̃_{0}(ℚG)-Conjecture). For every group G the map

K̃_{0}(ℤG) ⊗_{ℤ} ℚ → K̃_{0}(ℚG) ⊗_{ℤ} ℚ

induced by the change of coefficients is trivial.

Finally we mention the following variant of the Bass Conjecture.

Conjecture 3.10 (The Weak Bass Conjecture). Let P be a finitely generated projective ℤG-module. The value of the Hattori-Stallings rank of ℂG ⊗_{ℤG} P at the conjugacy class of the identity element is given by

HS_{ℂG}(ℂG ⊗_{ℤG} P)((1)) = dim_{ℤ}(ℤ ⊗_{ℤG} P).

Here ℤ is considered as a ℤG-module via the augmentation.

The K-theoretic Farrell-Jones Conjecture implies all four conjectures above. More precisely we have the following proposition.

Proposition 3.11. (i) The map

colim_{H∈Sub _{FIN} (G)} K_{0}(ℂH) ⊗_{ℤ} ℚ → K_{0}(ℂG) ⊗_{ℤ} ℚ

is always injective. If the map is also surjective (compare Conjecture 3.3) then the Hattori-Stallings rank induces an isomorphism

K_{0}(ℂG) ⊗_{ℤ} ℂ ≅ class_{0}(G)_{f}

and in particular the Strong Bass Conjecture for K_{0}(ℂG) and hence also the Strong Bass Conjecture for K_{0}(ℤG) hold.

(ii) The surjectivity of the map

A_{VCY}: H_{0}^{G}(E_{VCY}(G); K_{ℤ}) ⊗_{ℤ} ℚ → K_{0}(ℤG) ⊗_{ℤ} ℚ

implies the Rational K̃_{0}(ℤG)-to-K̃_{0}(ℚG) Conjecture and hence also the Strong Bass Conjecture for K_{0}(ℤG).

(iii) The Strong Bass Conjecture for K_{0}(ℂG) implies the Strong Bass Conjecture for K_{0}(ℤG). The Strong Bass Conjecture for K_{0}(ℤG) implies the Weak Bass Conjecture.

Proof. (i) follows from the following commutative diagram, compare [198, Lemma 2.15 on page 220].

\xymatrix{colim_{H∈Sub _{FIN} (G)} K_{0}(ℂH) ⊗_{ℤ} ℂ \ar[d]_{≅} \ar[rr] &  & K_{0}(ℂG) ⊗_{ℤ} ℂ \ar[d] \\ colim_{H∈Sub _{FIN} (G)} class_{0}(H) \ar[r]^{≅} & class_{0}(G)_{f} \ar[r]^{i} & class_{0}(G).}

Here the vertical maps are induced by the Hattori-Stallings rank, the map i is the natural inclusion and in particular injective and we have the indicated isomorphisms.

(ii) According to Proposition 2.17 the surjectivity of the map A_{VCY} appearing in (ii) implies the surjectivity of the corresponding assembly map A_{FIN} (rationalized and with ℤ as coefficient ring) for the family of finite subgroups. The map A_{FIN} is natural with respect to the change of the coefficient ring from ℤ to ℚ. By Theorem 8.5 we know that for every coefficient ring R there is an isomorphism from

⨁_{p,q,p+q=0} ⨁_{(C)∈(FCY)} H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} Θ_{C} · K_{q}(RC) ⊗_{ℤ} ℚ

to the 0-dimensional part of the left hand side of the rationalized FIN - assembly map A_{FIN}. The isomorphism is natural with respect to a change of coefficient rings. To see that the Rational K̃_{0}(ℤG)-to-K̃_{0}(ℚG) Conjecture follows, it hence suffices to show that the summand corresponding to C = {1} and p = q = 0 is the only one where the map induced from ℤ → ℚ is possibly non-trivial. But K_{q}(ℚC) = 0 if q < 0, because ℚC is semisimple and hence regular, and for a finite cyclic group C ≠ {1} we have by [198, Lemma 7.4]

Θ_{C} · K_{0}(ℤC) ⊗_{ℤ} ℚ = coker ( ⨁_{D⊊C} K_{0}(ℤD) ⊗_{ℤ} ℚ → K_{0}(ℤC) ⊗_{ℤ} ℚ) = 0,

since by a result of Swan K_{0}(ℤ) ⊗_{ℤ} ℚ → K_{0}(ℤH) ⊗_{ℤ} ℚ is an isomorphism for a finite group H, see [298, Proposition 9.1].

(iii) As already mentioned the first statement follows from [191, Theorem 4.1 on page 96]. The second statement follows from the formula

∑_{(g)∈con(G)} HS_{ℂG}(ℂ ⊗_{ℤ} P)(g) = dim_{ℤ}(ℤ ⊗_{ℤG} P).

⊓ ⊔

The next result is due to Berrick, Chatterji and Mislin [36, Theorem 5.2]. The Bost Conjecture is a variant of the Baum-Connes Conjecture and is explained in Subsection 4.1.3.

Theorem 3.12. If the assembly map appearing in the Bost Conjecture 4.2 is rationally surjective, then the Strong Bass Conjecture for K_{0}(ℂG) (compare 3.7) is true.

We now discuss some further questions and facts that seem to be relevant in the context of the Bass Conjectures.

Remark 3.13 (Integral K̃_{0}(ℤG)-to-K̃_{0}(ℚG)-Conjecture). We do not know a counterexample to the Integral K̃_{0}(ℤG)-to-K̃_{0}(ℚG) Conjecture, i.e. to the statement that the map

K̃_{0}(ℤG) → K̃_{0}(ℚG)

itself is trivial. But we also do not know a proof which shows that the Ktheoretic Farrell-Jones Conjecture implies this integral version. Note that the Integral K̃_{0}(ℤG)-to-K̃_{0}(ℚG) Conjecture would imply that the following diagram commutes.

K_{0}(ℤG) ⟶ K_{0}(ℚG)
p_{∗}↓  ^{i}↑
K_{0}(ℤ) ⟶_{≅}^{dim_{ℤ}} ℤ.

Here p_{∗} is induced by the projection G → {1} and i sends 1 ∈ ℤ to the class of ℚG.

Remark 3.14 (The passage from K̃_{0}(ℤG) to K̃_{0}(N (G))). Let N (G) denote the group von Neumann algebra of G. It is known that for every group G the composition

K̃_{0}(ℤG) → K̃_{0}(ℚG) → K̃_{0}(ℂG) → K̃_{0}(C_{r}^{∗}(G)) → K̃_{0}(N (G))

is the zero-map (see for instance [202, Theorem 9.62 on page 362]). Since the group von Neumann algebra N (G) is not functorial under arbitrary group homomorphisms such as G → {1}, this does not imply that the diagram

K_{0}(ℤG) ⟶ K_{0}(N (G))
p_{∗}↓  ^{i}↑
K_{0}(ℤ) ⟶_{≅}^{dim_{ℤ}}  ℤ

commutes. However, commutativity would follow from the Weak Bass Conjecture 3.10. For a discussion of these questions see [93].

More information and further references about the Bass Conjecture can be found for instance in [24], [36, Section 7], [42], [92], [93], [118], [191] [202, Subsection 9.5.2], and [225, page 66ff].

3.2 Applications VII

3.2.1 Novikov Conjectures

In Subsection 1.9.1 we discussed the Novikov Conjectures. Recall that one possible reformulation of the original Novikov Conjecture says that for every group G the rationalized classical assembly map in L-theory

A: H_{n}(BG; L^{p}(ℤ)) ⊗_{ℤ} ℚ → L_{n}^{p}(ℤG) ⊗_{ℤ} ℚ

is injective. Since A can be identified with A_{{1}→ALL} and we know from Subsection 2.2.8 that the relative assembly map

A_{{1}→FIN} : H_{n}^{G}(E_{{1}}(G); L_{ℤ}^{p}) ⊗_{ℤ} ℚ → H_{n}^{G}(E_{FIN} (G); L_{ℤ}^{p}) ⊗_{ℤ} ℚ

is injective we obtain the following proposition.

Proposition 3.15. The rational injectivity of the assembly map appearing in L-theoretic Farrell-Jones Conjecture (Conjecture 2.2) implies the L-theoretic Novikov Conjecture (Conjecture 1.51) and hence the original Novikov Conjecture 1.52.

Similarly the Baum-Connes Conjecture 2.3 implies the injectivity of the rationalized classical assembly map

A: H_{n}(BG; K^{top}) ⊗_{ℤ} ℚ → K_{n}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ.

In the next subsection we discuss how one can relate assembly maps for topological K-theory with L-theoretic assembly maps. The results imply in particular the following proposition.

Proposition 3.16. The rational injectivity of the assembly map appearing in the Baum-Connes Conjecture (Conjecture 2.3) implies the Novikov Conjecture (Conjecture 1.52).

Finally we would like to mention that by combining the results about the rationalization of A_{{1}→FIN} from Subsection 2.2.8 with the splitting result about A_{FIN →VCY} from Subsection 2.2.5 we obtain the following result

Proposition 3.17. The rational injectivity of the assembly map appearing in the Farrell-Jones Conjecture for algebraic K-theory (Conjecture 2.2) implies the K-theoretic Novikov Conjecture, i.e. the injectivity of

A: H_{n}(BG; K(R)) ⊗_{ℤ} ℚ → K_{n}(RG) ⊗_{ℤ} ℚ.

Remark 3.18 (Integral Injectivity Fails). In general the classical assembly maps A = A_{{1}} themselves, i.e. without rationalizing, are not injective. For example one can use the Atiyah-Hirzebruch spectral sequence to see that for G = ℤ/5

H_{1}(BG; K^{top}) and H_{1}(BG; L^{⟨−∞⟩}(ℤ))

contain 5-torsion, whereas for every finite group G the topological K-theory of ℂG is torsionfree and the torsion in the L-theory of ℤG is always 2-torsion, compare Proposition 8.1 (i) and Proposition 8.3 (i).

3.2.2 Relating Topological K-Theory and L-Theory

For every real C^{∗}-algebra A there is an isomorphism L_{n}^{p}(A)[1/2] ⟶^{≅} K_{n}(A)[1/2] [269]. This can be used to compare L-theory to topological K-theory and leads to the following result.

Proposition 3.19. Let F ⊆ FIN be a family of finite subgroups of G. If the topological K-theory assembly map

A_{F}: H_{n}^{G}(E_{F}(G); K^{top})[\frac{1}{2}] → K_{n}(C_{r}^{∗}(G))[\frac{1}{2}]

is injective, then for an arbitrary decoration j also the map

A_{F}: H_{n}^{G}(E_{F}(G); L_{ℤ}^{⟨j⟩})[\frac{1}{2}] → L_{n}^{⟨j⟩}(ℤG)[\frac{1}{2}]

is injective.

Proof. First recall from Remark 1.22 that after inverting 2 there is no difference between the different decorations and we can hence work with the p-decoration. One can construct for any subfamily F ⊆ FIN the following commutative diagram [200, Section 7.5]

H_{n}^{G}(E_{F}(G); L_{ℤ}^{p}[1/2]) ⟶^{A_{F}^{1}} L_{n}^{p}(ℤG)[1/2]
i_{1}↓≅  ^{j_{1}}↓≅
H_{n}^{G}(E_{F}(G); L_{ℚ}^{p}[1/2]) ⟶^{A_{F}^{2}} L_{n}^{p}(ℚG)[1/2]
i_{2}↓≅  ^{j_{2}}↓
H_{n}^{G}(E_{F}(G); L_{ℝ}^{p}[1/2]) ⟶^{A_{F}^{3}} L_{n}^{p}(ℝG)[1/2]
i_{3}↓≅  ^{j_{3}}↓
H_{n}^{G}(E_{F}(G); L_{C_{r}^{∗}(?;ℝ)}^{p}[1/2]) ⟶^{A_{F}^{4}} L_{n}^{p}(C_{r}^{∗}(G; ℝ))[1/2]
i_{4}↓≅  ^{j_{4}}↓≅
H_{n}^{G}(E_{F}(G); K_{ℝ}^{top}[1/2]) ⟶^{A_{F}^{5}} K_{n}(C_{r}^{∗}(G; ℝ))[1/2]
i_{5}↓  ^{j_{5}}↓
H_{n}^{G}(E_{F}(G); K_{ℂ}^{top}[1/2]) ⟶^{A_{F}^{6}} K_{n}(C_{r}^{∗}(G))[1/2]

Here

L_{ℤ}^{p}[1/2], L_{ℚ}^{p}[1/2], L_{ℝ}^{p}[1/2], L_{C_{r}^{∗}(?;ℝ)}[1/2],
K_{ℝ}^{top}[1/2] and K_{ℂ}^{top}[1/2]

are covariant Or(G)-spectra (compare Section 6.2 and in particular Proposition 6.7) such that the n-th homotopy group of their evaluations at G/H are given by

L_{n}^{p}(ℤH)[1/2], L_{n}^{p}(ℚH)[1/2], L_{n}^{p}(ℝH)[1/2], L_{n}^{p}(C_{r}^{∗}(H; ℝ))[1/2],
K_{n}(C_{r}^{∗}(H; ℝ))[1/2] respectively K_{n}(C_{r}^{∗}(H)[1/2].

All horizontal maps are assembly maps induced by the projection pr: E_{F}(G) → pt. The maps i_{k} and j_{k} for k = 1, 2, 3 are induced from a change of rings. The isomorphisms i_{4} and j_{4} come from the general isomorphism for any real C^{∗}- algebra A

L_{n}^{p}(A)[1/2] ⟶^{≅} K_{n}(A)[1/2]

and its spectrum version [269, Theorem 1.11 on page 350]. The maps i_{1}, j_{1}, i_{2} are isomorphisms by [256, page 376] and [258, Proposition 22.34 on page 252]. The map i_{3} is bijective since for a finite group H we have ℝH = C_{r}^{∗}(H; ℝ). The maps i_{5} and j_{5} are given by extending the scalars from ℝ to ℂ by induction. For every real C^{∗}-algebra A the composition

K_{n}(A)[1/2] → K_{n}(A ⊗_{ℝ} ℂ)[1/2] → K_{n}(M_{2}(A))[1/2]

is an isomorphism and hence j_{5} is split injective. An Or(G)-spectrum version of this argument yields that also i_{5} is split injective. ⊓ ⊔

Remark 3.20. One may conjecture that the right vertical maps j_{2} and j_{3} are isomorphisms and try to prove this directly. Then if we invert 2 everywhere the Baum-Connes Conjecture 2.3 for the real reduced group C^{∗}-algebra, would be equivalent to the Farrell-Jones Isomorphism Conjecture for L_{∗}(ℤG)[1/2].

3.3 Applications VIII

3.3.1 The Modified Trace Conjecture

Denote by Λ^{G} the subring of ℚ which is obtained from ℤ by inverting all orders |H| of finite subgroups H of G, i.e.

Λ^{G} = ℤ [|H|^{−1} | H ⊂ G, |H| < ∞] .  (15)

The following conjecture generalizes Conjecture 1.37 to the case where the group need no longer be torsionfree. For the standard trace compare (7).

Conjecture 3.21 (Modified Trace Conjecture for a group G). Let G be a group. Then the image of the homomorphism induced by the standard trace

tr_{C_{r}^{∗}(G)} : K_{0}(C_{r}^{∗}(G)) → ℝ  (16)

is contained in Λ^{G}.

The following result is proved in [203, Theorem 0.3].

Theorem 3.22. Let G be a group. Then the image of the composition

K_{0}^{G}(E_{FIN} (G)) ⊗_{ℤ} Λ^{G} ⟶^{A_{FIN} ⊗_{ℤ}id} K_{0}(C_{r}^{∗}(G)) ⊗_{ℤ} Λ^{G} ⟶^{tr_{C^{∗}_{r}(G)}} ℝ

is Λ^{G}. Here A_{FIN} is the map appearing in the Baum-Connes Conjecture 2.3. In particular the Baum-Connes Conjecture 2.3 implies the Modified Trace Conjecture.

The original version of the Trace Conjecture due to Baum and Connes [27, page 21] makes the stronger statement that the image of tr_{C_{r}^{∗}(G)} : K_{0}(C_{r}^{∗}(G)) → ℝ is the additive subgroup of ℚ generated by all numbers \frac{1}{|H|}, where H ⊂ G runs though all finite subgroups of G. Roy has constructed a counterexample to this version in [274] based on her article [275]. The examples of Roy do not contradict the Modified Trace Conjecture 3.21 or the Baum-Connes Conjecture 2.3.

3.3.2 The Stable Gromov-Lawson-Rosenberg Conjecture

The Stable Gromov-Lawson-Rosenberg Conjecture is a typical conjecture relating Riemannian geometry to topology. It is concerned with the question when a given manifold admits a metric of positive scalar curvature. To discuss its relation with the Baum-Connes Conjecture we will need the real version of the Baum-Connes Conjecture, compare Subsection 4.1.1.

Let Ω_{n}^{Spin}(BG) be the bordism group of closed Spin-manifolds M of dimension n with a reference map to BG. Let C_{r}^{∗}(G; ℝ) be the real reduced group C^{∗}-algebra and let KO_{n}(C_{r}^{∗}(G; ℝ)) = K_{n}(C_{r}^{∗}(G; ℝ)) be its topological K-theory. We use KO instead of K as a reminder that we here use the real reduced group C^{∗}-algebra. Given an element [u: M → BG] ∈ Ω_{n}^{Spin}(BG), we can take the C_{r}^{∗}(G; ℝ)-valued index of the equivariant Dirac operator associated to the G-covering \overline{M} → M determined by u. Thus we get a homomorphism

ind_{C_{r}^{∗}(G;ℝ)} : Ω_{n}^{Spin}(BG) → KO_{n}(C_{r}^{∗}(G; ℝ)).  (17)

A Bott manifold is any simply connected closed Spin-manifold B of dimension 8 whose Â-genus Â(B) is 8. We fix such a choice, the particular choice does not matter for the sequel. Notice that ind_{C_{r}^{∗}({1};ℝ)}(B) ∈ KO_{8}(ℝ) ≅ ℤ is a generator and the product with this element induces the Bott periodicity isomorphisms KO_{n}(C_{r}^{∗}(G; ℝ)) ⟶^{≅} KO_{n+8}(C_{r}^{∗}(G; ℝ)). In particular

ind_{C_{r}^{∗}(G;ℝ)}(M) = ind_{C_{r}^{∗}(G;ℝ)}(M × B),  (18)

if we identify KO_{n}(C_{r}^{∗}(G; ℝ)) = KO_{n+8}(C_{r}^{∗}(G; ℝ)) via Bott periodicity.

Conjecture 3.23 (Stable Gromov-Lawson-Rosenberg Conjecture). Let M be a closed connected Spin-manifold of dimension n ≥ 5. Let u_{M}: M → Bπ_{1}(M) be the classifying map of its universal covering. Then M × B^{k} carries for some integer k ≥ 0 a Riemannian metric with positive scalar curvature if and only if

ind_{C_{r}^{∗}(π_{1}(M);ℝ)}([M, u_{M}]) = 0 ∈ KO_{n}(C_{r}^{∗}(π_{1}(M); ℝ)).

If M carries a Riemannian metric with positive scalar curvature, then the index of the Dirac operator must vanish by the Bochner-Lichnerowicz formula [267]. The converse statement that the vanishing of the index implies the existence of a Riemannian metric with positive scalar curvature is the hard part of the conjecture. The following result is due to Stolz. A sketch of the proof can be found in [297, Section 3], details are announced to appear in a different paper.

Theorem 3.24. If the assembly map for the real version of the Baum-Connes Conjecture (compare Subsection 4.1.1) is injective for the group G, then the Stable Gromov-Lawson-Rosenberg Conjecture 3.23 is true for all closed Spinmanifolds of dimension ≥ 5 with π_{1}(M) ≅ G.

The requirement dim(M) ≥ 5 is essential in the Stable Gromov-Lawson-Rosenberg Conjecture, since in dimension four new obstructions, the Seiberg-Witten invariants, occur. The unstable version of this conjecture says that M carries a Riemannian metric with positive scalar curvature if and only if ind_{C_{r}^{∗}(π_{1}(M);ℝ)}([M, u_{M}]) = 0. Schick [278] constructs counterexamples to the unstable version using minimal hypersurface methods due to Schoen and Yau (see also [91]). It is not known at the time of writing whether the unstable version is true for finite fundamental groups. Since the Baum-Connes Conjecture 2.3 is true for finite groups (for the trivial reason that E_{FIN} (G) = pt for finite groups G), Theorem 3.24 implies that the Stable Gromov-Lawson Conjecture 3.23 holds for finite fundamental groups (see also [270]).

The index map appearing in (17) can be factorized as a composition

ind_{C_{r}^{∗}(G;ℝ)} : Ω_{n}^{Spin}(BG) ⟶^{D} KO_{n}(BG) ⟶^{A} KO_{n}(C_{r}^{∗}(G; ℝ)),  (19)

where D sends [M, u] to the class of the G-equivariant Dirac operator of the G-manifold \overline{M} given by u and A = A_{\{1\}} is the real version of the classical assembly map. The homological Chern character defines an isomorphism

KO_{n}(BG) ⊗_{ℤ} ℚ ⟶^{≅} ⨁_{p∈ℤ} H_{n+4p}(BG; ℚ).

Recall that associated to M there is the Â-class

Â(M) ∈ ∏_{p≥0} H^{p}(M; ℚ)  (20)

which is a certain polynomial in the Pontrjagin classes. The map D appearing in (19) sends the class of u: M → BG to u_{∗}(Â(M) ∩ [M]), i.e. the image of the Poincaré dual of Â(M) under the map induced by u in rational homology. Hence D([M, u]) = 0 if and only if u_{∗}(Â(M) ∩ [M]) vanishes. For x ∈ ∏_{k≥0} H^{k}(BG; ℚ) define the higher Â-genus of (M, u) associated to x to be

Â_{x}(M, u) = ⟨Â(M) ∪ u^{∗}x, [M]⟩ = ⟨x, u_{∗}(Â(M) ∩ [M])⟩ ∈ ℚ.  (21)

The vanishing of Â(M) is equivalent to the vanishing of all higher Â-genera Â_{x}(M, u). The following conjecture is a weak version of the Stable Gromov-Lawson-Rosenberg Conjecture.

Conjecture 3.25 (Homological Gromov-Lawson-Rosenberg Conjecture). Let G be a group. Then for any closed Spin-manifold M, which admits a Riemannian metric with positive scalar curvature, the Â-genus Â_{x}(M, u) vanishes for all maps u: M → BG and elements x ∈ ∏_{k≥0} H^{k}(BG; ℚ).

From the discussion above we obtain the following result.

Proposition 3.26. If the assembly map

KO_{n}(BG) ⊗_{ℤ} ℚ → KO_{n}(C_{r}^{∗}(G; ℝ)) ⊗_{ℤ} ℚ

is injective for all n ∈ ℤ, then the Homological Gromov-Lawson-Rosenberg Conjecture holds for G.

4 Generalizations and Related Conjectures

4.1 Variants of the Baum-Connes Conjecture

4.1.1 The Real Version

There is an obvious real version of the Baum-Connes Conjecture, which predicts that for all n ∈ ℤ and groups G the assembly map

A_{FIN}^{ℝ} : H_{n}^{G}(E_{F}(G); K_{ℝ}^{top}) → KO_{n}(C_{r}^{∗}(G; ℝ))

is an isomorphism. Here H_{n}^{G}(−; K_{ℝ}^{top}) is an equivariant homology theory whose distinctive feature is that H_{n}^{G}(G/H; K_{ℝ}^{top}) ≅ KO_{n}(C_{r}^{∗}(H; ℝ)). Recall that we write KO_{n}(−) only to remind ourselves that the C^{∗}-algebra we apply it to is a real C^{∗}-algebra, like for example the real reduced group C^{∗}-algebra C_{r}^{∗}(G; ℝ). The following result appears in [31].

Proposition 4.1. The Baum-Connes Conjecture 2.3 implies the real version of the Baum-Connes Conjecture.

In the proof of Proposition 3.19 we have already seen that after inverting 2 the “real assembly map” is a retract of the complex assembly map. In particular with 2-inverted or after rationalizing also injectivity results or surjectivity results about the complex Baum-Connes assembly map yield the corresponding results for the real Baum-Connes assembly map.

4.1.2 The Version for Maximal Group C^{∗}-Algebras

For a group G let C_{max}^{∗}(G) be its maximal group C^{∗}-algebra, compare [242, 7.1.5 on page 229]. The maximal group C^{∗}-algebra has the advantage that every homomorphism of groups ϕ: G → H induces a homomorphism C_{max}^{∗}(G) → C_{max}^{∗}(H) of C^{∗}-algebras. This is not true for the reduced group C^{∗}-algebra C_{r}^{∗}(G). Here is a counterexample: since C_{r}^{∗}(F) is a simple algebra if F is a non-abelian free group [245], there is no unital algebra homomorphism C_{r}^{∗}(F) → C_{r}^{∗}({1}) = ℂ.

One can construct a version of the Baum-Connes assembly map using an equivariant homology theory H_{n}^{G}(−; K_{max}^{top}) which evaluated on G/H yields the K-theory of C_{max}^{∗}(H) (use Proposition 6.7 and a suitable modification of K^{top}, compare Section 6.3).

Since on the left hand side of a FIN -assembly map only the maximal group C^{∗}-algebras for finite groups H matter, and clearly C_{max}^{∗}(H) = ℂH = C_{r}^{∗}(H) for such H, this left hand side coincides with the left hand side of the usual Baum-Connes Conjecture. There is always a C^{∗}-homomorphism p: C_{max}^{∗}(G) → C_{r}^{∗}(G) (it is an isomorphism if and only if G is amenable [242, Theorem 7.3.9 on page 243]) and hence we obtain the following factorization of the usual Baum-Connes assembly map

\xymatrix{ & K_{n}(C_{max}^{∗}(G)) \ar[d]^{K_{n}(p)} \\ H_{n}^{G}(E_{FIN} (G); K^{top}) \ar[r]_{A_{FIN}} \ar@{-->}[ur]^{A_{FIN}^{max}} & K_{n}(C_{r}^{∗}(G))}  (22)

It is known that the map A_{FIN}^{max} is in general not surjective. The Baum-Connes Conjecture would imply that the map is A_{FIN}^{max} is always injective, and that it is surjective if and only if the vertical map K_{n}(p) is injective.

A countable group G is called K-amenable if the map p: C_{max}^{∗}(G) → C_{r}^{∗}(G) induces a KK-equivalence (compare [78]). This implies in particular that the vertical map K_{n}(p) is an isomorphism for all n ∈ ℤ. Note that for K-amenable groups the Baum-Connes Conjecture holds if and only if the “maximal” version of the assembly map A_{FIN}^{max} is an isomorphism for all n ∈ ℤ. A-T-menable groups are K-amenable, compare Theorem 5.1. But K_{0}(p) is not injective for every infinite group which has property (T) such as for example SL_{n}(ℤ) for n ≥ 3, compare for instance the discussion in [163]. There are groups with property (T) for which the Baum-Connes Conjecture is known (compare Subsection 5.1.2) and hence there are counterexamples to the conjecture that A_{FIN}^{max} is an isomorphism.

In Theorem 1.48 and Remark 1.49 we discussed applications of the maximal C^{∗}-algebra version of the Baum-Connes Conjecture.

4.1.3 The Bost Conjecture

Some of the strongest results about the Baum-Connes Conjecture are proven using the so called Bost Conjecture (see [186]). The Bost Conjecture is the version of the Baum-Connes Conjecture, where one replaces the reduced group C^{∗}-algebra C_{r}^{∗}(G) by the Banach algebra l^{1}(G) of absolutely summable functions on G. Again one can use the spectra approach (compare Subsection 6.2 and 6.3 and in particular Proposition 6.7) to produce a variant of equivariant K-homology denoted H_{n}^{G}(−; K_{l^{1}}^{top}) which this time evaluated on G/H yields K_{n}(l^{1}(H)), the topological K-theory of the Banach algebra l^{1}(H).

As explained in the beginning of Chapter 2, we obtain an associated assembly map and we believe that it coincides with the one defined using a Banach-algebra version of KK-theory in [186].

Conjecture 4.2 (Bost Conjecture). Let G be a countable group. Then the assembly map

A_{FIN}^{l^{1}} : H_{n}^{G}(E_{FIN} (G); K_{l^{1}}^{top}) → K_{n}(l^{1}(G))

is an isomorphism.

Again the left hand side coincides with the left hand side of the Baum-Connes assembly map because for finite groups H we have l^{1}(H) = ℂH = C_{r}^{∗}(H). There is always a homomorphism of Banach algebras q: l^{1}(G) → C_{r}^{∗}(G) and one obtains a factorization of the usual Baum-Connes assembly map

\xymatrix{ & K_{n}(l^{1}(G)) \ar[d]^{K_{n}(q)} \\ H_{n}^{G}(E_{FIN} (G); K^{top}) \ar[ur]^{A_{FIN}^{l^{1}}} \ar[r]^{A_{FIN}} & K_{n}(C_{r}^{∗}(G)).}

Every group homomorphism G → H induces a homomorphism of Banach algebras l^{1}(G) → l^{1}(H). So similar as in the maximal group C^{∗}-algebra case this approach repairs the lack of functoriality for the reduced group C^{∗}-algebra.

The disadvantage of l^{1}(G) is however that indices of operators tend to take values in the topological K-theory of the group C^{∗}-algebras, not in K_{n}(l^{1}(G)). Moreover the representation theory of G is closely related to the group C^{∗}- algebra, whereas the relation to l^{1}(G) is not well understood.

For more information about the Bost Conjecture 4.2 see [186], [288].

4.1.4 The Baum-Connes Conjecture with Coefficients

The Baum-Connes Conjecture 2.3 can be generalized to the Baum-Connes Conjecture with Coefficients. Let A be a separable C^{∗}-algebra with an action of the countable group G. Then there is an assembly map

KK_{n}^{G}(E_{FIN} (G); A) → K_{n}(A ⋊ G)  (23)

defined in terms of equivariant KK-theory, compare Sections 7.3 and 7.4.

Conjecture 4.3 (Baum-Connes Conjecture with Coefficients). For every separable C^{∗}-algebra A with an action of a countable group G and every n ∈ ℤ the assembly map (23) is an isomorphism.

There are counterexamples to the Baum-Connes Conjecture with Coefficients, compare Remark 5.3. If we take A = ℂ with the trivial action, the map (23) can be identified with the assembly map appearing in the ordinary Baum-Connes Conjecture 2.3.

Remark 4.4 (A Spectrum Level Description). There is a formulation of the Baum-Connes Conjecture with Coefficients in the framework explained in Section 6.2. Namely, construct an appropriate covariant functor K^{top}(A ⋊ G^{G}(−)): Or(G) → SPECTRAsuch that

π_{n}(K^{top}(A ⋊ G^{G}(G/H)) ≅ K_{n}(A ⋊ H)

holds for all subgroups H ⊆ G and all n ∈ ℤ, and consider the associated G-homology theory H_{∗}^{G}(−; K^{top}(A ⋊ G^{G}(−))). Then the map (23) can be identified with the map which the projection pr: E_{FIN} (G) → pt induces for this homology theory.

Remark 4.5 (Farrell-Jones Conjectures with Coefficients). One can also formulate a “Farrell-Jones Conjecture with Coefficients”. (This should not be confused with the Fibered Farrell-Jones Conjecture discussed in Subsection 4.2.2.) Fix a ring S and an action of G on it by isomorphisms of rings. Construct an appropriate covariant functor K(S ⋊ G^{G}(−)): Or(G) → SPECTRA such that

π_{n}(K(S ⋊ G^{G}(G/H))) ≅ K_{n}(S ⋊ H)

holds for all subgroups H ⊆ G and n ∈ ℤ, where S ⋊ H is the associated twisted group ring. Now consider the associated G-homology theory H_{∗}^{G}(−; K(S ⋊ G^{G}(−))). There is an analogous construction for L-theory. A Farrell-Jones Conjecture with Coefficients would say that the map induced on these homology theories by the projection pr: E_{VCY}(G) → pt is always an isomorphism. We do not know whether there are counterexamples to the Farrell-Jones Conjectures with Coefficients, compare Remark 5.3.

4.1.5 The Coarse Baum Connes Conjecture

We briefly explain the Coarse Baum-Connes Conjecture, a variant of the Baum-Connes Conjecture, which applies to metric spaces. Its importance lies in the fact that isomorphism results about the Coarse Baum-Connes Conjecture can be used to prove injectivity results about the classical assembly map for topological K-theory. Compare also Section 7.10.

Let X be a proper (closed balls are compact) metric space and H_{X} a separable Hilbert space with a faithful nondegenerate ∗-representation of C_{0}(X), the algebra of complex valued continuous functions which vanish at infinity.

A bounded linear operator T has a support supp T ⊂ X × X, which is defined as the complement of the set of all pairs (x, x′), for which there exist functions ϕ and ϕ′ ∈ C_{0}(X) such that ϕ(x) ≠ 0, ϕ′(x′) ≠ 0 and ϕ′T ϕ = 0. The operator T is said to be a finite propagation operator if there exists a constant α such that d(x, x′) ≤ α for all pairs in the support of T. The operator is said to be locally compact if ϕT and T ϕ are compact for every ϕ ∈ C_{0}(X). An operator is called pseudolocal if ϕT ψ is a compact operator for all pairs of continuous functions ϕ and ψ with compact and disjoint supports.

The Roe-algebra C^{∗}(X) = C(X, H_{X}) is the operator-norm closure of the ∗-algebra of all locally compact finite propagation operators on H_{X}. The algebra D^{∗}(X) = D^{∗}(X, H_{X}) is the operator-norm closure of the pseudolocal finite propagation operators. One can show that the topological K-theory of the quotient algebra D^{∗}(X)/C^{∗}(X) coincides up to an index shift with the analytically defined (non-equivariant) K-homology K_{∗}(X), compare Section 7.1. For a uniformly contractible proper metric space the coarse assembly map K_{n}(X) → K_{n}(C^{∗}(X)) is the boundary map in the long exact sequence associated to the short exact sequence of C^{∗}-algebras

0 → C^{∗}(X) → D^{∗}(X) → D^{∗}(X)/C^{∗}(X) → 0.

For general metric spaces one first approximates the metric space by spaces with nice local behaviour, compare [263]. For simplicity we only explain the case, where X is a discrete metric space. Let P_{d}(X) the Rips complex for a fixed distance d, i.e. the simplicial complex with vertex set X, where a simplex is spanned by every collection of points in which every two points are a distance less than d apart. Equip P_{d}(X) with the spherical metric, compare [335].

A discrete metric space has bounded geometry if for each r > 0 there exists a N(r) such that for all x the ball of radius r centered at x ∈ X contains at most N(r) elements.

Conjecture 4.6 (Coarse Baum-Connes Conjecture). Let X be a proper discrete metric space of bounded geometry. Then for n = 0, 1 the coarse assembly map

colim_{d} K_{n}(P_{d}(X)) → colim_{d} K_{n}(C^{∗}(P_{d}(X))) ≅ K_{n}(C^{∗}(X))

is an isomorphism.

The conjecture is false if one drops the bounded geometry hypothesis. A counterexample can be found in [336, Section 8]. Our interest in the conjecture stems from the following fact, compare [263, Chapter 8].

Proposition 4.7. Suppose the finitely generated group G admits a classifying space BG of finite type. If G considered as a metric space via a word length metric satisfies the Coarse Baum-Connes Conjecture 4.6 then the classical assembly map A: K_{∗}(BG) → K_{∗}(C_{r}^{∗}G) which appears in Conjecture 1.31 is injective.

The Coarse Baum-Connes Conjecture for a discrete group G (considered as a metric space) can be interpreted as a case of the Baum-Connes Conjecture with Coefficients 4.3 for the group G with a certain specific choice of coefficients, compare [339].

Further information about the coarse Baum-Connes Conjecture can be found for instance in [151], [152], [154], [263], [334], [340], [335], [337], and [338].

4.1.6 The Baum-Connes Conjecture for Non-Discrete Groups

Throughout this subsection let T be a locally compact second countable topological Hausdorff group. There is a notion of a classifying space for proper T - actions \underline{E}T (see [28, Section 1 and 2] [304, Section I.6], [207, Section 1]) and one can define its equivariant topological K-theory K_{n}^{T} (\underline{E}T). The definition of a reduced C^{∗}-algebra C_{r}^{∗}(T) and its topological K-theory K_{n}(C_{r}^{∗}(T)) makes sense also for T. There is an assembly map defined in terms of equivariant index theory

A_{K}: K_{n}^{T} (\underline{E}T) → K_{n}(C_{r}^{∗}(T)).  (24)

The Baum-Connes Conjecture for T says that this map is bijective for all n ∈ ℤ [28, Conjecture 3.15 on page 254].

Now consider the special case where T is a connected Lie group. Let K be the family of compact subgroups of T. There is a notion of a T -CW - complex and of a classifying space E_{K}(T) defined as in Subsection 2.1.1 and 2.1.3. The classifying space E_{K}(T) yields a model for \underline{E}T. Let K ⊂ T be a maximal compact subgroup. It is unique up to conjugation. The space T/K is contractible and in fact a model for \underline{E}T (see [1, Appendix, Theorem A.5], [2, Corollary 4.14], [207, Section 1]). One knows (see [28, Proposition 4.22], [170])

K_{n}^{T} (\underline{E}T) = K_{n}^{T} (T/K) = \{ \begin{matrix}R_{ℂ}(K) & n = dim(T/K) mod 2, \\ 0 & n = 1 + dim(T/K) mod 2, \end{matrix}

where R_{ℂ}(K) is the complex representation ring of K.

Next we consider the special case where T is a totally disconnected group. Let KO be the family of compact-open subgroups of T. A T -CW -complex and a classifying space E_{KO}(T) for T and KO are defined similar as in Subsection 2.1.1 and 2.1.3. Then E_{KO}(T) is a model for \underline{E}T since any compact subgroup is contained in a compact-open subgroup, and the Baum-Connes Conjecture says that the assembly map yields for n ∈ ℤ an isomorphism

A_{KO}: K_{n}^{T} (E_{KO}(T)) → K_{n}(C_{r}^{∗}(T)).  (25)

For more information see [30].

4.2 Variants of the Farrell-Jones Conjecture

4.2.1 Pseudoisotopy Theory

An important variant of the Farrell-Jones Conjecture deals with the pseudoisotopy spectrum functor P, which we already discussed briefly in Section 1.4.2. In fact it is this variant of the Farrell-Jones Conjecture (and its fibered version which will be explained in the next subsection) for which the strongest results are known at the time of writing.

In Proposition 6.8 we will explain that every functor E: GROUPOIDS→ SPECTRA , which sends equivalences of groupoids to stable weak equivalences of spectra, yields a corresponding equivariant homology theory H_{n}^{G}(−; E). Now whenever we have a functor F: SPACES→ SPECTRA , we can precompose it with the functor “classifying space” which sends a groupoid G to its classifying space BG. (Here BG is simply the realization of the nerve of G considered as a category.) In particular this applies to the pseudoisotopy functor P. Thus we obtain a homology theory H_{n}^{G}(−; P ∘ B) whose essential feature is that

H_{n}^{G}(G/H; P ∘ B) ≅ π_{n}(P(BH)),

i.e. evaluated at G/H one obtains the homotopy groups of the pseudoisotopy spectrum of the classifying space BH of the group H. As the reader may guess there is the following conjecture.

Conjecture 4.8 (Farrell-Jones Conjecture for Pseudoisotopies of Aspherical Spaces). For every group G and all n ∈ ℤ the assembly map

H_{n}^{G}(E_{VCY}(G); P ∘ B) → H_{n}^{G}(pt; P ∘ B) ≅ π_{n}(P(BG))

is an isomorphism. Similarly for P^{diff}, the pseudoisotopy functor which is defined using differentiable pseudoisotopies.

A formulation of a conjecture for spaces which are not necessarily aspherical will be given in the next subsection, see in particular Remark 4.13.

Remark 4.9 (Relating K-Theory and Pseudoisotopy Theory). We already outlined in Subsection 1.4.1 the relationship between K-theory and pseudoisotopies. The comparison in positive dimensions described there can be extended to all dimensions. Vogell constructs in [309] a version of A-theory using retractive spaces that are bounded over ℝ^{k} (compare Subsection 1.2.3 and 1.4.2). This leads to a functor A^{−∞} from spaces to non-connective spectra. Compare also [56], [310], [311] and [326]. We define Wh_{P L}^{−∞} via the fibration sequence

X_{+}∧A^{−∞}(pt) → A^{−∞}(X) → Wh_{P L}^{−∞}(X),

where the first map is the assembly map. The natural equivalence

Ω^{2}Wh_{P L}^{−∞}(X) ≃ P(X)

seems to be hard to trace down in the literature but should be true. We will assume it in the following discussion.

Precompose the functors above with the classifying space functor B to obtain functors from groupoids to spectra. The pseudoisotopy assembly map which appears in Conjecture 4.8 is an isomorphism if and only if the A-theory assembly map

H_{n+2}^{G}(E_{VCY}(G); A^{−∞} ∘ B) → H_{n+2}^{G}(pt; A^{−∞} ∘ B) ≅ π_{n+2}(A^{−∞}(BG))

is an isomorphism. This uses a 5-lemma argument and the fact that for a fixed spectrum E the assembly map

H_{n}^{G}(E_{F}(G); BG^{G}(−)_{+}∧E) → H_{n}^{G}(pt; BG^{G}(−)_{+}∧E)

is always bijective. There is a linearization map A^{−∞}(X) → K(ℤΠ(X)_{⊕}) (see the next subsection for the notation) which is always 2-connected and a rational equivalence if X is aspherical (recall that K denotes the non-connective K-theory spectrum). For finer statements about the linearization map, compare also [230].

The above discussion yields in particular the following, compare [111, 1.6.7 on page 261].

Proposition 4.10. The rational version of the K-theoretic Farrell-Jones Conjecture 2.2 is equivalent to the rational version of the Farrell-Jones Conjecture for Pseudoisotopies of Aspherical Spaces 4.8. If the assembly map in the conjecture for pseudoisotopies is (integrally) an isomorphism for n ≤ −1, then so is the assembly map in the K-theoretic Farrell-Jones Conjecture for n ≤ 1.

4.2.2 Fibered Versions

Next we present the more general fibered versions of the Farrell-Jones Conjectures. These fibered versions have better inheritance properties, compare Section 5.4.

In the previous section we considered functors F: SPACES→ SPECTRA , like P, P^{diff} and A^{−∞}, and the associated equivariant homology theories H_{n}^{G}(−; F∘B) (compare Proposition 6.8). Here B denotes the classifying space functor, which sends a groupoid G to its classifying space BG. In fact all equivariant homology theories we considered so far can be obtained in this fashion for special choices of F. Namely, let F be one of the functors

K(RΠ(−)_{⊕}), L^{⟨−∞⟩}(RΠ(−)_{⊕}) or K^{top}(C_{r}^{∗}Π(−)_{⊕}),

where Π(X) denotes the fundamental groupoid of a space, RG_{⊕} respectively C_{r}^{∗}G_{⊕} is the R-linear respectively the C^{∗}-category associated to a groupoid G and K, L^{⟨−∞⟩} and K^{top} are suitable functors which send additive respectively C^{∗}-categories to spectra, compare the proof of Theorem 6.9. There is a natural equivalence G → ΠBG. Hence, if we precompose the functors above with the classifying space functor B we obtain functors which are equivalent to the functors we have so far been calling

K_{R}, L_{R}^{⟨−∞⟩} and K^{top},

compare Theorem 6.9. Note that in contrast to these three cases the pseudoisotopy functor P depends on more than just the fundamental groupoid. However Conjecture 4.8 above only deals with aspherical spaces.

Given a G-CW-complex Z and a functor F from spaces to spectra we obtain a functor X ↦ F(Z ×_{G} X) which digests G-CW-complexes. In particular we can restrict it to the orbit category to obtain a functor

F(Z ×_{G} −): Or(G) → SPECTRA .

According to Proposition 6.7 we obtain a corresponding G-homology theory

H_{n}^{G}(−; F(Z ×_{G} −))

and associated assembly maps. Note that restricted to the orbit category the functor EG ×_{G} − is equivalent to the classifying space functor B and so H_{n}^{G}(−; F ∘ B) can be considered as a special case of this construction.

Conjecture 4.11 (Fibered Farrell-Jones Conjectures). Let R be a ring (with involution). Let F: SPACES→ SPECTRAbe one of the functors

K(RΠ(−)_{⊕}), L^{⟨−∞⟩}(RΠ(−)_{⊕}), P(−), P^{diff}(−) or A^{−∞}(−).

Then for every free G-CW-complex Z and all n ∈ ℤ the associated assembly map

H_{n}^{G}(E_{VCY}(G); F(Z ×_{G} −)) → H_{n}^{G}(pt; F(Z ×_{G} −)) ≅ π_{n}(F(Z/G))

is an isomorphism.

Remark 4.12 (A Fibered Baum-Connes Conjecture). With the family FIN instead of VCY and the functor F = K^{top}(C_{r}^{∗}Π(−)_{⊕}) one obtains a Fibered Baum-Connes Conjecture.

Remark 4.13 (The Special Case Z = X̃). Suppose Z = X̃ is the universal covering of a space X equipped with the action of its fundamental group G = π_{1}(X). Then in the algebraic K- and L-theory case the conjecture above specializes to the “ordinary” Farrell-Jones Conjecture 2.2. In the pseudoisotopy and A-theory case one obtains a formulation of an (unfibered) conjecture about π_{n}(P(X)) or π_{n}(A^{−∞}(X)) for spaces X which are not necessarily aspherical.

Remark 4.14 (Relation to the Original Formulation). In [111] Farrell and Jones formulate a fibered version of their conjectures for every (Serre) fibration Y → X over a connected CW-complex X. In our set-up this corresponds to choosing Z to be the total space of the fibration obtained from Y → X by pulling back along the universal covering projection X̃ → X. This space is a free G-space for G = π_{1}(X). Note that an arbitrary free G-CW - complex Z can always be obtained in this fashion from a map Z/G → BG, compare [111, Corollary 2.2.1 on page 264].

Remark 4.15 (Relating K-Theory and Pseudoisotopy Theory in the Fibered Case). The linearization map π_{n}(A^{−∞}(X)) → K_{n}(ℤΠ(X)) is always 2-connected, but for spaces which are not aspherical it need not be a rational equivalence. Hence the comparison results discussed in Remark 4.9 apply for the fibered versions only in dimensions n ≤ 1.

4.2.3 The Isomorphism Conjecture for NK-groups

In Remark 1.15 we defined the groups NK_{n}(R) for a ring R. They are the simplest kind of Nil-groups responsible for the infinite cyclic group. Since the functor K_{R} is natural with respect to ring homomorphism we can define NK_{R} as the (objectwise) homotopy cofiber of K_{R} → K_{R[t]}. There is an associated assembly map.

Conjecture 4.16 (Isomorphism Conjecture for NK-groups). The assembly map

H_{n}^{G}(E_{VCY}(G); NK_{R}) → H_{n}^{G}(pt; NK_{R}) ≅ NK_{n}(RG)

is always an isomorphism.

There is a weak equivalence K_{R[t]} ≃ K_{R} ∨ NK_{R} of functors from GROUPOIDS to SPECTRA . This implies for a fixed family F of subgroups of G and n ∈ ℤ that whenever two of the three assembly maps

A_{F}: H_{n}^{G}(E_{F}(G); K_{R[t]}) → K_{n}(R[t][G]),
A_{F}: H_{n}^{G}(E_{F}(G); K_{R}) → K_{n}(R[G]),
A_{F}: H_{n}^{G}(E_{F}(G); NK_{R}) → NK_{n}(RG)

are bijective, then so is the third (compare [19, Section 7]). Similarly one can define a functor E_{R} from the category GROUPOIDS to SPECTRAand weak equivalences

K_{R[t,t^{−1}]} → E_{R} ⟵ K_{R} ∨ ΣK_{R} ∨ NK_{R} ∨ NK_{R},

which on homotopy groups corresponds to the Bass-Heller-Swan decomposition (see Remark 1.15). One obtains a two-out-of-three statement as above with the K_{R[t]}-assembly map replaced by the K_{R[t,t^{−1}]}-assembly map.

4.2.4 Algebraic K-Theory of the Hecke Algebra

In Subsection 4.1.6 we mentioned the classifying space E_{KO}(G) for the family of compact-open subgroups and the Baum-Connes Conjecture for a totally disconnected group T. There is an analogous conjecture dealing with the algebraic K-theory of the Hecke algebra.

Let H(T) denote the Hecke algebra of T which consists of locally constant functions G → ℂ with compact support and inherits its multiplicative structure from the convolution product. The Hecke algebra H(T) plays the same role for T as the complex group ring ℂG for a discrete group G and reduces to this notion if T happens to be discrete. There is a T -homology theory H_{∗}^{T} with the property that for any open and closed subgroup H ⊆ T and all n ∈ ℤ we have H_{n}^{T} (T/H) = K_{n}(H(H)), where K_{n}(H(H)) is the algebraic K-group of the Hecke algebra H(H).

Conjecture 4.17 (Isomorphism Conjecture for the Hecke-Algebra). For a totally disconnected group T the assembly map

A_{KO}: H_{n}^{T} (E_{KO}(T)) → H^{T} (pt) = K_{n}(H(T))  (26)

induced by the projection pr: E_{KO}(T) → pt is an isomorphism for all n ∈ ℤ.

In the case n = 0 this reduces to the statement that

colim_{T/H∈Or _{KO}(T)} K_{0}(H(H)) → K_{0}(H(T)).  (27)

is an isomorphism. For n ≤ −1 one obtains the statement that K_{n}(H(G)) = 0. The group K_{0}(H(T)) has an interpretation in terms of the smooth representations of T. The G-homology theory can be constructed using an appropriate functor E: Or_{KO}(T) → SPECTRAand the recipe explained in Section 6.2. The desired functor E is given in [276].

5 Status of the Conjectures

In this section, we give the status, at the time of writing, of some of the conjectures mentioned earlier. We begin with the Baum-Connes Conjecture.

5.1 Status of the Baum-Connes-Conjecture

5.1.1 The Baum-Connes Conjecture with Coefficients

We begin with the Baum-Connes Conjecture with Coefficients 4.3. It has better inheritance properties than the Baum-Connes Conjecture 2.3 itself and contains it as a special case.

Theorem 5.1. (Baum-Connes Conjecture with Coefficients and a-T-menable Groups). The discrete group G satisfies the Baum-Connes Conjecture with Coefficients 4.3 and is K-amenable provided that G is a-T-menable.

This theorem is proved in Higson-Kasparov [149, Theorem 1.1], where more generally second countable locally compact topological groups are treated (see also [164]).

A group G is a-T-menable, or, equivalently, has the Haagerup property if G admits a metrically proper isometric action on some affine Hilbert space. Metrically proper means that for any bounded subset B the set {g ∈ G | gB ∩ B ≠ ∅} is finite. An extensive treatment of such groups is presented in [66]. Any a-T-menable group is countable. The class of a-T-menable groups is closed under taking subgroups, under extensions with finite quotients and under finite products. It is not closed under semi-direct products. Examples of a-T-menable groups are countable amenable groups, countable free groups, discrete subgroups of SO(n, 1) and SU(n, 1), Coxeter groups, countable groups acting properly on trees, products of trees, or simply connected CAT(0) cubical complexes. A group G has Kazhdan’s property (T) if, whenever it acts isometrically on some affine Hilbert space, it has a fixed point. An infinite a-T-menable group does not have property (T). Since SL(n, ℤ) for n ≥ 3 has property (T), it cannot be a-T-menable.

Using the Higson-Kasparov result Theorem 5.1 and known inheritance properties of the Baum-Connes Conjecture with Coefficients (compare Section 5.4 and [233],[234]) Mislin describes an even larger class of groups for which the conjecture is known [225, Theorem 5.23].

Theorem 5.2 (The Baum-Connes Conjecture with Coefficients and the Class of Groups LHET H). The discrete group G satisfies the Baum-Connes Conjecture with Coefficients 4.3 provided that G belongs to the class LHET H.

The class LHET H is defined as follows. Let HT H be the smallest class of groups which contains all a-T-menable groups and contains a group G if there is a 1-dimensional contractible G-CW -complex whose stabilizers belong already to HT H. Let HET H be the smallest class of groups containing HT H and containing a group G if either G is countable and admits a surjective map p: G → Q with Q and p^{−1}(F) in HET H for every finite subgroup F ⊆ Q or if G admits a 1-dimensional contractible G-CW -complex whose stabilizers belong already to HET H. Let LHET H be the class of groups G whose finitely generated subgroups belong to HET H.

The class LHET H is closed under passing to subgroups, under extensions with torsion free quotients and under finite products. It contains in particular one-relator groups and Haken 3-manifold groups (and hence all knot groups). All these facts of the class LHET H and more information can be found in [225].

Vincent Lafforgue has an unpublished proof of the Baum-Connes Conjecture with Coefficients 4.3 for word-hyperbolic groups.

Remark 5.3. There are counterexamples to the Baum-Connes Conjecture with (commutative) Coefficients 4.3 as soon as the existence of finitely generated groups containing arbitrary large expanders in their Cayley graph is shown [150, Section 7]. The existence of such groups has been claimed by Gromov [138], [139]. Details of the construction are described by Ghys in [134]. At the time of writing no counterexample to the Baum-Connes Conjecture 2.3 (without coefficients) is known to the authors.

5.1.2 The Baum-Connes Conjecture

Next we deal with the Baum-Connes Conjecture 2.3 itself. Recall that all groups which satisfy the Baum-Connes Conjecture with Coefficients 4.3 do in particular satisfy the Baum-Connes Conjecture 2.3.

Theorem 5.4 (Status of the Baum-Connes Conjecture). A group G satisfies the Baum-Connes Conjecture 2.3 if it satisfies one of the following conditions.

(i) It is a discrete subgroup of a connected Lie groups L, whose Levi-Malcev decomposition L = RS into the radical R and semisimple part S is such that S is locally of the form

S = K × SO(n_{1}, 1) × … × SO(n_{k}, 1) × SU(m_{1}, 1) × … × SU(m_{l}, 1)

for a compact group K. (ii) The group G has property (RD) and admits a proper isometric action on

a strongly bolic weakly geodesic uniformly locally finite metric space. (iii) G is a subgroup of a word hyperbolic group. (iv) G is a discrete subgroup of Sp(n, 1).

Proof. The proof under condition (i) is due to Julg-Kasparov [166]. The proof under condition (ii) is due to Lafforgue [183] (see also [288]). Word hyperbolic groups have property (RD) [84]. Any subgroup of a word hyperbolic group satisfies the conditions appearing in the result of Lafforgue and hence satisfies the Baum-Connes Conjecture 2.3 [222, Theorem 20]. The proof under condition (iv) is due to Julg [165]. ⊓ ⊔

Lafforgue’s result about groups satisfying condition (ii) yielded the first examples of infinite groups which have Kazhdan’s property (T) and satisfy the Baum-Connes Conjecture 2.3. Here are some explanations about condition (ii).

A length function on G is a function L: G → ℝ_{≥0} such that L(1) = 0, L(g) = L(g^{−1}) for g ∈ G and L(g_{1}g_{2}) ≤ L(g_{1}) + L(g_{2}) for g_{1}, g_{2} ∈ G holds. The word length metric L_{S} associated to a finite set S of generators is an example. A length function L on G has property (RD) (“rapid decay”) if there exist C, s > 0 such that for any u = ∑_{g∈G} λ_{g} · g ∈ ℂG we have

||ρ_{G}(u)||_{∞} ≤ C · (∑_{g∈G} |λ_{g}|^{2} · (1 + L(g))^{2s})^{1/2} ,

where ||ρ_{G}(u)||_{∞} is the operator norm of the bounded G-equivariant operator l^{2}(G) → l^{2}(G) coming from right multiplication with u. A group G has property (RD) if there is a length function which has property (RD). More information about property (RD) can be found for instance in [63], [184] and [307, Chapter 8]. Bolicity generalizes Gromov’s notion of hyperbolicity for metric spaces. We refer to [169] for a precise definition.

Remark 5.5. We do not know whether all groups appearing in Theorem 5.4 satisfy also the Baum-Connes Conjecture with Coefficients 4.3.

Remark 5.6. It is not known at the time of writing whether the Baum-Connes Conjecture is true for SL(n, ℤ) for n ≥ 3.

Remark 5.7 (The Status for Topological Groups). We only dealt with the Baum-Connes Conjecture for discrete groups. We already mentioned that Higson-Kasparov [149] treat second countable locally compact topological groups. The Baum-Connes Conjecture for second countable almost connected groups G has been proven by Chabert-Echterhoff-Nest [60] based on the work of Higson-Kasparov [149] and Lafforgue [186]. The Baum-Connes Conjecture with Coefficients 4.3 has been proved for the connected Lie groups L appearing in Theorem 5.4 (i) by [166] and for Sp(n, 1) by Julg [165].

5.1.3 The Injectivity Part of the Baum-Connes Conjecture

In this subsection we deal with injectivity results about the assembly map appearing in the Baum-Connes Conjecture 2.3. Recall that rational injectivity already implies the Novikov Conjecture 1.52 (see Proposition 3.16) and the Homological Stable Gromov-Lawson-Rosenberg Conjecture 3.25 (see Proposition 3.26 and 2.20).

Theorem 5.8 (Rational Injectivity of the Baum-Connes Assembly Map). The assembly map appearing in the Baum-Connes Conjecture 2.3 is rationally injective if G belongs to one of the classes of groups below.

(i) Groups acting properly isometrically on complete manifolds with nonpositive sectional curvature.

(ii) Discrete subgroups of Lie groups with finitely many path components. (iii) Discrete subgroups of p-adic groups.

Proof. The proof of assertions (i) and (ii) is due to Kasparov [171], the one of assertion (iii) to Kasparov-Skandalis [172]. ⊓ ⊔

A metric space (X, d) admits a uniform embedding into Hilbert space if there exist a separable Hilbert space H, a map f: X → H and non-decreasing functions ρ_{1} and ρ_{2} from [0, ∞) → ℝ such that ρ_{1}(d(x, y)) ≤ ||f(x) − f(y)|| ≤ ρ_{2}(d(x, y)) for x, y ∈ X and lim_{r→∞} ρ_{i}(r) = ∞ for i = 1, 2. A metric is proper if for each r > 0 and x ∈ X the closed ball of radius r centered at x is compact. The question whether a discrete group G equipped with a proper left G-invariant metric d admits a uniform embedding into Hilbert space is independent of the choice of d, since the induced coarse structure does not depend on d [289, page 808]. For more information about groups admitting a uniform embedding into Hilbert space we refer to [87], [140].

The class of finitely generated groups, which embed uniformly into Hilbert space, contains a subclass A, which contains all word hyperbolic groups, finitely generated discrete subgroups of connected Lie groups and finitely generated amenable groups and is closed under semi-direct products [338, Definition 2.1, Theorem 2.2 and Proposition 2.6]. Gromov [138], [139] has announced examples of finitely generated groups which do not admit a uniform embedding into Hilbert space. Details of the construction are described in Ghys [134].

The next theorem is proved by Skandalis-Tu-Yu [289, Theorem 6.1] using ideas of Higson [148].

Theorem 5.9 (Injectivity of the Baum-Connes Assembly Map). Let G be a countable group. Suppose that G admits a G-invariant metric for which G admits a uniform embedding into Hilbert space. Then the assembly map appearing in the Baum-Connes Conjecture with Coefficients 4.3 is injective.

We now discuss conditions which can be used to verify the assumption in Theorem 5.9.

Remark 5.10 (Linear Groups). A group G is called linear if it is a subgroup of GL_{n}(F) for some n and some field F. Guentner-Higson-Weinberger [140] show that every countable linear group admits a uniform embedding into Hilbert space and hence Theorem 5.9 applies.

Remark 5.11 (Groups Acting Amenably on a Compact Space). A continuous action of a discrete group G on a compact space X is called amenable if there exists a sequence

p_{n}: X → M^{1}(G) = {f: G → [0, 1] | ∑_{g∈G} f(g) = 1}

of weak-∗-continuous maps such that for each g ∈ G one has

lim_{n→∞} sup_{x∈X} ||g ∗ (p_{n}(x) − p_{n}(g · x))||_{1} = 0.

Note that a group G is amenable if and only if its action on the one-point-space is amenable. More information about this notion can be found for instance in [5], [6].

Higson-Roe [153, Theorem 1.1 and Proposition 2.3] show that a finitely generated group equipped with its word length metric admits an amenable action on a compact metric space, if and only if it belongs to the class A defined in [338, Definition 2.1], and hence admits a uniform embedding into Hilbert space. Hence Theorem 5.9 implies the result of Higson [148, Theorem 1.1] that the assembly map appearing in the Baum-Connes Conjecture with Coefficients 4.3 is injective if G admits an amenable action on some compact space.

Word hyperbolic groups and the class of groups mentioned in Theorem 5.8 (ii) fall under the class of groups admitting an amenable action on some compact space [153, Section 4].

Remark 5.12. Higson [148, Theorem 5.2] shows that the assembly map appearing in the Baum-Connes Conjecture with Coefficients 4.3 is injective if \underline{E}G admits an appropriate compactification. This is a C^{∗}-version of the result for K-and L-theory due to Carlsson-Pedersen [55], compare Theorem 5.27.

Remark 5.13. We do not know whether the groups appearing in Theorem 5.8 and 5.9 satisfy the Baum-Connes Conjecture 2.3.

Next we discuss injectivity results about the classical assembly map for topological K-theory.

The asymptotic dimension of a proper metric space X is the infimum over all integers n such that for any R > 0 there exists a cover U of X with the property that the diameter of the members of U is uniformly bounded and every ball of radius R intersects at most (n + 1) elements of U (see [137, page 28]).

The next result is due to Yu [337].

Theorem 5.14 (The C^{∗}-Theoretic Novikov Conjecture and Groups of Finite Asymptotic Dimension). Let G be a group which possesses a finite model for BG and has finite asymptotic dimension. Then the assembly map in the Baum-Connes Conjecture 1.31

K_{n}(BG) → K_{n}(C_{r}^{∗}(G))

is injective for all n ∈ ℤ.

5.1.4 The Coarse Baum-Connes Conjecture

The coarse Baum-Connes Conjecture was explained in Section 4.1.5. Recall the descent principle (Proposition 4.7): if a countable group can be equipped with a G-invariant metric such that the resulting metric space satisfies the Coarse Baum-Connes Conjecture, then the classical assembly map for topological K-theory is injective.

Recall that a discrete metric space has bounded geometry if for each r > 0 there exists a N(r) such that for all x the ball of radius N(r) centered at x ∈ X contains at most N(r) elements.

The next result is due to Yu [338, Theorem 2.2 and Proposition 2.6].

Theorem 5.15 (Status of the Coarse Baum-Connes Conjecture). The Coarse Baum-Connes Conjecture 4.6 is true for a discrete metric space X of bounded geometry if X admits a uniform embedding into Hilbert space. In particular a countable group G satisfies the Coarse Baum-Connes Conjecture 4.6 if G equipped with a proper left G-invariant metric admits a uniform embedding into Hilbert space.

Also Yu’s Theorem 5.14 is proven via a corresponding result about the Coarse Baum-Connes Conjecture.

5.2 Status of the Farrell-Jones Conjecture

Next we deal with the Farrell-Jones Conjecture.

5.2.1 The Fibered Farrell-Jones Conjecture

The Fibered Farrell-Jones Conjecture 4.11 was discussed in Subsection 4.2.2. Recall that it has better inheritance properties (compare Section 5.4) and contains the ordinary Farrell-Jones Conjecture 2.2 as a special case.

Theorem 5.16 (Status of the Fibered Farrell-Jones Conjecture).

(i) Let G be a discrete group which satisfies one of the following conditions.

(a) There is a Lie group L with finitely many path components and G is a cocompact discrete subgroup of L.

(b) The group G is virtually torsionfree and acts properly discontinuously, cocompactly and via isometries on a simply connected complete nonpositively curved Riemannian manifold.

Then

(1) The version of the Fibered Farrell-Jones Conjecture 4.11 for the topological and the differentiable pseudoisotopy functor is true for G.

(2) The version of the Fibered Farrell-Jones Conjecture 4.11 for K-theory and R = ℤ is true for G in the range n ≤ 1, i.e. the assembly map is bijective for n ≤ 1.

Moreover we have the following statements.

(ii) The version of the Fibered Farrell-Jones Conjecture 4.11 for K-theory and

R = ℤ is true in the range n ≤ 1 for braid groups. (iii) The L-theoretic version of the Fibered Farrell-Jones Conjecture 4.11 with

R = ℤ holds after inverting 2 for elementary amenable groups.

Proof. (i) For assertion (1) see [111, Theorem 2.1 on page 263], [111, Proposition 2.3] and [119, Theorem A]. Assertion (2) follows from (1) by Remark 4.15. (ii) See [119]. (iii) is proven in [117, Theorem 5.2]. For crystallographic groups see also [333]. ⊓ ⊔

A surjectivity result about the Fibered Farrell-Jones Conjecture for Pseudoisotopies appears as the last statement in Theorem 5.20.

The rational comparison result between the K-theory and the pseudoisotopy version (see Proposition 4.10) does not work in the fibered case, compare Remark 4.15. However, in order to exploit the good inheritance properties one can first use the pseudoisotopy functor in the fibered set-up, then specialize to the unfibered situation and finally do the rational comparison to K-theory.

Remark 5.17. The version of the Fibered Farrell-Jones Conjecture 4.11 for L-theory and R = ℤ seems to be true if G satisfies the condition (a) appearing in Theorem 5.16. Farrell and Jones [111, Remark 2.1.3 on page 263] say that they can also prove this version without giving the details.

Remark 5.18. Let G be a virtually poly-cyclic group. Then it contains a maximal normal finite subgroup N such that the quotient G/N is a discrete cocompact subgroup of a Lie group with finitely many path components [331, Theorem 3, Remark 4 on page 200]. Hence by Subsection 5.4.3 and Theorem 5.16 the version of the Fibered Farrell-Jones Conjecture 4.11 for the topological and the differentiable pseudoisotopy functor, and for K-theory and R = ℤ in the range n ≤ 1, is true for G. Earlier results of this type were treated for example in [100], [105].

5.2.2 The Farrell-Jones Conjecture

Here is a sample of some results one can deduce from Theorem 5.16.

Theorem 5.19 (The Farrell-Jones Conjecture and Subgroups of Lie groups). Suppose H is a subgroup of G, where G is a discrete cocompact subgroup of a Lie group L with finitely many path components. Then

(i) The version of the Farrell-Jones Conjecture for K-theory and R = ℤ is true for H rationally, i.e. the assembly map appearing in Conjecture 2.2 is an isomorphism after applying − ⊗_{ℤ} ℚ.

(ii) The version of the Farrell-Jones Conjecture for K-theory and R = ℤ is true for H in the range n ≤ 1, i.e. the assembly map appearing in Conjecture 2.2 is an isomorphism for n ≤ 1.

Proof. The results follow from Theorem 5.16, since the Fibered Farrell-Jones Conjecture 4.11 passes to subgroups [111, Theorem A.8 on page 289] (compare Section 5.4.2) and implies the Farrell-Jones Conjecture 2.2. ⊓ ⊔

We now discuss results for torsion free groups. Recall that for R = ℤ the K-theoretic Farrell-Jones Conjecture in dimensions ≤ 1 together with the Ltheoretic version implies already the Borel Conjecture 1.27 in dimension ≥ 5 (see Theorem 1.28).

A complete Riemannian manifold M is called A-regular if there exists a sequence of positive real numbers A_{0}, A_{1}, A_{2}, … such that ||∇^{n}K|| ≤ A_{n}, where ||∇^{n}K|| is the supremum-norm of the n-th covariant derivative of the curvature tensor K. Every locally symmetric space is A-regular since ∇K is identically zero. Obviously every closed Riemannian manifold is A-regular.

Theorem 5.20 (Status of the Farrell-Jones Conjecture for Torsionfree Groups). Consider the following conditions for the group G.

(i) G = π_{1}(M) for a complete Riemannian manifold M with non-positive sectional curvature which is A-regular.

(ii) G = π_{1}(M) for a closed Riemannian manifold M with non-positive sec-

tional curvature. (iii) G = π_{1}(M) for a complete Riemannian manifold with negatively pinched

sectional curvature. (iv) G is a torsion free discrete subgroup of GL(n, ℝ).

(v) G is a torsion free solvable discrete subgroup of GL(n, ℂ). (vi) G = π_{1}(X) for a non-positively curved finite simplicial complex X. (vii) G is a strongly poly-free group in the sense of Aravinda-Farrell-Roushon

[10, Definition 1.1]. The pure braid group satisfies this hypothesis.

Then

(1) Suppose that G satisfies one of the conditions (i) to (vii). Then the Ktheoretic Farrell-Jones Conjecture is true for R = ℤ in dimensions n ≤ 1. In particular Conjecture 1.3 holds for G.

(2) Suppose that G satisfies one of the conditions (i), (ii), (iii) or (iv). Then G satisfies the Farrell-Jones Conjecture for Torsion Free Groups and L-Theory 1.19 for R = ℤ.

(3) Suppose that G satisfies (ii). Then the Farrell-Jones Conjecture for Pseudoisotopies of Aspherical Spaces 4.8 holds for G.

(4) Suppose that G satisfies one of the conditions (i), (iii) or (iv). Then the assembly map appearing in the version of the Fibered Farrell-Jones Conjecture for Pseudoisotopies 4.11 is surjective, provided that the G-space Z appearing in Conjecture 4.11 is connected.

Proof. Note that (ii) is a special case of (i) because every closed Riemannian manifold is A-regular. If M is a pinched negatively curved complete Riemannian manifold, then there is another Riemannian metric for which M is negatively curved complete and A-regular. This fact is mentioned in [115, page 216] and attributed there to Abresch [3] and Shi [285]. Hence also (iii) can be considered as a special case of (i). The manifold M = G\GL(n, ℝ)/O(n) is a non-positively curved complete locally symmetric space and hence in particular A-regular. So (iv) is a special case of (i).

Assertion (1) under the assumption (i) is proved by Farrell-Jones in [115, Proposition 0.10 and Lemma 0.12]. The earlier work [110] treated the case (ii). Under assumption (v) assertion (1) is proven by Farrell-Linnell [117, Theorem 1.1]. The result under assumption (vi) is proved by Hu [156], under assumption (vii) it is proved by Aravinda-Farrell-Roushon [10, Theorem 1.3].

Assertion (2) under assumption (i) is proven by Farrell-Jones in [115]. The case (ii) was treated earlier in [112].

Assertion (3) is proven by Farrell-Jones in [110] and assertion (4) by Jones in [161]. ⊓ ⊔

Remark 5.21. As soon as certain collapsing results (compare [114], [116]) are extended to orbifolds, the results under (4) above would also apply to groups with torsion and in particular to SL_{n}(ℤ) for arbitrary n.

5.2.3 The Farrell-Jones Conjecture for Arbitrary Coefficients

The following result due to Bartels-Reich [21] deals with algebraic K-theory for arbitrary coefficient rings R. It extends Bartels-Farrell-Jones-Reich [19].

Theorem 5.22. Suppose that G is the fundamental group of a closed Riemannian manifold with negative sectional curvature. Then the K-theoretic part of the Farrell-Jones Conjecture 2.2 is true for any ring R, i.e. the assembly map

A_{VCY}: H_{n}^{G}(E_{VCY}(G); K_{R}) → K_{n}(RG)

is an isomorphism for all n ∈ Z.

Note that the assumption implies that G is torsion free and hence the family VCY reduces to the family CYC of cyclic subgroups. Recall that for a regular ring R the theorem above implies that the classical assembly

A: H_{n}(BG; K(R)) → K_{n}(RG)

is an isomorphism, compare Proposition 2.10 (i).

5.2.4 Injectivity Part of the Farrell-Jones Conjecture

The next result about the classical K-theoretic assembly map is due to Bökstedt-Hsiang-Madsen [38].

Theorem 5.23 (Rational Injectivity of the Classical K-Theoretic Assembly Map). Let G be a group such that the integral homology H_{j}(BG; ℤ) is finitely generated for each j ∈ ℤ. Then the rationalized assembly map

A: H_{n}(BG; K(ℤ)) ⊗_{ℤ} ℚ ≅ H_{n}^{G}(E_{{1}}(G); K_{ℤ}) ⊗_{ℤ} ℚ → K_{n}(ℤG) ⊗_{ℤ} ℚ

is injective for all n ∈ Z.

Because of the homological Chern character (see Remark 1.12) we obtain for the groups treated in Theorem 5.23 an injection

⨁_{s+t=n} H_{s}(BG; ℚ) ⊗_{ℚ} (K_{t}(ℤ) ⊗_{ℤ} ℚ) → K_{n}(ℤG) ⊗_{ℤ} ℚ.  (28)

Next we describe a generalization of Theorem 5.23 above from the trivial family {1} to the family FIN of finite subgroups due to Lück-Reich-Rognes-Varisco [208]. Let K_{ℤ}^{con} : GROUPOIDS→ SPECTRAbe the connective version of the functor K_{ℤ} of (37). In particular H_{n}(G/H; K_{ℤ}^{con}) is isomorphic to K_{n}(ℤH) for n ≥ 0 and vanishes in negative dimensions. For a prime p we denote by ℤ_{p} the p-adic integers. Let K_{n}(R; ℤ_{p}) denote the homotopy groups π_{n}(K^{con}(R)_{p}) of the p-completion of the connective K-theory spectrum of the ring R.

Theorem 5.24 (Rational Injectivity of the Farrell-Jones Assembly Map for Connective K-Theory). Suppose that the group G satisfies the following two conditions:

(H) For each finite cyclic subgroup C ⊆ G and all j ≥ 0 the integral homology

group H_{j}(BZ_{G}C; ℤ) of the centralizer Z_{G}C of C in G is finitely generated. (K) There exists a prime p such that for each finite cyclic subgroup C ⊆ G and

each j ≥ 1 the map induced by the change of coefficients homomorphism

K_{j}(ℤC; ℤ_{p}) ⊗_{ℤ} ℚ → K_{j}(ℤ_{p}C; ℤ_{p}) ⊗_{ℤ} ℚ

is injective.

Then the rationalized assembly map

A_{VCY}: H_{n}^{G}(E_{VCY}(G); K_{ℤ}^{con}) ⊗_{ℤ} ℚ → K_{n}(ℤG) ⊗_{ℤ} ℚ

is an injection for all n ∈ ℤ.

Remark 5.25. The methods of Chapter 8 apply also to K_{ℤ}^{con} and yield under assumption (H) and (K) an injection

⨁_{s+t=n, t≥0} ⨁_{(C)∈(FCY)} H_{s}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · K_{t}(ℤC) ⊗_{ℤ} ℚ

⟶ K_{n}(ℤG) ⊗_{ℤ} ℚ.

Notice that in the index set for the direct sum appearing in the source we require t ≥ 0. This reflects the fact that the result deals only with the connective K-theory spectrum. If one drops the restriction t ≥ 0 the Farrell-Jones Conjecture 2.2 predicts that the map is an isomorphism, compare Subsection 2.2.5 and Theorem 8.5. If we restrict the injection to the direct sum given by C = 1, we rediscover the map (28) whose injectivity follows already from Theorem 5.23.

The condition (K) appearing in Theorem 5.24 is conjectured to be true for all primes p (compare [280], [290] and [291]) but no proof is known. The weaker version of condition (K), where C is the trivial group is also needed in Theorem 5.23. But that case is known to be true and hence does not appear in its formulation. The special case of condition (K), where j = 1 is implied by the Leopoldt Conjecture for abelian fields (compare [229, IX, § 3]), which is known to be true [229, Theorem 10.3.16]. This leads to the following result.

Theorem 5.26 (Rational Contribution of Finite Subgroups to Wh(G)). Let G be a group. Suppose that for each finite cyclic subgroup C ⊆ G and each j ≤ 4 the integral homology group H_{j}(BZ_{G}C) of the centralizer Z_{G}C of C in G is finitely generated. Then the map

colim_{H∈Sub _{FIN} (G)} Wh(H) ⊗_{ℤ} ℚ → Wh(G) ⊗_{ℤ} ℚ.

is injective, compare Conjecture 3.5.

The result above should be compared to the result which is proven using Fuglede-Kadison determinants in [209, Section 5], [202, Theorem 9.38 on page 354]: for every (discrete) group G and every finite normal subgroup H ⊆ G the map Wh(H) ⊗_{ℤG} ℤ → Wh(G) induced by the inclusion H → G is rationally injective.

The next result is taken from Rosenthal [271], where the techniques and results of Carlsson-Pedersen [55] are extended from the trivial family {1} to the family of finite subgroups FIN.

Theorem 5.27. Suppose there exists a model E for the classifying space E_{FIN} (G) which admits a metrizable compactification \overline{E} to which the group action extends. Suppose \overline{E}^{H} is contractible and E^{H} is dense in \overline{E}^{H} for every finite subgroup H ⊂ G. Suppose compact subsets of E become small near \overline{E} − E. Then for every ring R the assembly map

A_{FIN} : H_{n}^{G}(E_{FIN} (G); K_{R}) → K_{n}(RG)

is split injective.

A compact subset K ⊂ E is said to become small near \overline{E} − E if for every neighbourhood U ⊂ \overline{E} of a point x ∈ \overline{E} − E there exists a neighbourhood V ⊂ \overline{E} such that g ∈ G and gK ∩ V ≠ ∅ implies gK ⊂ U. Presumably there is an analogous result for L^{⟨−∞⟩}-theory under the assumption that K_{−n}(RH) vanishes for finite subgroups H of G and n large enough. This would extend the corresponding result for the family F = {1} which appears in Carlsson-Pedersen [55].

We finally discuss injectivity results about assembly maps for the trivial family. The following result is due to Ferry-Weinberger [129, Corollary 2.3] extending earlier work of Farrell-Hsiang [99].

Theorem 5.28. Suppose G = π_{1}(M) for a complete Riemannian manifold of non-positive sectional curvature. Then the L-theory assembly map

A: H_{n}(BG; L_{ℤ}^{ϵ}) → L_{n}^{ϵ}(ℤG)

is injective for ϵ = h, s.

In fact Ferry-Weinberger also prove a corresponding splitting result for the classical A-theory assembly map. In [155] Hu shows that a finite complex of non-positive curvature is a retract of a non-positively curved P L-manifold and concludes split injectivity of the classical L-theoretic assembly map for R = ℤ.

The next result due to Bartels [17] is the algebraic K- and L-theory analogue of Theorem 5.14.

Theorem 5.29 (The K-and L-Theoretic Novikov Conjecture and Groups of Finite Asymptotic Dimension). Let G be a group which admits a finite model for BG. Suppose that G has finite asymptotic dimension. Then

(i) The assembly maps appearing in the Farrell-Jones Conjecture 1.11

A: H_{n}(BG; K(R)) → K_{n}(RG)

is injective for all n ∈ ℤ. (ii) If furthermore R carries an involution and K_{−j}(R) vanishes for suffi-

ciently large j, then the assembly maps appearing in the Farrell-Jones Conjecture 1.19

A: H_{n}(BG; L^{⟨−∞⟩}(R)) → L_{n}^{⟨−∞⟩}(RG)

is injective for all n ∈ ℤ.

Further results related to the Farrell-Jones Conjecture 2.2 can be found for instance in [9], [33].

5.3 List of Groups Satisfying the Conjecture

In the following table we list prominent classes of groups and state whether they are known to satisfy the Baum-Connes Conjecture 2.3 (with coefficients 4.3) or the Farrell-Jones Conjecture 2.2 (fibered 4.11). Some of the classes are redundant. A question mark means that the authors do not know about a corresponding result. The reader should keep in mind that there may exist results of which the authors are not aware. type of group Baum-Connes Conjecture 2.3 (with coefficients 4.3)

Farrell-Jones Conjecture 2.2 for K-theory for R = ℤ (fibered 4.11)

Farrell-Jones Conjecture 2.2 for L-theory for R = ℤ (fibered 4.11)

a-T-menable groups true with coefficients (see Theorem 5.1)

? injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

amenable groups true with coefficients (see Theorem 5.1)

? injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

elementary amenable groups true with coefficients (see Theorem 5.1)

? true fibered after inverting 2 (see Theorem 5.16)

virtually polycyclic true with coefficients (see Theorem 5.1) true rationally, true fibered in the range n ≤ 1 (compare Remark 5.18) true fibered after inverting 2 (see Theorem 5.16)

torsion free virtually solvable subgroups of GL(n, ℂ) true with coefficients (see Theorem 5.1) true in the range ≤ 1 [117, Theorem 1.1] true fibered after inverting 2 [117, Corollary 5.3] type of group Baum-Connes Conjecture 2.3 (with coefficients 4.3)

Farrell-Jones Conjecture 2.2 for K-theory for R = ℤ (fibered 4.11)

Farrell-Jones Conjecture 2.2 for L-theory for R = ℤ (fibered 4.11)

discrete subgroups of Lie groups with finitely many path components injectivity true (see Theorem 5.9 and Remark 5.11)

? injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

subgroups of groups which are discrete cocompact subgroups of Lie groups with finitely many path components injectivity true (see Theorem 5.9 and Remark 5.11) true rationally, true fibered in the range n ≤ 1 (see Theorem 5.16) probably true fibered (see Remark 5.17). Injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

linear groups injectivity is true (see Theorem 5.9 and Remark 5.10)

? injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

arithmetic groups injectivity is true (see Theorem 5.9 and Remark 5.10)

? injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

torsion free discrete subgroups of GL(n, ℝ) injectivity is true (see Theorem 5.9 and Remark 5.11) true in the range n ≤ 1 (see Theorem 5.20) true (see Theorem 5.20) type of group Baum-Connes Conjecture 2.3 (with coefficients 4.3)

Farrell-Jones Conjecture 2.2 for K-theory for R = ℤ (fibered 4.11)

Farrell-Jones Conjecture 2.2 for L-theory for R = ℤ (fibered 4.11)

Groups with finite BG and finite asymptotic dimension injectivity is true (see Theorem 5.14) injectivity is true for arbitrary coefficients R (see Theorem 5.29) injectivity is true for regular R as coefficients (see Theorem 5.29)

G acts properly and isometrically on a complete Riemannian manifold M with non-positive sectional curvature rational injectivity is true (see Theorem 5.8)

? rational injectivity is true (see Propositions 2.18 and 3.19)

π_{1}(M) for a complete Riemannian manifold M with non-positive sectional curvature rationally injective (see Theorem 5.8)

? injectivity true (see Theorem 5.28)

π_{1}(M) for a complete Riemannian manifold M with non-positive sectional curvature which is A-regular rationally injective (see Theorem 5.8) true in the range n ≤ 1, rationally surjective (see Theorem 5.20) true (see Theorem 5.20)

π_{1}(M) for a complete Riemannian manifold M with pinched negative sectional curvature rational injectivity is true (see Theorem 5.9) true in the range n ≤ 1, rationally surjective (see Theorem 5.20) true (see Theorem 5.20)

π_{1}(M) for a closed Riemannian manifold M with non-positive sectional curvature rationally injective (see Theorem 5.8) true fibered in the range n ≤ 1, true rationally (see Theorem 5.20) true (see Theorem 5.20)

π_{1}(M) for a closed Riemannian manifold M with negative sectional curvature true for all subgroups (see Theorem 5.4) true for all coefficients R (see Theorem 5.22) true (see Theorem 5.20) type of group Baum-Connes Conjecture 2.3 (with coefficients 4.3)

Farrell-Jones Conjecture 2.2 for K-theory and R = ℤ (fibered 4.11)

Farrell-Jones Conjecture 2.2 for L-theory for R = ℤ (fibered 4.11)

word hyperbolic groups true for all subgroups (see Theorem 5.4). Unpublished proof with coefficients by V. Lafforgue ? injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

one-relator groups true with coefficients (see Theorem 5.2) rational injectivity is true for the fibered version (see [20]) injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

torsion free onerelator groups true with coefficients (see Theorem 5.2) true for R regular [313, Theorem 19.4 on page 249 and Theorem 19.5 on page 250] true after inverting 2 [47, Corollary 8]

Haken 3-manifold groups (in particular knot groups) true with coefficients (see Theorem 5.2) true in the range n ≤ 1 for R regular [313, Theorem 19.4 on page 249 and Theorem 19.5 on page 250] true after inverting 2 [47, Corollary 8]

SL(n, ℤ), n ≥ 3 injectivity is true compare Remark 5.21 injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

Artin’s braid group B_{n} true with coefficients [225, Theorem 5.25], [277] true fibered in the range n ≤ 1, true rationally [119] injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

pure braid group C_{n} true with coefficients true in the range n ≤ 1 (see Theorem 5.20) injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

Thompson’s group F true with coefficients [94] ? injectivity is true after inverting 2 (see Propositions 2.18 and 3.19)

Remark 5.30. The authors have no information about the status of these conjectures for mapping class groups of higher genus or the group of outer automorphisms of free groups. Since all of these spaces have finite models for E_{FIN} (G) Theorem 5.24 applies in these cases.

5.4 Inheritance Properties

In this Subsection we list some inheritance properties of the various conjectures.

5.4.1 Directed Colimits

Let {G_{i} | i ∈ I} be a directed system of groups. Let G = colim_{i∈I} G_{i} be the colimit. We do not require that the structure maps are injective. If the Fibered Farrell-Jones Conjecture 4.11 is true for each G_{i}, then it is true for G [117, Theorem 6.1].

Suppose that {G_{i} | i ∈ I} is a system of subgroups of G directed by inclusion such that G = colim_{i∈I} G_{i}. If each G_{i} satisfies the Farrell-Jones Conjecture 2.2, the Baum-Connes Conjecture 2.3 or the Baum-Connes Conjecture with Coefficients 4.3, then the same is true for G [32, Theorem 1.1], [225, Lemma 5.3]. We do not know a reference in Farrell-Jones case. An argument in that case uses Lemma 2.7, the fact that K_{n}(RG) = colim_{i∈I} K_{n}(RG_{i}) and that for suitable models we have E_{F}(G) = ⋃_{i∈I} G ×_{G_{i}} E_{F∩G_{i}}(G_{i}).

5.4.2 Passing to Subgroups

The Baum-Connes Conjecture with Coefficients 4.3 and the Fibered Farrell-Jones Conjecture 4.11 pass to subgroups, i.e. if they hold for G, then also for any subgroup H ⊆ G. This claim for the Baum-Connes Conjecture with Coefficients 4.3 has been stated in [28], a proof can be found for instance in [59, Theorem 2.5]. For the Fibered Farrell-Jones Conjecture this is proved in [111, Theorem A.8 on page 289] for the special case R = ℤ, but the proof also works for arbitrary rings R.

It is not known whether the Baum-Connes Conjecture 2.3 or the Farrell-Jones Conjecture 2.2 itself passes to subgroups.

5.4.3 Extensions of Groups

Let p: G → K be a surjective group homomorphism. Suppose that the Baum-Connes Conjecture with Coefficients 4.3 or the Fibered Farrell-Jones Conjecture 4.11 respectively holds for K and for p^{−1}(H) for any subgroup H ⊂ K which is finite or virtually cyclic respectively. Then the Baum-Connes Conjecture with Coefficients 4.3 or the Fibered Farrell-Jones Conjecture 4.11 respectively holds for G. This is proved in [233, Theorem 3.1] for the Baum-Connes Conjecture with Coefficients 4.3, and in [111, Proposition 2.2 on page 263] for the Fibered Farrell-Jones Conjecture 4.11 in the case R = ℤ. The same proof works for arbitrary coefficient rings.

It is not known whether the corresponding statement holds for the Baum-Connes Conjecture 2.3 or the Farrell-Jones Conjecture 2.2 itself.

Let H ⊆ G be a normal subgroup of G. Suppose that H is a-T-menable. Then G satisfies the Baum-Connes Conjecture with Coefficients 4.3 if and only if G/H does [59, Corollary 3.14]. The corresponding statement is not known for the Baum-Connes Conjecture 2.3.

5.4.4 Products of Groups

The group G_{1} × G_{2} satisfies the Baum-Connes Conjecture with Coefficients 4.3 if and only if both G_{1} and G_{2} do [59, Theorem 3.17], [233, Corollary 7.12]. The corresponding statement is not known for the Baum-Connes Conjecture 2.3.

Let D_{∞} = ℤ/2∗ℤ/2 denote the infinite dihedral group. Whenever a version of the Fibered Farrell-Jones Conjecture 4.11 is known for G = ℤ×ℤ, G = ℤ× D_{∞} and D_{∞} × D_{∞}, then that version of the Fibered Farrell-Jones Conjecture is true for G_{1} × G_{2} if and only if it is true for G_{1} and G_{2}.

5.4.5 Subgroups of Finite Index

It is not known whether the Baum-Connes Conjecture 2.3, the Baum-Connes Conjecture with Coefficients 4.3, the Farrell-Jones Conjecture 2.2 or the Fibered Farrell-Jones Conjecture 4.11 is true for a group G if it is true for a subgroup H ⊆ G of finite index.

5.4.6 Groups Acting on Trees

Let G be a countable discrete group acting without inversion on a tree T. Then the Baum-Connes Conjecture with Coefficients 4.3 is true for G if and only if it holds for all stabilizers of the vertices of T. This is proved by Oyono-Oyono [234, Theorem 1.1]. This implies that Baum-Connes Conjecture with Coefficients 4.3 is stable under amalgamated products and HNN-extensions. Actions on trees in the context the Farrell-Jones Conjecture 2.2 will be treated in [20].

6 Equivariant Homology Theories

We already defined the notion of a G-homology theory in Subsection 2.1.4. If G-homology theories for different G are linked via a so called induction structure one obtains the notion of an equivariant homology theory. In this section we give a precise definition and we explain how a functor from the orbit category Or(G) to the category of spectra leads to a G-homology theory (see Proposition 6.7) and how more generally a functor from the category of groupoids leads to an equivariant homology theory (see Proposition 6.8). We then describe the main examples of such spectra valued functors which were already used in order to formulate the Farrell-Jones and the Baum-Connes Conjectures in Chapter 2.

6.1 The Definition of an Equivariant Homology Theory

The notion of a G-homology theory H_{∗}^{G} with values in Λ-modules for a commutative ring Λ was defined in Subsection 2.1.4. We now recall the axioms of an equivariant homology theory from [201, Section 1]. We will see in Section 6.3 that the G-homology theories we used in the formulation of the Baum-Connes and the Farrell-Jones Conjectures in Chapter 2 are in fact the values at G of suitable equivariant homology theories.

Let α: H → G be a group homomorphism. Given a H-space X, define the induction of X with α to be the G-space ind_{α} X which is the quotient of G × X by the right H-action (g, x) · h := (gα(h), h^{−1}x) for h ∈ H and (g, x) ∈ G × X. If α: H → G is an inclusion, we also write ind_{H}^{G} instead of ind_{α}.

An equivariant homology theory H_{∗}^{?} with values in Λ-modules consists of a G-homology theory H_{∗}^{G} with values in Λ-modules for each group G together with the following so called induction structure: given a group homomorphism α: H → G and a H-CW -pair (X, A) such that ker(α) acts freely on X, there are for each n ∈ ℤ natural isomorphisms

ind_{α}: H_{n}^{H}(X, A) ⟶^{≅} H_{n}^{G}(ind_{α}(X, A))

satisfying the following conditions.

(i) Compatibility with the boundary homomorphisms ∂_{n}^{G} ∘ ind_{α} = ind_{α} ∘∂_{n}^{H}.

(ii) Functoriality Let β: G → K be another group homomorphism such that ker(β ∘ α) acts freely on X. Then we have for n ∈ ℤ

ind_{β∘α} = H_{n}^{K}(f_{1}) ∘ ind_{β} ∘ ind_{α}: H_{n}^{H}(X, A) → H_{n}^{K}(ind_{β∘α}(X, A)),

where f_{1} : ind_{β} ind_{α}(X, A) ⟶^{≅} ind_{β∘α}(X, A), (k, g, x) ↦ (kβ(g), x) is

the natural K-homeomorphism. (iii) Compatibility with conjugation

For n ∈ ℤ, g ∈ G and a G-CW -pair (X, A) the homomorphism

ind_{c(g) : G→G}: H_{n}^{G}(X, A) → H_{n}^{G}(ind_{c(g) : G→G}(X, A))

agrees with H_{n}^{G}(f_{2}), where the G-homeomorphism

f_{2} : (X, A) → ind_{c(g) : G→G}(X, A)

sends x to (1, g^{−1}x) and c(g): G → G sends g′ to gg′g^{−1}.

This induction structure links the various homology theories for different groups G.

If the G-homology theory H_{∗}^{G} is defined or considered only for proper G-CW -pairs (X, A), we call it a proper G-homology theory H_{∗}^{G} with values in Λ-modules.

Example 6.1. Let K_{∗} be a homology theory for (non-equivariant) CW -pairs with values in Λ-modules. Examples are singular homology, oriented bordism theory or topological K-homology. Then we obtain two equivariant homology theories with values in Λ-modules, whose underlying G-homology theories for a group G are given by the following constructions

H_{n}^{G}(X, A) = K_{n}(G\X, G\A);
H_{n}^{G}(X, A) = K_{n}(EG ×_{G} (X, A)).

Example 6.2. Given a proper G-CW -pair (X, A), one can define the Gbordism group Ω_{n}^{G}(X, A) as the abelian group of G-bordism classes of maps f: (M, ∂M) → (X, A) whose sources are oriented smooth manifolds with cocompact orientation preserving proper smooth G-actions. The definition is analogous to the one in the non-equivariant case. This is also true for the proof that this defines a proper G-homology theory. There is an obvious induction structure coming from induction of equivariant spaces. Thus we obtain an equivariant proper homology theory Ω_{∗}^{?}.

Example 6.3. Let Λ be a commutative ring and let

M: GROUPOIDS→ Λ-MODULES

be a contravariant functor. For a group G we obtain a covariant functor

M^{G}: Or(G) → Λ-MODULES

by its composition with the transport groupoid functor G^{G} defined in (30). Let H_{∗}^{G}(−; M) be the G-homology theory given by the Bredon homology with coefficients in M^{G} as defined in Example 2.8. There is an induction structure such that the collection of the H^{G}(−; M) defines an equivariant homology theory H_{∗}^{?}(−; M). This can be interpreted as the special case of Proposition 6.8, where the covariant functor GROUPOIDS→ Ω-SPECTRAis the composition of M with the functor sending a Λ-module to the associated Eilenberg-MacLane spectrum. But there is also a purely algebraic construction.

The next lemma was used in the proof of the Transitivity Principle 2.9.

Lemma 6.4. Let H_{∗}^{?} be an equivariant homology theory with values in Λmodules. Let G be a group and let F a family of subgroups of G. Let Z be a G-CW -complex. Consider N ∈ ℤ ∪ {∞}. For H ⊆ G let F ∩ H be the family of subgroups of H given by {K ∩ H | K ∈ F}. Suppose for each H ⊂ G, which occurs as isotropy group in Z, that the map induced by the projection pr: E_{F∩H}(H) → pt

H_{n}^{H}(pr): H_{n}^{H}(E_{F∩H}(H)) → H_{n}^{H}(pt)

is bijective for all n ∈ ℤ, n ≤ N.

Then the map induced by the projection pr_{2} : E_{F}(G) × Z → Z

H_{n}^{G}(pr_{2}): H_{n}^{G}(E_{F}(G) × Z) → H_{n}^{G}(Z)

is bijective for n ∈ ℤ, n ≤ N.

Proof. We first prove the claim for finite-dimensional G-CW -complexes by induction over d = dim(Z). The induction beginning dim(Z) = −1, i.e. Z = ∅, is trivial. In the induction step from (d − 1) to d we choose a G-pushout

∐_{i∈I_{d}} G/H_{i} × S^{d−1} ⟶ Z_{d−1}
↓  ↓
∐_{i∈I_{d}} G/H_{i} × D^{d} ⟶ Z_{d}

If we cross it with E_{F}(G), we obtain another G-pushout of G-CW -complexes. The various projections induce a map from the Mayer-Vietoris sequence of the latter G-pushout to the Mayer-Vietoris sequence of the first G-pushout. By the Five-Lemma it suffices to prove that the following maps

H_{n}^{G}(pr_{2}): H_{n}^{G} (E_{F}(G) × ∐_{i∈I_{d}} G/H_{i} × S^{d−1}) → H_{n}^{G} ( ∐_{i∈I_{d}} G/H_{i} × S^{d−1});
H_{n}^{G}(pr_{2}): H_{n}^{G}(E_{F}(G) × Z_{d−1}) → H_{n}^{G}(Z_{d−1});
H_{n}^{G}(pr_{2}): H_{n}^{G} (E_{F}(G) × ∐_{i∈I_{d}} G/H_{i} × D^{d}) → H_{n}^{G} ( ∐_{i∈I_{d}} G/H_{i} × D^{d})

are bijective for n ∈ ℤ, n ≤ N. This follows from the induction hypothesis for the first two maps. Because of the disjoint union axiom and G-homotopy invariance of H_{∗}^{?} the claim follows for the third map if we can show for any H ⊆ G which occurs as isotropy group in Z that the map

H_{n}^{G}(pr_{2}): H_{n}^{G}(E_{F}(G) × G/H) → H^{G}(G/H)  (29)

is bijective for n ∈ ℤ, n ≤ N. The G-map

G ×_{H} res_{G}^{H} E_{F}(G) → G/H × E_{F}(G) (g, x) ↦ (gH, gx)

is a G-homeomorphism where res_{G}^{H} denotes the restriction of the G-action to an H-action. Obviously res_{G}^{H} E_{F}(G) is a model for E_{F∩H}(H). We conclude from the induction structure that the map (29) can be identified with the map

H_{n}^{G}(pr): H_{n}^{H}(E_{F∩H}(H)) → H^{H}(pt)

which is bijective for all n ∈ ℤ, n ≤ N by assumption. This finishes the proof in the case that Z is finite-dimensional. The general case follows by a colimit argument using Lemma 2.7. ⊓ ⊔

6.2 Constructing Equivariant Homology Theories

Recall that a (non-equivariant) spectrum yields an associated (non-equivariant) homology theory. In this section we explain how a spectrum over the orbit category of a group G defines a G-homology theory. We would like to stress that our approach using spectra over the orbit category should be distinguished from approaches to equivariant homology theories using spectra with G-action or the more complicated notion of G-equivariant spectra in the sense of [190], see for example [53] for a survey. The latter approach leads to a much richer structure but only works for compact Lie groups.

We briefly fix some conventions concerning spectra. We always work in the very convenient category SPACES of compactly generated spaces (see [295], [330, I.4]). In that category the adjunction homeomorphism map(X×Y, Z) ⟶^{≅} map(X, map(Y, Z)) holds without any further assumptions such as local compactness and the product of two CW -complexes is again a CW -complex. Let SPACES^{+} be the category of pointed compactly generated spaces. Here the objects are (compactly generated) spaces X with base points for which the inclusion of the base point is a cofibration. Morphisms are pointed maps. If X is a space, denote by X_{+} the pointed space obtained from X by adding a disjoint base point. For two pointed spaces X = (X, x) and Y = (Y, y) define their smash product as the pointed space

X ∧ Y = X × Y/({x} × Y ∪ X × {y}),

and the reduced cone as

cone(X) = X × [0, 1]/(X × {1} ∪ {x} × [0, 1]).

A spectrum E = {(E(n), σ(n)) | n ∈ ℤ} is a sequence of pointed spaces {E(n) | n ∈ ℤ} together with pointed maps called structure maps σ(n): E(n) ∧ S^{1} ⟶ E(n + 1). A map of spectra f : E → E′ is a sequence of maps f(n): E(n) → E′(n) which are compatible with the structure maps σ(n), i.e. we have f(n + 1) ∘ σ(n) = σ′(n) ∘ (f(n) ∧ id_{S^{1}}) for all n ∈ ℤ. Maps of spectra are sometimes called functions in the literature, they should not be confused with the notion of a map of spectra in the stable category (see [4, III.2.]). The category of spectra and maps will be denoted SPECTRA . Recall that the homotopy groups of a spectrum are defined by

π_{i}(E) = colim_{k→∞} π_{i+k}(E(k)),

where the system π_{i+k}(E(k)) is given by the composition

π_{i+k}(E(k)) ⟶^{S} π_{i+k+1}(E(k) ∧ S^{1}) ⟶^{σ(k)_{∗}} π_{i+k+1}(E(k + 1))

of the suspension homomorphism S and the homomorphism induced by the structure map. A weak equivalence of spectra is a map f : E → F of spectra inducing an isomorphism on all homotopy groups.

Given a spectrum E and a pointed space X, we can define their smash product X ∧ E by (X ∧ E)(n) := X ∧ E(n) with the obvious structure maps. It is a classical result that a spectrum E defines a homology theory by setting

H_{n}(X, A; E) = π_{n} ((X_{+} ∪_{A_{+}} cone(A_{+})) ∧ E) .

We want to extend this to G-homology theories. This requires the consideration of spaces and spectra over the orbit category. Our presentation follows [82], where more details can be found.

In the sequel C is a small category. Our main example is the orbit category Or(G), whose objects are homogeneous G-spaces G/H and whose morphisms are G-maps.

Definition 6.5. A covariant (contravariant) C-space X is a covariant (contravariant) functor

X: C → SPACES.

A map between C-spaces is a natural transformation of such functors. Analogously a pointed C-space is a functor from C to SPACES^{+} and a C-spectrum a functor to SPECTRA .

Example 6.6. Let Y be a left G-space. Define the associated contravariant Or(G)-space map_{G}(−, Y) by

map_{G}(−, Y): Or(G) → SPACES, G/H ↦ map_{G}(G/H, Y) = Y^{H}.

If Y is pointed then map_{G}(−, Y) takes values in pointed spaces.

Let X be a contravariant and Y be a covariant C-space. Define their balanced product to be the space

X ×_{C} Y := ∐_{c∈ob(C)} X(c) × Y (c)/ ∼

where ∼ is the equivalence relation generated by (xϕ, y) ∼ (x, ϕy) for all morphisms ϕ: c → d in C and points x ∈ X(d) and y ∈ Y (c). Here xϕ stands for X(ϕ)(x) and ϕy for Y (ϕ)(y). If X and Y are pointed, then one defines analogously their balanced smash product to be the pointed space

X ∧_{C} Y = ⋁_{c∈ob(C)} X(c) ∧ Y (c)/ ∼ .

In [82] the notation X ⊗_{C} Y was used for this space. Doing the same construction level-wise one defines the balanced smash product X ∧_{C} E of a contravariant pointed C-space and a covariant C-spectrum E.

The proof of the next result is analogous to the non-equivariant case. Details can be found in [82, Lemma 4.4], where also cohomology theories are treated.

Proposition 6.7 (Constructing G-Homology Theories). Let E be a covariant Or(G)-spectrum. It defines a G-homology theory H_{∗}^{G}(−; E) by

H_{n}^{G}(X, A; E) = π_{n} (map_{G} (−, (X_{+} ∪_{A_{+}} cone(A_{+}))) ∧_{Or (G)} E) .

In particular we have

H_{n}^{G}(G/H; E) = π_{n}(E(G/H)).

Recall that we seek an equivariant homology theory and not only a Ghomology theory. If the Or(G)-spectrum in Proposition 6.7 is obtained from a GROUPOIDS-spectrum in a way we will now describe, then automatically we obtain the desired induction structure.

Let GROUPOIDS be the category of small groupoids with covariant functors as morphisms. Recall that a groupoid is a category in which all morphisms are isomorphisms. A covariant functor f: G_{0} → G_{1} of groupoids is called injective, if for any two objects x, y in G_{0} the induced map mor_{G_{0}}(x, y) → mor_{G_{1}}(f(x), f(y)) is injective. Let GROUPOIDS^{inj} be the subcategory of GROUPOIDS with the same objects and injective functors as morphisms. For a G-set S we denote by G^{G}(S) its associated transport groupoid. Its objects are the elements of S. The set of morphisms from s_{0} to s_{1} consists of those elements g ∈ G which satisfy gs_{0} = s_{1}. Composition in G^{G}(S) comes from the multiplication in G. Thus we obtain for a group G a covariant functor

G^{G}: Or(G) → GROUPOIDS^{inj},  G/H ↦ G^{G}(G/H).  (30)

A functor of small categories F: C → D is called an equivalence if there exists a functor G: D → C such that both F ∘ G and G ∘ F are naturally equivalent to the identity functor. This is equivalent to the condition that F induces a bijection on the set of isomorphisms classes of objects and for any objects x, y ∈ C the map mor_{C}(x, y) → mor_{D}(F (x), F (y)) induced by F is bijective.

Proposition 6.8 (Constructing Equivariant Homology Theories). Consider a covariant GROUPOIDS^{inj}-spectrum

E: GROUPOIDS^{inj} → SPECTRA .

Suppose that E respects equivalences, i.e. it sends an equivalence of groupoids to a weak equivalence of spectra. Then E defines an equivariant homology theory H_{∗}^{?}(−; E), whose underlying G-homology theory for a group G is the Ghomology theory associated to the covariant Or(G)-spectrum E∘G^{G}: Or(G) → SPECTRAin the previous Proposition 6.7, i.e.

H_{∗}^{G}(X, A; E) = H_{∗}^{G}(X, A; E ∘ G^{G}).

In particular we have

H_{n}^{G}(G/H; E) ≅ H_{n}^{H}(pt; E) ≅ π_{n}(E(I(H))),

where I(H) denotes H considered as a groupoid with one object. The whole construction is natural in E.

Proof. We have to specify the induction structure for a homomorphism α: H → G. We only sketch the construction in the special case where α is injective and A = ∅. The details of the full proof can be found in [276, Theorem 2.10 on page 21].

The functor induced by α on the orbit categories is denoted in the same way

α: Or(H) → Or(G), H/L ↦ ind_{α}(H/L) = G/α(L).

There is an obvious natural equivalence of functors Or(H) → GROUPOIDS^{inj}

T: G^{H} → G^{G} ∘ α.

Its evaluation at H/L is the equivalence of groupoids G^{H}(H/L) → G^{G}(G/α(L)) which sends an object hL to the object α(h)α(L) and a morphism given by h ∈ H to the morphism α(h) ∈ G. The desired isomorphism

ind_{α}: H_{n}^{H}(X; E ∘ G^{H}) → H_{n}^{G}(ind_{α} X; E ∘ G^{G})

is induced by the following map of spectra

map_{H}(−, X_{+}) ∧_{Or (H)} E ∘ G^{H} ⟶^{id ∧E(T)} map_{H}(−, X_{+}) ∧_{Or (H)} E ∘ G^{G} ∘ α

⟵^{≃} (α_{∗} map_{H}(−, X_{+})) ∧_{Or (G)} E ∘ G^{G} ⟵^{≃} map_{G}(−, ind_{α} X_{+}) ∧_{Or (G)} E ∘ G^{G}.

Here α_{∗} map_{H}(−, X_{+}) is the pointed Or(G)-space which is obtained from the pointed Or(H)-space map_{H}(−, X_{+}) by induction, i.e. by taking the balanced product over Or(H) with the Or(H)-Or(G) bimodule mor_{Or (G)}(??, α(?)) [82, Definition 1.8]. Notice that E ∘ G^{G} ∘ α is the same as the restriction of the Or(G)-spectrum E ∘ G^{G} along α which is often denoted by α^{∗}(E ∘ G^{G}) in the literature [82, Definition 1.8]. The second map is given by the adjunction homeomorphism of induction α_{∗} and restriction α^{∗} (see [82, Lemma 1.9]). The third map is the homeomorphism of Or(G)-spaces which is the adjoint of the obvious map of Or(H)-spaces map_{H}(−, X_{+}) → α^{∗} map_{G}(−, ind_{α} X_{+}) whose evaluation at H/L is given by ind_{α}. ⊓ ⊔ 6.3 K- and L-Theory Spectra over Groupoids

Let RINGS be the category of associative rings with unit. An involution on a R is a map R → R, r ↦ \overline{r} satisfying \overline{1} = 1, \overline{x + y} = \overline{x} + \overline{y} and \overline{x · y} = \overline{y} · \overline{x} for all x, y ∈ R. Let RINGS^{inv} be the category of rings with involution. Let C^{∗}-ALGEBRASbe the category of C^{∗}-algebras. There are classical functors for j ∈ −∞ ⨿ {j ∈ ℤ | j ≤ 2}

K: RINGS → SPECTRA ;  (31)
L^{⟨j⟩}: RINGS^{inv} → SPECTRA ;  (32)
K^{top} : C^{∗}-ALGEBRAS→ SPECTRA .  (33)

The construction of such a non-connective algebraic K-theory functor goes back to Gersten [133] and Wagoner [312]. The spectrum for quadratic algebraic L-theory is constructed by Ranicki in [258]. In a more geometric formulation it goes back to Quinn [250]. In the topological K-theory case a construction using Bott periodicity for C^{∗}-algebras can easily be derived from the Kuiper-Mingo Theorem (see [281, Section 2.2]). The homotopy groups of these spectra give the algebraic K-groups of Quillen (in high dimensions) and of Bass (in negative dimensions), the decorated quadratic L-theory groups, and the topological K-groups of C^{∗}-algebras.

We emphasize again that in all three cases we need the non-connective versions of the spectra, i.e. the homotopy groups in negative dimensions are non-trivial in general. For example the version of the Farrell-Jones Conjecture where one uses connective K-theory spectra is definitely false in general, compare Remark 1.15.

Now let us fix a coefficient ring R (with involution). Then sending a group G to the group ring RG yields functors R(−): GROUPS→ RINGS, respectively R(−): GROUPS→ RINGS^{inv}, where GROUPSdenotes the category of groups. Let GROUPS^{inj} be the category of groups with injective group homomorphisms as morphisms. Taking the reduced group C^{∗}-algebra defines a functor C_{r}^{∗}: GROUPS^{inj} → C^{∗}-ALGEBRAS. The composition of these functors with the functors (31), (32) and (33) above yields functors

KR(−): GROUPS→ SPECTRA ;  (34)
L^{⟨j⟩}R(−): GROUPS→ SPECTRA ;  (35)
K^{top}C_{r}^{∗}(−): GROUPS^{inj} → SPECTRA .  (36)

They satisfy

π_{n}(KR(G)) = K_{n}(RG);
π_{n}(L^{⟨j⟩}R(G)) = L_{n}^{⟨j⟩}(RG);
π_{n}(K^{top}C_{r}^{∗}(G)) = K_{n}(C_{r}^{∗}(G)),

for all groups G and n ∈ ℤ. The next result essentially says that these functors can be extended to groupoids.

Theorem 6.9 (K- and L-Theory Spectra over Groupoids). Let R be a ring (with involution). There exist covariant functors

K_{R}: GROUPOIDS → SPECTRA ;  (37)
L_{R}^{⟨j⟩}: GROUPOIDS → SPECTRA ;  (38)
K^{top} : GROUPOIDS^{inj} → SPECTRA  (39)

with the following properties:

(i) If F: G_{0} → G_{1} is an equivalence of (small) groupoids, then the induced maps K_{R}(F), L_{R}^{⟨j⟩}(F) and K^{top}(F) are weak equivalences of spectra.

(ii) Let I: GROUPS→ GROUPOIDS be the functor sending G to G considered as a groupoid, i.e. to G^{G}(G/G). This functor restricts to a functor GROUPS^{inj} → GROUPOIDS^{inj}. There are natural transformations from KR(−) to K_{R} ∘I, from L^{⟨j⟩}R(−) to L_{R}^{⟨j⟩} ∘I and from KC_{r}^{∗}(−) to K^{top} ∘I such that the evaluation of each of these natural transformations at a given group is an equivalence of spectra.

(iii) For every group G and all n ∈ ℤ we have

π_{n}(K_{R} ∘ I(G)) ≅ K_{n}(RG);
π_{n}(L_{R}^{⟨j⟩} ∘ I^{inv}(G)) ≅ L_{n}^{⟨j⟩}(RG);
π_{n}(K^{top} ∘ I(G)) ≅ K_{n}(C_{r}^{∗}(G)).

Proof. We only sketch the strategy of the proof. More details can be found in [82, Section 2].

Let G be a groupoid. Similar to the group ring RG one can define an R-linear category RG by taking the free R-modules over the morphism sets of G. Composition of morphisms is extended R-linearly. By formally adding finite direct sums one obtains an additive category RG_{⊕}. Pedersen-Weibel [237] (compare also [51]) define a non-connective algebraic K-theory functor which digests additive categories and can hence be applied to RG_{⊕}. For the comparison result one uses that for every ring R (in particular for RG) the Pedersen-Weibel functor applied to R_{⊕} (a small model for the category of finitely generated free R-modules) yields the non-connective K-theory of the ring R and that it sends equivalences of additive categories to equivalences of spectra. In the L-theory case RG_{⊕} inherits an involution and one applies the construction of [258, Example 13.6 on page 139] to obtain the L^{⟨1⟩} = L^{h}- version. The versions for j ≤ 1 can be obtained by a construction which is analogous to the Pedersen-Weibel construction for K-theory, compare [55, Section 4]. In the C^{∗}-case one obtains from G a C^{∗}-category C_{r}^{∗}(G) and assigns to it its topological K-theory spectrum. There is a construction of the topological K-theory spectrum of a C^{∗}-category in [82, Section 2]. However, the construction given there depends on two statements, which appeared in [130, Proposition 1 and Proposition 3], and those statements are incorrect, as already pointed out by Thomason in [302]. The construction in [82, Section 2] can easily be fixed but instead we recommend the reader to look at the more recent construction of Joachim [159]. ⊓ ⊔

6.4 Assembly Maps in Terms of Homotopy Colimits

In this section we describe a homotopy-theoretic formulation of the Baum-Connes and Farrell-Jones Conjectures. For the classical assembly maps which in our set-up correspond to the trivial family such formulations were described in [328].

For a group G and a family F of subgroups we denote by Or_{F}(G) the restricted orbit category. Its objects are homogeneous spaces G/H with H ∈ F. Morphisms are G-maps. If F = ALL we get back the (full) orbit category, i.e. Or(G) = Or_{ALL}(G).

Meta-Conjecture 6.10 (Homotopy-Theoretic Isomorphism Conjecture). Let G be a group and F a family of subgroups. Let E: Or(G) → SPECTRAbe a covariant functor. Then

A_{F}: hocolim_{Or _{F} (G)} E|_{Or _{F} (G)} → hocolim_{Or (G)} E ≃ E(G/G)

is a weak equivalence of spectra.

Here hocolim is the homotopy colimit of a covariant functor to spectra, which is itself a spectrum. The map A_{F} is induced by the obvious functor Or_{F}(G) → Or(G). The equivalence hocolim_{Or (G)} E ≃ E(G/G) comes from the fact that G/G is a final object in Or(G). For information about homotopycolimits we refer to [40], [82, Section 3] and [88].

Remark 6.11. If we consider the map on homotopy groups that is induced by the map A_{F} which appears in the Homotopy-Theoretic Isomorphism Conjecture above, then we obtain precisely the map with the same name in Meta-Conjecture 2.1 for the homology theory H_{∗}^{G}(−; E) associated with E in Proposition 6.7, compare [82, Section 5]. In particular the Baum-Connes Conjecture 2.3 and the Farrell-Jones Conjecture 2.2 can be seen as special cases of Meta-Conjecture 6.10.

Remark 6.12 (Universal Property of the Homotopy-Theoretic Assembly Map). The Homotopy-Theoretic Isomorphism Conjecture 6.10 is in some sense the most conceptual formulation of an Isomorphism Conjecture because it has a universal property as the universal approximation from the left by a (weakly) excisive F-homotopy invariant functor. This is explained in detail in [82, Section 6]. This universal property is important if one wants to identify different models for the assembly map, compare e.g. [19, Section 6] and [142].

6.5 Naturality under Induction

Consider a covariant functor E: GROUPOIDS → SPECTRAwhich respects equivalences. Let H_{∗}^{?}(−; E) be the associated equivariant homology theory (see Proposition 6.8). Then for a group homomorphism α: H → G and H-CW -pair (X, A) we obtain a homomorphism

ind_{α}: H_{n}^{H}(X, A; E) → H_{n}^{G}(ind_{α}(X, A); E)

which is natural in (X, A). Note that we did not assume that ker(α) acts freely on X. In fact the construction sketched in the proof of Proposition 6.8 still works even though ind_{α} may not be an isomorphism as it is the case if ker(α) acts freely. We still have functoriality as described in (ii) towards the beginning of Section 6.1.

Now suppose that H and G are families of subgroups for H and G such that α(K) ∈ G holds for all K ∈ H. Then we obtain a G-map f: ind_{α} E_{H}(H) → E_{G}(G) from the universal property of E_{G}(G). Let p: ind_{α} pt = G/α(H) → pt be the projection. Let I: GROUPS→ GROUPOIDS be the functor sending G to G^{G}(G/G). Then the following diagram, where the horizontal arrows are induced from the projections to the one point space, commutes for all n ∈ ℤ.

H_{n}^{H}(E_{H}(H); E) ⟶^{A_{H}} H_{n}^{H}(pt; E) = π_{n}(E(I(H)))
H_{n}^{G}(f)∘ind_{α}↓  ↓^{H_{n}^{G}(p)∘ind_{α}=π_{n}(E(I(α)))}
H_{n}^{G}(E_{G}(G); E) ⟶_{A_{G}} H_{n}^{G}(pt; E) = π_{n}(E(I(G))).

If we take the special case E = K_{R} and H = G = VCY, we get the following commutative diagram, where the horizontal maps are the assembly maps appearing in the Farrell-Jones Conjecture 2.2 and α_{∗} is the change of rings homomorphism (induction) associated to α.

H_{n}^{H}(E_{VCY}(H); K_{R}) ⟶^{A_{VCY}} K_{n}(RH)
H_{n}^{G}(f)∘ind_{α}↓  ↓^{α_{∗}}
H_{n}^{G}(E_{VCY}(G); K_{R}) ⟶_{A_{VCY}} K_{n}(RG).

We see that we can define a kind of induction homomorphism on the source of the assembly maps which is compatible with the induction structure given on their target. We get analogous diagrams for the L-theoretic version of the Farrell-Jones-Isomorphism Conjecture 2.2, for the Bost Conjecture 4.2 and for the Baum-Connes Conjecture for maximal group C^{∗}-algebras (see (22) in Subsection 4.1.2).

Remark 6.13. The situation for the Baum-Connes Conjecture 2.3 itself, where one has to work with reduced C^{∗}-algebras, is more complicated. Recall that not every group homomorphism α: H → G induces a homomorphisms of C^{∗}-algebras C_{r}^{∗}(H) → C_{r}^{∗}(G). (It does if ker(α) is finite.) But it turns out that the source H_{n}^{H}(E_{FIN} (H); K^{top}) always admits such a homomorphism. The point is that the isotropy groups of E_{FIN} (H) are all finite and the spectra-valued functor K^{top} extends from GROUPOIDS^{inj} to the category GROUPOIDS^{finker}, which has small groupoids as objects but as morphisms only those functors f: G_{0} → G_{1} with finite kernels (in the sense that for each object x ∈ G_{0} the group homomorphism aut_{G_{0}}(x) → aut_{G_{1}}(f(x)) has finite kernel). This is enough to get for any group homomorphism α: H → G an induced map ind_{α}: H_{n}^{H}(X, A; K^{top}) → H_{n}^{G}(ind_{α}(X, A); K^{top}) provided that X is proper. Hence one can define an induction homomorphism for the source of the assembly map as above.

In particular the Baum-Connes Conjecture 2.3 predicts that for any group homomorphism α: H → G there is an induced induction homomorphism α_{∗}: K_{n}(C_{r}^{∗}(H)) → K_{n}(C_{r}^{∗}(G)) on the targets of the assembly maps although there is no induced homomorphism of C^{∗}-algebras C_{r}^{∗}(H) → C_{r}^{∗}(G) in general.

7 Methods of Proof

In Chapter 2, we formulated the Baum-Connes Conjecture 2.2 and the Farrell-Jones Conjecture 2.3 in abstract homological terms. We have seen that this formulation was very useful in order to understand formal properties of assembly maps. But in order to actually prove cases of the conjectures one needs to interpret the assembly maps in a way that is more directly related to geometry or analysis. In this chapter we wish to explain such approaches to the assembly maps. We briefly survey some of the methods of proof that are used to attack the Baum-Connes Conjecture 2.3 and the Farrell-Jones Conjecture 2.2.

7.1 Analytic Equivariant K-Homology

Recall that the covariant functor K^{top} : GROUPOIDS^{inj} → SPECTRAintroduced in (39) defines an equivariant homology theory H_{∗}^{?}(−; K^{top}) in the sense of Section 6.1 such that

H_{n}^{G}(G/H; K^{top}) = H_{n}^{H}(pt; K^{top}) = \{ \begin{matrix}R(H) & for even n; \\ 0 & for odd n, \end{matrix}

holds for all groups G and subgroups H ⊆ G (see Proposition 6.8). Next we want to give for a proper G-CW -complex X an analytic definition of H_{n}^{G}(X; K^{top}).

Consider a locally compact proper G-space X. Recall that a G-space X is called proper if for each pair of points x and y in X there are open neighborhoods V_{x} of x and W_{y} of y in X such that the subset {g ∈ G | gV_{x}∩W_{y} ≠ ∅} of G is finite. A G-CW -complex X is proper if and only if all its isotropy groups are finite [197, Theorem 1.23]. Let C_{0}(X) be the C^{∗}-algebra of continuous functions f: X → ℂ which vanish at infinity. The C^{∗}-norm is the supremum norm. A generalized elliptic G-operator is a triple (U, ρ, F), which consists of a unitary representation U: G → B(H) of G on a Hilbert space H, a ∗- representation ρ: C_{0}(X) → B(H) such that ρ(f ∘ l_{g^{−1}}) = U(g) ∘ ρ(f) ∘ U(g)^{−1} holds for g ∈ G, and a bounded selfadjoint G-operator F: H → H such that the operators ρ(f)(F^{2} − 1) and [ρ(f), F] are compact for all f ∈ C_{0}(X). Here B(H) is the C^{∗}-algebra of bounded operators H → H, l_{g^{−1}} : H → H is given by multiplication with g^{−1}, and [ρ(f), F] = ρ(f) ∘ F − F ∘ ρ(f). We also call such a triple (U, ρ, F) an odd cycle. If we additionally assume that H comes with a ℤ/2-grading such that ρ preserves the grading if we equip C_{0}(X) with the trivial grading, and F reverses it, then we call (U, ρ, F) an even cycle. This means that we have an orthogonal decomposition H = H_{0} ⊕ H_{1} such that U, ρ and F look like

U = ( \begin{matrix}U_{0} & 0 \\ 0 & U_{1} \end{matrix} )  ρ = ( \begin{matrix}ρ_{0} & 0 \\ 0 & ρ_{1} \end{matrix} )  F = ( \begin{matrix}0 & P^{∗} \\ P & 0 \end{matrix} ) .  (40)

An important example of an even cocycle is described in Section 7.5. A cycle (U, ρ, f) is called degenerate, if for each f ∈ C_{0}(X) we have [ρ(f), F] = ρ(f)(F^{2} − 1) = 0. Two cycles (U_{0}, ρ_{0}, F_{0}) and (U_{1}, ρ_{1}, F_{1}) of the same parity are called homotopic, if U_{0} = U_{1}, ρ_{0} = ρ_{1} and there exists a norm continuous path F_{t}, t ∈ [0, 1] in B(H) such that for each t ∈ [0, 1] the triple (U_{0}, ρ_{0}, F_{t}) is again a cycle of the same parity. Two cycles (U_{0}, ρ_{0}, F_{0}) and (U_{1}, ρ_{1}, F_{1}) are called equivalent, if they become homotopic after taking the direct sum with degenerate cycles of the same parity. Let K_{n}^{G}(C_{0}(X)) for even n be the set of equivalence classes of even cycles and K_{n}^{G}(C_{0}(X)) for odd n be the set of equivalence classes of odd cycles. These become abelian groups by the direct sum. The neutral element is represented by any degenerate cycle. The inverse of an even cycle is represented by the cycle obtained by reversing the grading of H. The inverse of an odd cycle (U, ρ, F) is represented by (U, ρ, −F).

A proper G-map f: X → Y induces a map of C^{∗}-algebras C_{0}(f): C_{0}(Y) → C_{0}(X) by composition and thus in the obvious way a homomorphism of abelian groups K_{0}^{G}(f): K_{0}^{G}(C_{0}(X)) → K_{0}^{G}(C_{0}(Y)). It depends only on the proper G-homotopy class of f. One can show that this construction defines a G-homology theory on the category of finite proper G-CW -complexes. It extends to a G-homology theory K_{∗}^{G} for all proper G-CW -complexes by

K_{n}^{G}(X) = colim_{Y ∈I(X)} K_{n}^{G}(C_{0}(Y))  (41)

where I(X) is the set of proper finite G-CW -subcomplexes Y ⊆ X directed by inclusion. This definition is forced upon us by Lemma 2.7. The groups K_{n}^{G}(X) and K_{n}^{G}(C_{0}(X)) agree for finite proper G-CW -complexes, in general they are different.

The cycles were introduced by Atiyah [11]. The equivalence relation, the group structure and the homological properties of K_{n}^{G}(X) were established by Kasparov [168]. More information about analytic K-homology can be found in Higson-Roe [154].

7.2 The Analytic Assembly Map

For for every G-CW-complex X the projection pr: X → pt induces a map

H_{n}^{G}(X; K^{top}) → H_{n}^{G}(pt; K^{top}) = K_{n}(C_{r}^{∗}(G)).  (42)

In the case where X is the proper G-space E_{FIN} (G) we obtain the assembly map appearing in the Baum-Connes Conjecture 2.3. We explain its analytic analogue

ind_{G}: K_{n}^{G}(X) → K_{n}(C_{r}^{∗}(G)).  (43)

Note that we need to assume that X is a proper G-space since K_{n}^{G}(X) was only defined for such spaces. It suffices to define the map for a finite proper G-CW -complex X. In this case it assigns to the class in K_{n}^{G}(X) = K_{n}^{G}(C_{0}(X)) represented by a cycle (U, ρ, F) its G-index in K_{n}(C_{r}^{∗}(G)) in the sense of Mishencko-Fomenko [223]. At least in the simple case, where G is finite, we can give its precise definition. The odd K-groups vanish in this case and K_{0}(C_{r}^{∗}(G)) reduces to the complex representation ring R(G). If we write F in matrix form as in (40) then P: H → H is a G-equivariant Fredholm operator. Hence its kernel and cokernel are G-representations and the G-index of F is defined as [ker(P)] − [coker(P)] ∈ R(G). In the case of an infinite group the kernel and cokernel are a priori not finitely generated projective modules over C_{r}^{∗}(G), but they are after a certain pertubation. Moreover the choice of the pertubation does not affect [ker(P)] − [coker(P)] ∈ K_{0}(C_{r}^{∗}(G)).

The identification of the two assembly maps (42) and (43) has been carried out in Hambleton-Pedersen [142] using the universal characterization of the assembly map explained in [82, Section 6]. In particular for a proper G-CW - complex X we have an identification H_{n}^{G}(X; K^{top}) ≅ K_{n}^{G}(X). Notice that H_{n}^{G}(X; K^{top}) is defined for all G-CW -complexes, whereas K_{n}^{G}(X) has only been introduced for proper G-CW -complexes.

Thus the Baum-Connes Conjecture 2.3 gives an index-theoretic interpretations of elements in K_{0}(C_{r}^{∗}(G)) as generalized elliptic operators or cycles (U, ρ, F). We have explained already in Subsection 1.8.1 an application of this interpretation to the Trace Conjecture for Torsionfree Groups 1.37 and in Subsection 3.3.2 to the Stable Gromov-Lawson-Rosenberg Conjecture 3.23.

7.3 Equivariant KK-theory

Kasparov [170] developed equivariant KK-theory, which we will briefly explain next. It is one of the basic tools in the proofs of theorems about the Baum-Connes Conjecture 2.3.

A G-C^{∗}-algebra A is a C^{∗}-algebra with a G-action by ∗-automorphisms. To any pair of separable G-C^{∗}-algebras (A, B) Kasparov assigns abelian groups KK_{n}^{G}(A, B). If G is trivial, we write briefly KK_{n}(A, B). We do not give the rather complicated definition but state the main properties.

If we equip ℂ with the trivial G-action, then KK_{n}^{G}(C_{0}(X), ℂ) reduces to the abelian group K_{n}^{G}(C_{0}(X)) introduced in Section 7.1. The topological K-theory K_{n}(A) of a C^{∗}-algebra coincides with KK_{n}(ℂ, A). The equivariant KK-groups are covariant in the second and contravariant in the first variable under homomorphism of C^{∗}-algebras. One of the main features is the bilinear Kasparov product

KK_{i}^{G}(A, B) × KK_{j}^{G}(B, C) → KK_{i+j}(A, C), (α, β) ↦ α ⊗_{B} β. (44)

It is associative and natural. A homomorphism α: A → B defines an element in KK_{0}(A, B). There are natural descent homomorphisms

j_{G}: KK_{n}^{G}(A, B) → KK_{n}(A ⋊_{r} G, B ⋊_{r} G),  (45)

where A ⋊_{r} G and B ⋊_{r} G denote the reduced crossed product C^{∗}-algebras.

7.4 The Dirac-Dual Dirac Method

A G-C^{∗}-algebra A is called proper if there exists a locally compact proper G-space X and a G-homomorphism σ: C_{0}(X) → B(A), f ↦ σ_{f} satisfying σ_{f}(ab) = aσ_{f}(b) = σ_{f}(a)b for f ∈ C_{0}(X), a, b ∈ A and for every net {f_{i} | i ∈ I}, which converges to 1 uniformly on compact subsets of X, we have lim_{i∈I} ∥ σ_{f_{i}}(a) − a ∥ = 0 for all a ∈ A. A locally compact G-space X is proper if and only if C_{0}(X) is proper as a G-C^{∗}-algebra.

Given a proper G-CW -complex X and a G-C^{∗}-algebra A, we put

KK_{n}^{G}(X; A) = colim_{Y ∈I(X)} KK_{n}^{G}(C_{0}(Y), A),  (46)

where I(Y) is the set of proper finite G-CW -subcomplexes Y ⊆ X directed by inclusion. We have KK_{n}^{G}(X; ℂ) = K_{n}^{G}(X). There is an analytic index map

ind_{G}^{A}: KK_{n}^{G}(X; A) → K_{n}(A ⋊_{r} G),  (47)

which can be identified with the assembly map appearing in the Baum-Connes Conjecture with Coefficients 4.3. The following result is proved in Tu [305] extending results of Kasparov-Skandalis [169], [172].

Theorem 7.1. The Baum-Connes Conjecture with coefficients 4.3 holds for a proper G-C^{∗}-algebra A, i.e. ind_{G}^{A}: KK_{n}^{G}(E_{FIN} (G); A) → K_{n}(A ⋊ G) is bijective.

Now we are ready to state the Dirac-dual Dirac method which is the key strategy in many of the proofs of the Baum-Connes Conjecture 2.3 or the Baum-Connes Conjecture with coefficients 4.3.

Theorem 7.2 (Dirac-Dual Dirac Method). Let G be a countable (discrete) group. Suppose that there exist a proper G-C^{∗}-algebra A, elements α ∈ KK_{i}^{G}(A, ℂ), called the Dirac element, and β ∈ KK_{i}^{G}(ℂ, A), called the dual Dirac element, satisfying

β ⊗_{A} α = 1 ∈ KK_{0}^{G}(ℂ, ℂ).

Then the Baum-Connes Conjecture 2.3 is true, or, equivalently, the analytic index map

ind_{G}: K_{n}^{G}(X) → K_{n}(C_{r}^{∗}(G))

of 43 is bijective.

Proof. The index map ind_{G} is a retract of the bijective index map ind_{G}^{A} from Theorem 7.1. This follows from the following commutative diagram

\xymatrix{K_{n}^{G}(E_{FIN} (G)) \ar[d]^{_{ind_{G}}} \ar[r]^{−⊗_{ℂ}β} & KK_{n}^{G}(E_{FIN} (G); A) \ar[d]^{_{ind{A \atop G}}} \ar[r]^{−⊗_{A}α} & K_{n}^{G}(E_{FIN} (G)) \ar[d]^{_{ind_{G}}} \\ K_{n}(C_{r}^{∗}(G)) \ar[r]^{−⊗_{C∗ r(G)}j_{G}(β)} & K_{n}(A ⋊_{r} G) \ar[r]^{−⊗_{A⋊r} j_{G}(α)} & K_{n}(C_{r}^{∗}(G))}

and the fact that the composition of both the top upper horizontal arrows and lower upper horizontal arrows are bijective. ⊓ ⊔

7.5 An Example of a Dirac Element

In order to give a glimpse of the basic ideas from operator theory we briefly describe how to define the Dirac element α in the case where G acts by isometries on a complete Riemannian manifold M. Let T_{ℂ}M be the complexified tangent bundle and let Cliff(T_{ℂ}M) be the associated Clifford bundle. Let A be the proper G-C^{∗}-algebra given by the sections of Cliff(T_{ℂ}M) which vanish at infinity. Let H be the Hilbert space L^{2}(∧T_{ℂ}^{∗}M) of L^{2}-integrable differential forms on T_{ℂ}M with the obvious ℤ/2-grading coming from even and odd forms. Let U be the obvious G-representation on H coming from the G-action on M. For a 1-form ω on M and u ∈ H define a ∗-homomorphism ρ: A → B(H) by

ρ_{ω}(u) := ω ∧ u + i_{ω}(u).

Now D = (d + d^{∗}) is a symmetric densely defined operator H → H and defines a bounded selfadjoint operator F: H → H by putting F = \frac{D}{\sqrt{1+D^{2}}} . Then (U, ρ, F) is an even cocycle and defines an element α ∈ K_{0}^{G}(M) = KK_{0}^{G}(C_{0}(M), ℂ). More details of this construction and the construction of the dual Dirac element β under the assumption that M has non-positive curvature and is simply connected, can be found for instance in [307, Chapter 7.6 Banach KK-Theory

Skandalis showed that the Dirac-dual Dirac method cannot work for all groups [287] as long as one works with KK-theory in the unitary setting. The problem is that for a group with property (T) the trivial and the regular unitary representation cannot be connected by a continuous path in the space of unitary representations, compare also the discussion in [163]. This problem can be circumvented if one drops the condition unitary and works with a variant of KK-theory for Banach algebras as worked out by Lafforgue [183], [185], [186].

7.7 Controlled Topology and Algebra

To a topological problem one can often associate a notion of “size”. We describe a prototypical example. Let M be a Riemannian manifold. Recall that an h-cobordism W over M = ∂^{−}W admits retractions r^{±}: W × I → W, (x, t) ↦ r_{t}^{±}(x, t) which retract W to ∂^{±}W, i.e. which satisfy r_{0}^{±} = id_{W} and r_{1}^{±}(W) ⊂ ∂^{±}W. Given ϵ > 0 we say that W is ϵ-controlled if the retractions can be chosen in such a way that for every x ∈ W the paths (called tracks of the h-cobordism) p_{x}^{±}: I → M, t ↦ r_{1}^{−} ∘ r_{t}^{±}(x) both lie within an ϵ-neighbourhood of their starting point. The usefulness of this concept is illustrated by the following theorem [124].

Theorem 7.3. Let M be a compact Riemannian manifold of dimension ≥ 5. Then there exists an ϵ = ϵ_{M} > 0, such that every ϵ-controlled h-cobordism over M is trivial.

If one studies the s-Cobordism Theorem 1.5 and its proof one is naturally lead to algebraic analogues of the notions above. A (geometric) R-module over the space X is by definition a family M = (M_{x})_{x∈X} of free R-modules indexed by points of X with the property that for every compact subset K ⊂ X the module ⊕_{x∈K}M_{x} is a finitely generated R-module. A morphism ϕ from M to N is an R-linear map ϕ = (ϕ_{y,x}): ⊕_{x∈X} M_{x} → ⊕_{y∈X}N_{y}. Instead of specifying fundamental group data by paths (analogues of the tracks of the h-cobordism) one can work with modules and morphisms over the universal covering X̃, which are invariant under the operation of the fundamental group G = π_{1}(X) via deck transformations, i.e. we require that M_{gx} = M_{x} and ϕ_{gy,gx} = ϕ_{y,x}. Such modules and morphisms form an additive category which we denote by C^{G}(X̃; R). In particular one can apply to it the non-connective K-theory functor K (compare [237]). In the case where X is compact the category is equivalent to the category of finitely generated free RG-modules and hence π_{∗}KC^{G}(X̃; R) ≅ K_{∗}(RG). Now suppose X̃ is equipped with a G-invariant metric, then we will say that a morphism ϕ = (ϕ_{y,x}) is ϵ-controlled if ϕ_{y,x} = 0, whenever x and y are further than ϵ apart. (Note that ϵ-controlled morphisms do not form a category because the composition of two such morphisms will in general be 2ϵ-controlled.)

Theorem 7.3 has the following algebraic analogue [251] (see also Section 4 in [240]).

Theorem 7.4. Let M be a compact Riemannian manifold with fundamental group G. There exists an ϵ = ϵ_{M} > 0 with the following property. The K_{1}- class of every G-invariant automorphism of modules over M̃ which together with its inverse is ϵ-controlled lies in the image of the classical assembly map

H_{1}(BG; KR) → K_{1}(RG) ≅ K_{1}(C^{G}(M̃; R)).

To understand the relation to Theorem 7.3 note that for R = ℤ such an ϵ-controlled automorphism represents the trivial element in the Whitehead group which is in bijection with the h-cobordisms over M, compare Theorem 1.5.

There are many variants to the simple concept of “metric ϵ-control” we used above. In particular it is very useful to not measure size directly in M but instead use a map p: M → X to measure size in some auxiliary space X. (For example we have seen in Subsection 1.2.3 and 1.4.2 that “bounded” control over ℝ^{k} may be used in order to define or describe negative K-groups.)

Before we proceed we would like to mention that there are analogous control-notions for pseudoisotopies and homotopy equivalences. The tracks of a pseudoisotopy f: M × I → M × I are defined as the paths in M which are given by the composition

\xymatrix{p_{x}: I = \{x\} × I ⊂ M × I \ar[r]^{f} & M × I \ar[r]^{p} & M}

for each x ∈ M, where the last map is the projection onto the M-factor. Suppose f: N → M is a homotopy equivalence, g: M → N its inverse and h_{t} and h′_{t} are homotopies from f ∘ g to id_{M} respectively from g ∘ f to id_{N} then the tracks are defined to be the paths in M that are given by t ↦ h_{t}(x) for x ∈ M and t ↦ f ∘ h′_{t}(y) for y ∈ N. In both cases, for pseudoisotopies and for homotopy equivalences, the tracks can be used to define ϵ-control.

7.8 Assembly as Forget Control

If instead of a single problem over M one defines a family of problems over M × [1, ∞) and requires the control to tend to zero for t → ∞ in a suitable sense, then one obtains something which is a homology theory in M. Relaxing the control to just bounded families yields the classical assembly map. This idea appears in [252] in the context of pseudoisotopies and in a more categorical fashion suitable for higher algebraic K-theory in [55] and [241]. We spell out some details in the case of algebraic K-theory, i.e. for geometric modules.

Let M be a Riemannian manifold with fundamental group G and let S(1/t) be the space of all functions [1, ∞) → [0, ∞), t ↦ δ_{t} such that t ↦ t · δ_{t} is bounded. Similarly let S(1) be the space of all functions t ↦ δ_{t} which are bounded. Note that S(1/t) ⊂ S(1). A G-invariant morphism ϕ over M̃×[1, ∞) is S-controlled for S = S(1) or S(1/t) if there exists an α > 0 and a δ_{t} ∈ S (both depending on the morphism) such that ϕ_{(x,t),(x′,t′)} ≠ 0 implies that |t − t′| ≤ α and dM̃(x, x′) ≤ δ_{min{t,t′}}. We denote by C^{G}(M̃× [1, ∞), S; R) the category of all S-controlled morphisms. Furthermore C^{G}(M̃ × [1, ∞), S; R)^{∞} denotes the quotient category which has the same objects, but where two morphisms are identified, if their difference factorizes over an object which lives over M̃ × [1, N] for some large but finite number N. This passage to the quotient category is called “taking germs at infinity”. It is a special case of a Karoubi quotient, compare [51].

Theorem 7.5 (Classical Assembly as Forget Control). Suppose M is aspherical, i.e. M is a model for BG, then for all n ∈ ℤ the map

π_{n}(KC^{G}(M̃ × [1, ∞), S(1/t); R)^{∞}) → π_{n}(KC^{G}(M̃ × [1, ∞), S(1); R)^{∞})

can be identified up to an index shift with the classical assembly map that appears in Conjecture 1.11, i.e. with

H_{n−1}(BG; K(R)) → K_{n−1}(RG).

Note that the only difference between the left and the right hand side is that on the left morphism are required to become smaller in a 1/t-fashion, whereas on the right hand side they are only required to stay bounded in the [1, ∞)-direction.

Using so called equivariant continuous control (see [7] and [19, Section 2] for the equivariant version) one can define an equivariant homology theory which applies to arbitrary G-CW-complexes. This leads to a “forget-control description” for the generalized assembly maps that appear in the Farrell-Jones Conjecture 2.2. Alternatively one can use stratified spaces and stratified Riemannian manifolds in order to describe generalized assembly maps in terms of metric control. Compare [111, 3.6 on p.270] and [252, Appendix].

7.9 Methods to Improve Control

From the above description of assembly maps we learn that the problem of proving surjectivity results translates into the problem of improving control. A combination of many different techniques is used in order to achieve such control-improvements. We discuss some prototypical arguments which go back to [98] and [102] and again restrict attention to K-theory. Of course this can only give a very incomplete impression of the whole program which is mainly due to Farrell-Hsiang and Farrell-Jones. The reader should consult [120] and [160] for a more detailed survey.

We restrict to the basic case, where M is a compact Riemannian manifold with negative sectional curvature. In order to explain a contracting property of the geodesic flow Φ: ℝ × SM̃ → SM̃ on the unit sphere bundle SM̃, we introduce the notion of foliated control. We think of SM̃ as a manifold equipped with the one-dimensional foliation by the flow lines of Φ and equip it with its natural Riemannian metric. Two vectors v and w in SM̃ are called foliated (α, δ)-controlled if there exists a path of length α inside one flow line such that v lies within distance δ/2 of the starting point of that path and w lies within distance δ/2 of its endpoint.

Two vectors v and w ∈ SM̃ are called asymptotic if the distance between their associated geodesic rays is bounded. These rays will then determine the same point on the sphere at infinity which can be introduced to compactify M̃ to a disk. Recall that the universal covering of a negatively curved manifold is diffeomorphic to ℝ^{n}. Suppose v and w are α-controlled asymptotic vectors, i.e. their distance is smaller than α > 0. As a consequence of negative sectional curvature the vectors Φ_{t}(v) and Φ_{t}(w) are foliated (Cα, δ_{t})-controlled, where C > 1 is a constant and δ_{t} > 0 tends to zero when t tends to ∞. So roughly speaking the flow contracts the directions transverse to the flow lines and leaves the flow direction as it is, at least if we only apply it to asymptotic vectors.

This property can be used in order to find foliated (α, δ)-controlled representatives of K-theory classes with arbitrary small δ if one is able to define a suitable transfer from M to SM̃, which yields representatives whose support is in an asymptotic starting position for the flow. Here one needs to take care of the additional problem that in general such a transfer may not induce an isomorphism in K-theory.

Finally one is left with the problem of improving foliated control to ordinary control. Corresponding statements are called “Foliated Control Theorems”. Compare [18], [101], [103], [104] and [108].

If such an improvement were possible without further hypothesis, we could prove that the classical assembly map, i.e. the assembly map with respect to the trivial family is surjective. We know however that this is not true in general. It fails for example in the case of topological pseudoisotopies or for algebraic K-theory with arbitrary coefficients. In fact the geometric arguments that are involved in a “Foliated Control Theorem” need to exclude flow lines in SM̃ which correspond to “short” closed geodesic loops in SM. But the techniques mentioned above can be used in order to achieve ϵ-control for arbitrary small ϵ > 0 outside of a suitably chosen neighbourhood of “short” closed geodesics. This is the right kind of control for the source of the assembly map which involves the family of cyclic subgroups. (Note that a closed a loop in M determines the conjugacy class of a maximal infinite cyclic subgroup inside G = π_{1}(M).) We see that even in the torsionfree case the class of cyclic subgroups of G naturally appears during the proof of a surjectivity result.

Another source for processes which improve control are expanding selfmaps. Think for example of an n-torus ℝ^{n}/ℤ^{n} and the self-map f_{s} which is induced by m_{s}: ℝ^{n} → ℝ^{n}, x → sx for a large positive integer s. If one pulls an automorphism back along such a map one can improve control, but unfortunately the new automorphism describes a different K-theory class.

Additional algebraic arguments nevertheless make this technique very successful. Compare for example [98]. Sometimes a clever mixture between flows and expanding self-maps is needed in order to achieve the goal, compare [105]. Recent work of Farrell-Jones (see [114], [115], [116] and [162]) makes use of a variant of the Cheeger-Fukaya-Gromov collapsing theory.

Remark 7.6 (Algebraicizing the Farrell-Jones Approach). In this Subsection we sketched some of the geometric ideas which are used in order to obtain control over an h-cobordism, a pseudisotopy or an automorphism of a geometric module representing a single class in K_{1}. In Subsection 7.8 we used families over the cone M × [1, ∞) in order to described the whole algebraic K-theory assembly map at once in categorical terms without ever referring to a single K-theory element. The recent work [21] shows that the geometric ideas can be adapted to this more categorical set-up, at least in the case where the group is the fundamental group of a Riemannian manifold with strictly negative curvature. However serious difficulties had to be overcome in order to achieve this. One needs to formulate and prove a Foliated Control Theorem in this context and also construct a transfer map to the sphere bundle for higher K-theory which is in a suitable sense compatible with the control structures.

7.10 The Descent Principle

In Theorem 7.5 we described the classical assembly map as a forget control map using G-invariant geometric modules over M̃ × [1, ∞). If in that context one does not require the modules and morphisms to be invariant under the G-action one nevertheless obtains a forget control functor between additive categories for which we introduce the notation

D(1/t) = C(M̃ × [1, ∞), S(1/t); R)^{∞} → D(1) = C(M̃ × [1, ∞), S(1); R)^{∞}.

Applying K-theory yields a version of a “coarse” assembly map which is the algebraic K-theory analogue of the map described in Section 4.1.5. A crucial feature of such a construction is that the left hand side can be interpreted as a locally finite homology theory evaluated on M̃. It is hence an invariant of the proper homotopy type of M̃. Compare [7] and [326]. It is usually a lot easier to prove that this coarse assembly map is an equivalence. Suppose for example that M has non-positive curvature, choose a point x_{0} ∈ M (this will destroy the G-invariance) and with increasing t ∈ [1, ∞) move the modules along geodesics towards x_{0}. In this way one can show that the coarse assembly map is an isomorphism. Such coarse assembly maps exist also in the context of algebraic L-theory and topological K-theory, compare [151], [263].

Results about these maps (compare e.g. [17], [55], [336], [338]) lead to injectivity results for the classical assembly map by the “descent principle” (compare [52], [55], [263]) which we will now briefly describe in the context of algebraic K-theory. (We already stated an analytic version in Section 4.1.5.) For a spectrum E with G-action we denote by E^{hG} the homotopy fixed points.

Since there is a natural map from fixed points to homotopy fixed points we obtain a commutative diagram

\xymatrix{K(D(1/t))^{G} \ar[r] \ar[d] & K(D(1))^{G} \ar[d] \\ K(D(1/t))^{hG} \ar[r] & K(D(1))^{hG}.}

If one uses a suitable model K-theory commutes with taking fixed points and hence the upper horizontal map can be identified with the classical assembly map by Theorem 7.5. Using that K-theory commutes with infinite products [54], one can show by an induction over equivariant cells, that the vertical map on the left is an equivalence. Since we assume that the map K(D(1/t)) → K(D(1)) is an equivalence, a standard property of the homotopy fixed point construction implies that the lower horizontal map is an equivalence. It follows that the upper horizontal map and hence the classical assembly map is split injective. A version of this argument which involves the assembly map for the family of finite subgroups can be found in [271].

7.11 Comparing to Other Theories

Every natural transformation of G-homology theories leads to a comparison between the associated assembly maps. For example one can compare topological K-theory to periodic cyclic homology [72], i.e. for every Banach algebra completion A(G) of ℂG inside C_{r}^{∗}(G) there exists a commutative diagram

\xymatrix{K_{∗}(BG) \ar[d] \ar[r] & K_{∗}(A(G)) \ar[d] \\ H_{∗}(BG; HP_{∗}(ℂ)) \ar[r] & HP_{∗}(A(G)).}

This is used in [72] to prove injectivity results for word hyperbolic groups. Similar diagrams exist for other cyclic theories (compare for example [246]).

A suitable model for the cyclotomic trace trc: K_{n}(RG) → T C_{n}(RG) from (connective) algebraic K-theory to topological cyclic homology [38] leads for every family F to a commutative diagram

\xymatrix{H_{n}(E_{F}(G); K_{ℤ}^{con}) \ar[d] \ar[r] & K_{n}^{con}(ℤG) \ar[d] \\ H_{n}(E_{F}(G); TC_{ℤ}) \ar[r] & T C_{n}(ℤG).}

Injectivity results about the left hand and the lower horizontal map lead to injectivity results about the upper horizontal map. This is the principle behind Theorem 5.23 and 5.24.

8 Computations

Our ultimate goal is to compute K- and L-groups such as K_{n}(RG), L_{n}^{⟨−∞⟩}(RG) and K_{n}(C_{r}^{∗}(G)). Assuming that the Baum-Connes Conjecture 2.3 or the Farrell-Jones Conjecture 2.2 is true, this reduces to the computation of the left hand side of the corresponding assembly map, i.e. to H_{n}^{G}(E_{FIN} (G); K^{top}), H_{n}^{G}(E_{VCY}(G); K_{R}) and H_{n}^{G}(E_{VCY}(G); L_{R}^{⟨−∞⟩}). This is much easier since here we can use standard methods from algebraic topology such as spectral sequences, Mayer-Vietoris sequences and Chern characters. Nevertheless such computations can be pretty hard. Roughly speaking, one can obtain a general reasonable answer after rationalization, but integral computations have only been done case by case and no general pattern is known.

8.1 K- and L- Groups for Finite Groups

In all these computations the answer is given in terms of the values of K_{n}(RG), L_{n}^{⟨−∞⟩}(RG) and K_{n}(C_{r}^{∗}(G)) for finite groups G. Therefore we briefly recall some of the results known for finite groups focusing on the case R = ℤ

8.1.1 Topological K-Theory for Finite Groups

Let G be a finite group. By r_{F} (G), we denote the number of isomorphism classes of irreducible representations of G over the field F. By r_{ℝ}(G; ℝ), r_{ℝ}(G; ℂ), respectively r_{ℝ}(G; ℍ) we denote the number of isomorphism classes of irreducible real G-representations V, which are of real, complex respectively of quaternionic type, i.e. aut_{ℝG}(V) is isomorphic to the field of real numbers ℝ, complex numbers ℂ or quaternions ℍ. Let RO(G) respectively R(G) be the real respectively the complex representation ring.

Notice that ℂG = l^{1}(G) = C_{r}^{∗}(G) = C_{max}^{∗}(G) holds for a finite group, and analogous for the real versions.

Proposition 8.1. Let G be a finite group.

(i) We have

K_{n}(C_{r}^{∗}(G)) ≅ \{ \begin{matrix}R(G) & ≅ & ℤ^{r_{ℂ}(G)} & for n even; \\ 0 & & & for n odd. \end{matrix}

(ii) There is an isomorphism of topological K-groups

K_{n}(C_{r}^{∗}(G; ℝ)) ≅ K_{n}(ℝ)^{r_{ℝ}(G;ℝ)} × K_{n}(ℂ)^{r_{ℝ}(G;ℂ)} × K_{n}(ℍ)^{r_{ℝ}(G;ℍ)}.

Moreover K_{n}(ℂ) is 2-periodic with values ℤ, 0 for n = 0, 1, K_{n}(ℝ) is 8-periodic with values ℤ, ℤ/2, ℤ/2, 0, ℤ, 0, 0, 0 for n = 0, 1, …, 7 and K_{n}(ℍ) = K_{n+4}(ℝ) for n ∈ ℤ.

Proof. One gets isomorphisms of C^{∗}-algebras

C_{r}^{∗}(G) ≅ ∏_{j=1}^{r_{ℂ}(G)} M_{n_{i}}(ℂ)

and

C_{r}^{∗}(G; ℝ) ≅ ∏_{i=1}^{r_{ℝ}(G;ℝ)} M_{m_{i}}(ℝ) × ∏_{i=1}^{r_{ℝ}(G;ℂ)} M_{n_{i}}(ℂ) × ∏_{i=1}^{r_{ℝ}(G;ℍ)} M_{p_{i}}(ℍ)

from [283, Theorem 7 on page 19, Corollary 2 on page 96, page 102, page 106]. Now the claim follows from Morita invariance and the well-known values for K_{n}(ℝ), K_{n}(ℂ) and K_{n}(ℍ) (see for instance [301, page 216]). ⊓ ⊔

To summarize, the values of K_{n}(C_{r}^{∗}(G)) and K_{n}(C_{r}^{∗}(G; ℝ)) are explicitly known for finite groups G and are in the complex case in contrast to the real case always torsion free.

8.1.2 Algebraic K-Theory for Finite Groups

Here are some facts about the algebraic K-theory of integral group rings of finite groups.

Proposition 8.2. Let G be a finite group.

(i) K_{n}(ℤG) = 0 for n ≤ −2. (ii) We have

K_{−1}(ℤG) ≅ ℤ^{r} ⊕ (ℤ/2)^{s},

where

r = 1 − r_{ℚ}(G) + ∑_{p | |G|} r_{ℚ_{p}}(G) − r_{𝔽_{p}}(G)

and the sum runs over all primes dividing the order of G. (Recall that r_{F} (G) denotes the number of isomorphism classes of irreducible representations of G over the field F.) There is an explicit description of the integer s in terms of global and local Schur indices [58]. If G contains a normal abelian subgroup of odd index, then s = 0.

(iii) The group K̃_{0}(ℤG) is finite. (iv) The group Wh(G) is a finitely generated abelian group and its rank is

r_{ℝ}(G) − r_{ℚ}(G).

(v) The groups K_{n}(ℤG) are finitely generated for all n ∈ ℤ. (vi) We have K_{−1}(ℤG) = 0, K̃_{0}(ℤG) = 0 and Wh(G) = 0 for the following

finite groups G = {1}, ℤ/2, ℤ/3, ℤ/4, ℤ/2 × ℤ/2, D_{6}, D_{8}, where D_{m} is the dihedral group of order m. If p is a prime, then K_{−1}(ℤ[ℤ/p]) = K_{−1}(ℤ[ℤ/p × ℤ/p]) = 0. We have

K_{−1}(ℤ[ℤ/6]) ≅ ℤ, K̃_{0}(ℤ[ℤ/6]) = 0, Wh(ℤ/6) = 0
K_{−1}(ℤ[D_{12}]) ≅ ℤ, K̃_{0}(ℤ[D_{12}]) = 0, Wh(D_{12}) = 0.

(vii) Let Wh_{2}(G) denote the cokernel of the assembly map

H_{2}(BG; K(ℤ)) → K_{2}(ℤG).

We have Wh_{2}(G) = 0 for G = {1}, ℤ/2, ℤ/3 and ℤ/4. Moreover | Wh_{2}(ℤ/6)| ≤ 2, | Wh_{2}(ℤ/2 × ℤ/2)| ≥ 2 and Wh_{2}(D_{6}) = ℤ/2.

Proof. (i) and (ii) are proved in [58]. (iii) is proved in [298, Proposition 9.1 on page 573]. (iv) This is proved for instance in [232].

(v) See [181], [248]. (vi) and (vii) The computation K_{−1}(ℤG) = 0 for G = ℤ/p or ℤ/p × ℤ/p can be found in [22, Theorem 10.6, p. 695] and is a special case of [58].

The vanishing of K̃_{0}(ℤG) is proven for G = D_{6} in [262, Theorem 8.2] and for G = D_{8} in [262, Theorem 6.4]. The cases G = ℤ/2, ℤ/3, ℤ/4, ℤ/6, and (ℤ/2)^{2} are treated in [79, Corollary 5.17]. Finally, K̃_{0}(ℤD_{12}) = 0 follows from [79, Theorem 50.29 on page 266] and the fact that ℚD_{12} as a ℚ-algebra splits into copies of ℚ and matrix algebras over ℚ, so its maximal order has vanishing class group by Morita equivalence.

The claims about Wh_{2}(ℤ/n) for n = 2, 3, 4, 6 and for Wh_{2}((ℤ/2)^{2}) are taken from [85, Proposition 5], [89, p.482] and [296, p. 218 and 221].

We get K_{2}(ℤD_{6}) ≅ (ℤ/2)^{3} from [296, Theorem 3.1]. The assembly map H_{2}(Bℤ/2; K(ℤ)) → K_{2}(ℤ[ℤ/2]) is an isomorphism by [89, Theorem on p. 482]. Now construct a commutative diagram

H_{2}(Bℤ/2; K(ℤ)) ⟶^{≅} H_{2}(BD_{6}; K(ℤ))
≅↓  ↓
K_{2}(ℤ[ℤ/2]) ⟶ K_{2}(ℤD_{6})

whose lower horizontal arrow is split injective and whose upper horizontal arrow is an isomorphism by the Atiyah-Hirzebruch spectral sequence. Hence the right vertical arrow is split injective and Wh_{2}(D_{6}) = ℤ/2. ⊓ ⊔

Let us summarize. We already mentioned that a complete computation of K_{n}(ℤ) is not known. Also a complete computation of K̃_{0}(ℤ[ℤ/p]) for arbitrary primes p is out of reach (see [221, page 29,30]). There is a complete formula for K_{−1}(ℤG) and K_{n}(ℤG) = 0 for n ≤ −2 and one has a good understanding of Wh(G) (see [232]). We have already mentioned Borel’s formula for K_{n}(ℤ)⊗_{ℤ}ℚ for all n ∈ ℤ (see Remark 1.14). For more rational information see also 8.9.

8.1.3 Algebraic L-Theory for Finite Groups

Here are some facts about L-groups of finite groups.

Proposition 8.3. Let G be a finite group. Then

(i) For each j ≤ 1 the groups L_{n}^{⟨j⟩}(ℤG) are finitely generated as abelian groups and contain no p-torsion for odd primes p. Moreover, they are finite for odd n.

(ii) For every decoration ⟨j⟩ we have

L_{n}^{⟨j⟩}(ℤG)[1/2] ≅ L_{n}^{⟨j⟩}(ℝG)[1/2] ≅ \{ \begin{matrix}ℤ[1/2]^{r_{ℝ}(G)} & n ≡ 0 (4); \\ ℤ[1/2]^{r_{ℂ}(G)} & n ≡ 2 (4); \\ 0 & n ≡ 1, 3 (4). \end{matrix}

(iii) If G has odd order and n is odd, then L_{n}^{ϵ}(ℤG) = 0 for ϵ = p, h, s.

Proof. (i) See for instance [143]. (ii) See [258, Proposition 22.34 on page 253]. (iii) See [13] or [143]. ⊓ ⊔

The L-groups of ℤG are pretty well understood for finite groups G. More information about them can be found in [143].

8.2 Rational Computations for Infinite Groups

Next we state what is known rationally about the K- and L-groups of an infinite (discrete) group, provided the Baum-Connes Conjecture 2.3 or the relevant version of the Farrell-Jones Conjecture 2.2 is known.

In the sequel let (FCY) be the set of conjugacy classes (C) for finite cyclic subgroups C ⊆ G. For H ⊆ G let N_{G}H = {g ∈ G | gHg^{−1} = H} be its normalizer, let Z_{G}H = {g ∈ G | ghg^{−1} = h for h ∈ H} be its centralizer, and put

W_{G}H := N_{G}H/(H · Z_{G}H),

where H · Z_{G}H is the normal subgroup of N_{G}H consisting of elements of the form hu for h ∈ H and u ∈ Z_{G}H. Notice that W_{G}H is finite if H is finite.

Recall that the Burnside ring A(G) of a finite group is the Grothendieck group associated to the abelian monoid of isomorphism classes of finite Gsets with respect to the disjoint union. The ring multiplication comes from the cartesian product. The zero element is represented by the empty set, the unit is represented by G/G = pt. For a finite group G the abelian groups K_{q}(C_{r}^{∗}(G)), K_{q}(RG) and L^{⟨−∞⟩}(RG) become modules over A(G) because these functors come with a Mackey structure and [G/H] acts by ind_{H}^{G} ∘ res_{G}^{H}.

We obtain a ring homomorphism

χ^{G}: A(G) → ∏_{(H)∈FIN} ℤ, [S] ↦ (|S^{H}|)_{(H)∈FIN}

which sends the class of a finite G-set S to the element given by the cardinalities of the H-fixed point sets. This is an injection with finite cokernel. This leads to an isomorphism of ℚ-algebras

χ_{ℚ}^{G} := χ^{G} ⊗_{ℤ} id_{ℚ}: A(G) ⊗_{ℤ} ℚ ⟶^{≅} ∏_{(H)∈(FIN)} ℚ.  (48)

For a finite cyclic group C let

θ_{C} ∈ A(C) ⊗_{ℤ} ℤ[1/|C|]  (49)

be the element which is sent under the isomorphism χ_{ℚ}^{C}: A(C) ⊗_{ℤ} ℚ ⟶^{≅} ∏_{(H)∈FIN} ℚ of (48) to the element, whose entry is one if (H) = (C) and is zero if (H) ≠ (C). Notice that θ_{C} is an idempotent. In particular we get a direct summand θ_{C} · K_{q}(C_{r}^{∗}(C)) ⊗_{ℤ} ℚ in K_{q}(C_{r}^{∗}(C)) ⊗_{ℤ} ℚ and analogously for K_{q}(RC) ⊗_{ℤ} ℚ and L^{⟨−∞⟩}(RG) ⊗_{ℤ} ℚ.

8.2.1 Rationalized Topological K-Theory for Infinite Groups

The next result is taken from [203, Theorem 0.4 and page 127]. Recall that Λ^{G} is the ring ℤ ⊆ Λ^{G} ⊆ ℚ which is obtained from ℤ by inverting the orders of the finite subgroups of G.

Theorem 8.4 (Rational Computation of Topological K-Theory for Infinite Groups). Suppose that the group G satisfies the Baum-Connes Conjecture 2.3. Then there is an isomorphism

⨁_{p+q=n} ⨁_{(C)∈(FCY)} K_{p}(BZ_{G}C) ⊗_{ℤ[W_{G}C]} θ_{C} · K_{q}(C_{r}^{∗}(C)) ⊗_{ℤ} Λ^{G}

⟶^{≅} K_{n}(C_{r}^{∗}(G)) ⊗_{ℤ} Λ^{G}.

If we tensor with ℚ, we get an isomorphism

⨁_{p+q=n} ⨁_{(C)∈(FCY)} H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · K_{q}(C_{r}^{∗}(C)) ⊗_{ℤ} ℚ.

⟶^{≅} K_{n}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ.

8.2.2 Rationalized Algebraic K-Theory for Infinite Groups

Recall that for algebraic K-theory of the integral group ring we know because of Proposition 2.17 that in the Farrell-Jones Conjecture we can reduce to the family of finite subgroups. A reduction to the family of finite subgroups also works if the coefficient ring is a regular ℚ-algebra, compare 2.14. The next result is a variation of [201, Theorem 0.4].

Theorem 8.5 (Rational Computation of Algebraic K-Theory). Suppose that the group G satisfies the Farrell-Jones Conjecture 2.2 for the algebraic K-theory of RG, where either R = ℤ or R is a regular ring with ℚ ⊂ R. Then we get an isomorphism

⨁_{p+q=n} ⨁_{(C)∈(FCY)} H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · K_{q}(RC) ⊗_{ℤ} ℚ

⟶^{≅} K_{n}(RG) ⊗_{ℤ} ℚ.

Remark 8.6. If in Theorem 8.5 we assume the Farrell-Jones Conjecture for the algebraic K-theory of RG but make no assumption on the coefficient ring R, then we still obtain that the map appearing there is split injective.

Example 8.7 (The Comparison Map from Algebraic to Topological K-theory). If we consider R = ℂ as coefficient ring and apply −⊗_{ℤ}ℂ instead of − ⊗_{ℤ} ℚ , the formulas simplify. Suppose that G satisfies the Baum-Connes Conjecture 2.3 and the Farrell-Jones Conjecture 2.2 for algebraic K-theory with ℂ as coefficient ring. Recall that con(G)_{f} is the set of conjugacy classes (g) of elements g ∈ G of finite order. We denote for g ∈ G by ⟨g⟩ the cyclic subgroup generated by g.

Then we get the following commutative square, whose horizontal maps are isomorphisms and whose vertical maps are induced by the obvious change of theory homorphisms (see [201, Theorem 0.5])

⨁_{p+q=n} ⨁_{(g)∈con(G)_{f}} H_{p}(Z_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}(ℂ) ⟶^{≅} K_{n}(ℂG) ⊗_{ℤ} ℂ
↓  ↓
⨁_{p+q=n} ⨁_{(g)∈con(G)_{f}} H_{p}(Z_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}^{top}(ℂ) ⟶_{≅} K_{n}(C_{r}^{∗}(G)) ⊗_{ℤ} ℂ

The Chern character appearing in the lower row of the commutative square above has already been constructed by different methods in [26]. The construction in [201] works also for ℚ (and even smaller rings) and other theories like algebraic K- and L-theory. This is important for the proof of Theorem 3.22 and to get the commutative square above.

Example 8.8 (A Formula for K_{0}(ℤG) ⊗_{ℤ} ℚ). Suppose that the Farrell-Jones Conjecture is true rationally for K_{0}(ℤG), i.e. the assembly map

A_{VCY}: H_{0}^{G}(E_{VCY}(G); K_{ℤ}) ⊗_{ℤ} ℚ → K_{0}(ℤG) ⊗_{ℤ} ℚ

is an isomorphism. Then we obtain

K_{0}(ℤG) ⊗_{ℤ} ℚ ≅
K_{0}(ℤ) ⊗_{ℤ} ℚ ⊕ ⨁_{(C)∈(FCY)} H_{1}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · K_{−1}(RC) ⊗_{ℤ} ℚ .

Notice that K̃_{0}(ℤG)⊗_{ℤ} ℚ contains only contributions from K_{−1}(ℤC)⊗_{ℤ} ℚ for finite cyclic subgroups C ⊆ G.

Remark 8.9. Note that these statements are interesting already for finite groups. For instance Theorem 8.4 yields for a finite group G and R = ℂ an isomorphism

⨁_{(C)∈(FCY)} Λ_{G} ⊗_{Λ_{G}[W_{G}C]} θ_{C} · R(C) ⊗_{ℤ} Λ_{G} ≅ R(G) ⊗_{ℤ} Λ_{G}.

which in turn implies Artin’s Theorem discussed in Remark 3.6.

8.2.3 Rationalized Algebraic L-Theory for Infinite Groups

Here is the L-theory analogue of the results above. Compare [201, Theorem 0.4].

Theorem 8.10 (Rational Computation of Algebraic L-Theory for Infinite Groups). Suppose that the group G satisfies the Farrell-Jones Conjecture 2.2 for L-theory. Then we get for all j ∈ ℤ, j ≤ 1 an isomorphism

⨁_{p+q=n} ⨁_{(C)∈(FCY)} H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · L_{q}^{⟨j⟩}(RC) ⊗_{ℤ} ℚ

⟶^{≅} L_{n}^{⟨j⟩}(RG) ⊗_{ℤ} ℚ.

Remark 8.11 (Separation of Variables). Notice that in Theorem 8.4, 8.5 and 8.10 we see again the principle we called separation of variables in Remark 1.13. There is a group homology part which is independent of the coefficient ring R and the K- or L-theory under consideration and a part depending only on the values of the theory under consideration on RC or C_{r}^{∗}(C) for all finite cyclic subgroups C ⊆ G.

8.3 Integral Computations for Infinite Groups

As mentioned above, no general pattern for integral calculations is known or expected. We mention at least one situation where a certain class of groups can be treated simultaneously. Let MFI be the subset of FIN consisting of elements in FIN which are maximal in FIN. Consider the following assertions on the group G.

(M) M_{1}, M_{1} ∈ MFI, M_{1} ∩ M_{2} ≠ 1 ⇒ M_{1} = M_{2}; (NM) M ∈ MFI ⇒ N_{G}M = M; (VCL_{I}) If V is an infinite virtually cyclic subgroup of G, then V is of type I

(see Lemma 2.15); (FJK_{N}) The Isomorphism Conjecture of Farrell-Jones for algebraic K-theory

2.2 is true for ℤG in the range n ≤ N for a fixed element N ∈ ℤ ⨿ {∞}, i.e. the assembly map A: H_{n}^{G}(E_{VCY}(G); K_{R}) ⟶^{≅} K_{n}(RG) is bijective for n ∈ ℤ with n ≤ N.

Let K̃_{n}(C_{r}^{∗}(H)) be the cokernel of the map K_{n}(C_{r}^{∗}({1})) → K_{n}(C_{r}^{∗}(H)) and \overline{L}_{n}^{⟨j⟩}(RG) be the cokernel of the map L_{n}^{⟨j⟩}(R) → L_{n}^{⟨j⟩}(RG). This coincides with L̃_{n}^{⟨j⟩}(R), which is the cokernel of the map L_{n}^{⟨j⟩}(ℤ) → L_{n}^{⟨j⟩}(R) if R = ℤ but not in general. Denote by Wh_{n}^{R}(G) the n-th Whitehead group of RG which is the (n − 1)-th homotopy group of the homotopy fiber of the assembly map BG_{+} ∧ K(R) → K(RG). It agrees with the previous defined notions if R = ℤ. The next result is taken from [83, Theorem 4.1].

Theorem 8.12. Let ℤ ⊆ Λ ⊆ ℚ be a ring such that the order of any finite subgroup of G is invertible in Λ. Let (MFI) be the set of conjugacy classes (H) of subgroups of G such that H belongs to MFI. Then:

(i) If G satisfies (M), (NM) and the Baum-Connes Conjecture 2.3, then for n ∈ ℤ there is an exact sequence of topological K-groups

0 → ⨁_{(H)∈(MFI)} K̃_{n}(C_{r}^{∗}(H)) → K_{n}(C_{r}^{∗}(G)) → K_{n}(G\E_{FIN} (G)) → 0,

which splits after applying − ⊗_{ℤ} Λ. (ii) If G satisfies (M), (NM), (VCL_{I}) and the L-theory part of the Farrell-

Jones Conjecture 2.2, then for all n ∈ ℤ there is an exact sequence

… → H_{n+1}(G\E_{FIN} (G); L^{⟨−∞⟩}(R)) → ⨁_{(H)∈(MFI)} \overline{L}_{n}^{⟨−∞⟩}(RH)

→ L_{n}^{⟨−∞⟩}(RG) → H_{n}(G\E_{FIN} (G); L^{⟨−∞⟩}(R)) → …

It splits after applying − ⊗_{ℤ} Λ, more precisely

L_{n}^{⟨−∞⟩}(RG) ⊗_{ℤ} Λ → H_{n}(G\E_{FIN} (G); L^{⟨−∞⟩}(R)) ⊗_{ℤ} Λ

is a split-surjective map of Λ-modules. (iii) If G satisfies (M), (NM) and the Farrell-Jones Conjecture 2.2 for L_{n}(RG)[1/2],

then the conclusion of assertion (ii) still holds if we invert 2 everywhere. Moreover, in the case R = ℤ the sequence reduces to a short exact sequence

0 → ⨁_{(H)∈(MFI)} L̃_{n}^{⟨j⟩}(ℤH)[\frac{1}{2}] → L_{n}^{⟨j⟩}(ℤG)[\frac{1}{2}]

→ H_{n}(G\E_{FIN} (G); L(ℤ)[\frac{1}{2}] → 0,

which splits after applying − ⊗_{ℤ[ \frac{1}{2} ]} Λ[\frac{1}{2}]. (iv) If G satisfies (M), (NM), and (FJK_{N}), then there is for n ∈ ℤ, n ≤ N an

isomorphism

H_{n}(E_{VCY}(G), E_{FIN} (G); K_{R}) ⊕ ⨁_{(H)∈(MFI)} Wh_{n}^{R}(H) ⟶^{≅} Wh_{n}^{R}(G),

where Wh_{n}^{R}(H) → Wh_{n}^{R}(G) is induced by the inclusion H → G.

Remark 8.13 (Role of G\E_{FIN} (G)). Theorem 8.12 illustrates that for such computations a good understanding of the geometry of the orbit space G\E_{FIN} (G) is necessary.

Remark 8.14. In [83] it is explained that the following classes of groups do satisfy the assumption appearing in Theorem 8.12 and what the conclusions are in the case R = ℤ. Some of these cases have been treated earlier in [34], [212].

• Extensions 1 → ℤ^{n} → G → F → 1 for finite F such that the conjugation action of F on ℤ^{n} is free outside 0 ∈ ℤ^{n};

• Fuchsian groups F;

• One-relator groups G.

Theorem 8.12 is generalized in [204] in order to treat for instance the semidirect product of the discrete three-dimensional Heisenberg group by ℤ/4. For this group G\E_{FIN} (G) is S^{3}.

A calculation for 2-dimensional crystallographic groups and more general cocompact NEC-groups is presented in [212] (see also [236]). For these groups the orbit spaces G\E_{FIN} (G) are compact surfaces possibly with boundary.

Example 8.15. Let F be a cocompact Fuchsian group with presentation

F = ⟨a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{t} |

c_{1}^{γ_{1}} = … = c_{t}^{γ_{t}} = c_{1}^{−1} ⋯ c_{t}^{−1}[a_{1}, b_{1}] ⋯ [a_{g}, b_{g}] = 1⟩

for integers g, t ≥ 0 and γ_{i} > 1. Then G\E_{FIN} (G) is a closed orientable surface of genus g. The following is a consequence of Theorem 8.12 (see [212] for more details).

• There are isomorphisms

K_{n}(C_{r}^{∗}(F)) ≅ \{ \begin{matrix}(2 + ∑_{i=1}^{t}(γ_{i} − 1)) · ℤ & n = 0; \\ (2g) · ℤ & n = 1. \end{matrix}

• The inclusions of the maximal subgroups ℤ/γ_{i} = ⟨c_{i}⟩ induce an isomorphism

⨁_{i=1}^{t} Wh_{n}(ℤ/γ_{i}) ⟶^{≅} Wh_{n}(F)

for n ≤ 1.

• There are isomorphisms

L_{n}(ℤF)[1/2] ≅ \{ \begin{matrix}(1 + ∑_{i=1}^{t} [\frac{γ_{i}}{2} ]) · ℤ[1/2] & n ≡ 0 (4); \\ (2g) · ℤ[1/2] & n ≡ 1 (4); \\ (1 + ∑_{i=1}^{t} [\frac{γ_{i}−1}{2}]) · ℤ[1/2] & n ≡ 2 (4); \\ 0 & n ≡ 3 (4), \end{matrix}

where [r] for r ∈ ℝ denotes the largest integer less than or equal to r. From now on suppose that each γ_{i} is odd. Then the number m above is odd and we get for for ϵ = p and s

L_{n}^{ϵ}(ℤF) ≅ \{ \begin{matrix}ℤ/2 ⨁ (1 + ∑_{i=1}^{t} \frac{γ_{i}−1}{2}) · ℤ & n ≡ 0 (4); \\ (2g) · ℤ & n ≡ 1 (4); \\ ℤ/2 ⨁ (1 + ∑_{i=1}^{t} \frac{γ_{i}−1}{2}) · ℤ & q ≡ 2 (4); \\ (2g) · ℤ/2 & n ≡ 3 (4). \end{matrix}

For ϵ = h we do not know an explicit formula. The problem is that no general formula is known for the 2-torsion contained in L̃_{2q}^{h}(ℤ[ℤ/m]), for m odd, since it is given by the term Ĥ^{2}(ℤ/2; K̃_{0}(ℤ[ℤ/m])), see [14, Theorem 2].

Information about the left hand side of the Farrell-Jones assembly map for algebraic K-theory in the case where G is SL_{3}(ℤ) can be found in [306].

8.4 Techniques for Computations

We briefly outline some methods that are fundamental for computations and for the proofs of some of the theorems above.

8.4.1 Equivariant Atiyah-Hirzebruch Spectral Sequence

Let H_{∗}^{G} be a G-homology theory with values in Λ-modules. Then there are two spectral sequences which can be used to compute it. The first one is the rather obvious equivariant version of the Atiyah-Hirzebuch spectral sequence. It converges to H_{n}^{G}(X) and its E^{2}-term is given in terms of Bredon homology

E_{p,q}^{2} = H_{p}^{G}(X; H_{q}^{G}(G/H))

of X with respect to the coefficient system, which is given by the covariant functor Or(G) → Λ-MODULES, G/H ↦ H_{q}^{G}(G/H). More details can be found for instance in [82, Theorem 4.7].

8.4.2 p-Chain Spectral Sequence

There is another spectral sequence, the p-chain spectral sequence [83]. Consider a covariant functor E: Or(G) → SPECTRA . It defines a G-homology theory H_{∗}^{G}(−; E) (see Proposition 6.7). The p-chain spectral sequence converges to H_{n}^{G}(X) but has a different setup and in particular a different E^{2}- term than the equivariant Atiyah-Hirzebruch spectral sequence. We describe the E^{1}-term for simplicity only for a proper G-CW -complex.

A p-chain is a sequence of conjugacy classes of finite subgroups

(H_{0}) < … < (H_{p})

where (H_{i−1}) < (H_{i}) means that H_{i−1} is subconjugate, but not conjugate to (H_{i}). Notice for the sequel that the group of automorphism of G/H in Or(G) is isomorphic to NH/H. To such a p-chain there is associated the NH_{p}/H_{p}- NH_{0}/H_{0}-set

S((H_{0}) < … < (H_{p})) = map(G/H_{p−1}, G/H_{p})^{G}×_{NH_{p−1}/H_{p−1}}

… ×_{NH_{1}/H_{1}} map(G/H_{0}, G/H_{1})^{G}.

The E^{1}-term E_{p,q}^{1} of the p-chain spectral sequence is

⨁_{(H_{0})<...<(H_{p})} π_{q} ((X^{H_{p}} ×_{NH_{p}/H_{p}} S((H_{0}) < … < (H_{p})))_{+} ∧_{NH_{0}/H_{0}} E(G/H_{0}))

where Y_{+} means the pointed space obtained from Y by adjoining an extra base point. There are many situations where the p-chain spectral sequence is much more useful than the equivariant Atiyah-Hirzebruch spectral sequence. Sometimes a combination of both is necessary to carry through the desired calculation.

8.4.3 Equivariant Chern Characters

Equivariant Chern characters have been studied in [201] and [203] and allow to compute equivariant homology theories for proper G-CW -complexes. The existence of the equivariant Chern character says that under certain conditions the Atiyah-Hirzebruch spectral sequence collapses and, indeed, the source of the equivariant Chern character is canonically isomorphic to ⨁_{p+q} E_{p,q}^{2}, where E_{p,q}^{2} is the E^{2}-term of the equivariant Atiyah-Hirzebruch spectral sequence.

The results of Section 8.2 are essentially proved by applying the equivariant Chern character to the source of the assembly map for the family of finite subgroups.

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Notation

A(G), 114 A(X), 16 A_{F}, 33 Â_{x}(M, u), 59 BG, 14 class_{0}(G), 50 class_{0}(G)_{f}, 50 con(G), 50 con(G)_{f}, 50 cone(X), 92 C^{∗}-ALGEBRAS, 96 C_{max}^{∗}(G), 61 C_{r}^{∗}(G), 24 C_{r}^{∗}(G; ℝ), 25 C_{0}(X), 101 EG, 37 E_{F}(G), 37 \underline{E}G, 37 G_{n}(R), 49 (g), 50 ⟨g⟩, 116 GROUPS, 96 GROUPS^{inj}, 96 GROUPOIDS, 94 GROUPOIDS^{finker}, 100 GROUPOIDS^{inj} , 94 H_{∗}^{G}(−; E), 94 H_{∗}^{?}(−; E), 95 H_{∗}^{G}(X; M), 39 HS_{ℂG}, 51 inn(K), 47 K_{n}(A), 24 KK_{n}(A, B), 103 KK_{n}^{G}(A, B), 103 K̃_{n}(C_{r}^{∗}(H)), 118 K_{n}(Nil(R)), 15 K_{n}(R), 7 K̃_{n}(R), 8 K_{n}^{G}(C_{0}(X)), 101 K_{n}^{G}(X), 101 K_{∗}(Y), 23 KO_{n}(C_{r}^{∗}(G; ℝ)), 25 KO_{∗}(Y), 25 L_{n}^{⟨−∞⟩}(R), 18 L̃_{n}^{⟨j⟩}(R), 118 \overline{L}_{n}^{⟨j⟩}(RG), 118 L^{⟨j⟩}(R), 18 map_{G}(−, Y), 93 N_{G}H, 114 NK_{n}(R), 15 Nil_{n−1}(R), 15 Or(G), 39 Or_{F}(G), 98 õ(X), 11 P (M), 16 P(M), 17 P_{b}(M; ℝ^{k}), 17 pt, 14 RINGS, 96 RINGS^{inv}, 96 r_{F} (G), 111 r_{ℝ}(G; ℝ), 111 r_{ℝ}(G; ℂ), 111 r_{ℝ}(G; ℍ), 111 R(G), 111 RO(G), 111 sign_{x}(M), 31 SPACES, 92 SPACES^{+}, 92 SPECTRA , 93 Sub_{F}(G), 47 tr_{C_{r}^{∗}(G)}, 26 tr_{ℂG}^{u}, 50 (U, ρ, F), 101 W_{G}H, 114 Wh(G), 8 Wh_{0}^{R}(G), 8 Wh_{1}^{R}(G), 8 Wh_{n}^{R}(G), 118 Wh^{Diff}(X), 17 Wh^{P L}(X), 16 X_{+}, 92 X ∧ E, 93 X ∧ Y, 92 X ×_{C} Y, 93 X ∧_{C} Y, 94 X ∧_{C} E, 94 Z_{G}H, 114 ℤ_{p}, 80 η(M), 28 η^{(2)}(M̃), 28 Λ^{G}, 57 π_{i}(E), 93 ρ^{(2)}(M), 28 τ^{(2)}(M) ∈ ℝ, 29 Â(M), 59 ALL, 36 CYC, 36 FCY, 36 (FCY), 114 FIN, 36 G^{G}(S), 94 H(G), 70 H_{∗}^{G}, 37 H_{∗}^{?}, 90 L(M), 30 MFI, 117 P(M), 16 P^{Diff}(M), 17 VCY, 36 VCY_{I}, 36 {1}, 36

Index

CW -complex finite, 11 finitely dominated, 11

Â-class, 59 Â-genus

higher, 59 A-regular, 77 Alexander trick, 11 amenable group action, 74 Artin’s, 48 aspherical, 21 assembly map, 33

analytic, 102

relative, 39 asymptotic dimension, 75

balanced product

of C-spaces, 93 balanced smash product

of pointed C-spaces, 94 Bass-Heller-Swan-decomposition, 15 Baum-Connes Conjecture with Coeffi-

cients and a-T-menable Groups, 70

Bott manifold, 58 bounded geometry, 64 Bounded h-Cobordism Theorem, 13 bounded map, 12 Bredon homology, 39 Burnside ring, 114

C^{∗}-algebra G-C^{∗}-algebra, 103 maximal complex group C^{∗}-algebra,

61 reduced complex group C^{∗}-algebra,

24 reduced real group C^{∗}-algebra, 25

category

of compactly generated spaces, 92 centralizer, 114 classifying space for a family, 37 for proper G-actions, 37 of a group, 14

cobordism, 9 bounded, 12 diffeomorphic relative M^{−}, 10 trivial, 10 cocompact, 36 cone

reduced cone, 92 Conjecture

Atiyah-Conjecture for Torsion Free Groups, 27 Strong, 28 Bass Conjecture Strong Bass Conjecture for K_{0}(ℂG), 51 Strong Bass Conjecture for K_{0}(ℤG), 51 Weak Bass Conjecture, 52 Baum-Connes Conjecture fibered version, 68 for Torsion Free Groups, 23 real version, 25 with Coefficients, 63 Borel Conjecture, 21 Bost Conjecture, 62 Embedding Conjecture, 28 Farrell-Jones Conjecture fibered version, 68 for K_{0}(ℚG), 47 for K_{n}(ℤG) for n ≤ −1, 46 for Low Dimensional K-Theory and Torsion Free Groups, 8 for Pseudoisotopies of Aspherical Spaces, 66 for Torsion Free Groups and K-Theory, 13 for Torsion Free Groups and L-Theory, 18 Vanishing of Low Dimensional K-Theory for Torsionfree Groups and Integral Coefficients, 9 with Coefficients, 63 Gromov-Lawson-Rosenberg Conjec-

ture Homological Gromov-Lawson-Rosenberg Conjecture, 60 Stable Gromov-Lawson-Rosenberg Conjecture, 58

Homotopy Invariance of the L^{2}-Rho-Invariant for Torsionfree Groups, 29

Homotopy-theoretic Isomorphism

Conjecture for (G, E, F), 98 Idempotent Conjecture, 27 Integral K_{0}(ℤG)-to-K_{0}(ℚG)-

Conjecture, 53 Isomorphism Conjecture for

NK-groups, 69 Isomorphism Conjecture for the

Hecke-Algebra, 70 Kadison Conjecture, 27 Kaplansky Conjecture, 27 Meta-Conjecture, 33 Novikov Conjecture, 31 for K and L-theory, 30 Rational K_{0}(ℤG)-to-K_{0}(ℚG)-

Conjecture, 51 Trace Conjecture for Torsion Free Groups, 26 Modified Trace Conjecture, 57 Unit-Conjecture, 28 Zero-Divisor-Conjecture, 27 Zero-in-the-spectrum Conjecture, 32

cycle degenerate, 101 equivalent, 101 even, 101 homotopic, 101 odd, 101

decoration, 18 descent homomorphism, 103 Dirac element, 104

dual, 104 Dirac-Dual Dirac Method, 104 Dirac-dual Dirac method, 103 disjoint union axiom, 38

elliptic operator

generalized elliptic G-operator, 101 equivalence of categories, 94 eta-invariant, 28 exotic sphere, 11

family of subgroups, 36 finiteness obstruction, 11 Fundamental Theorem of algebraic

K-theory, 15

G-CW -complex, 36 finite, 36 proper, 37

group a-T-menable, 71 having property (RD), 73 having property (T), 71 having the Haagerup property, 71 K-amenable, 61 linear, 74 virtually cyclic, 37 virtually cyclic of the first kind, 37 virtually poly-cyclic, 49

h-cobordism, 10

bounded, 13 Hattori-Stallings-rank, 50 Hecke algebra, 70 homology theory

equivariant G-homology theory, 89 G-homology theory, 37 proper G-homology theory, 90 Homotopy Groups of Diff(M), 23 Homotopy Groups of Top(M), 23 Homotopy invariance of τ^{(2)}(M) and

ρ^{(2)}(M), 29

induction

of equivariant spaces, 89 induction structure, 89 Injectivity of the Baum-Connes

Assembly Map, 74

K- and L-Theory Spectra over

Groupoids, 97 K-groups

algebraic K-groups of a ring, 7

reduced algebraic K-groups of a ring,

8 topological K-groups of a C^{∗}-algebra,

24 topological K-homology groups of a

space, 23 K-theory spectrum

algebraic K-theory spectrum of a

ring, 13 over groupoids, 97

Kasparov product, 103 KK-theory

equivariant, 102

L-class, 30 L-groups, 18 L-theory spectrum

algebraic L-theory spectrum of a

ring, 18 over groupoids, 97

L^{2}-eta-invariant, 28 L^{2}-Rho-invariant, 28 length function, 72

manifold parametrized over ℝ^{k}, 12 Meta-Conjecture, 33 metrically proper, 71

Negative Homotopy Groups of

Pseudoisotopies, 18 Nil-groups, 15 normalizer, 114

orbit category, 39 restricted, 98 p-chain, 120 Poincaré Conjecture, 10 principle

separation of variables, 14, 117 proper

G-space, 100

G-C^{∗}-algebra, 103 metric space, 73

Properties of the Finiteness Obstruc-

tion, 12 pseudoisotopy, 16

Rational Computation of Algebraic K-Theory for Infinite Groups, 116, 117

Rational Computation of Topological

K-Theory for Infinite Groups, 115 Rational Contribution of Finite Subgroups to Wh(G), 81 Rational Injectivity of the Baum-Connes

Assembly Map, 73 Rational Injectivity of the Classical

K-Theoretic Assembly Map, 79 Rational Injectivity of the Farrell-Jones Assembly Map for Connective K-Theory, 80 relative assembly map, 39 representation ring, 111 ring

integral domain, 27

Noetherian, 8 regular, 8

ring with involution, 96 Rothenberg sequence, 19

s-Cobordism Theorem, 10 Shaneson-splitting, 20 signature, 31 higher, 31 smash product, 92

of a space with a spectrum, 93 space

C-space, 93 spectral sequence

equivariant Atiyah-Hirzebruch spectral sequence, 120 p-chain spectral sequence, 120

spectrum, 92 homotopy groups of a spectrum, 93 map of spectra, 92 structure maps of a spectrum, 92

Status of the Baum-Connes Conjecture,

72 Status of the Coarse Baum-Connes

Conjecture, 75 Status of the Farrell-Jones Conjecture

for Torsionfree Groups, 78 Status of the Fibered Farrell-Jones

Conjecture, 76

tensor product

of pointed C-space with a C-spectrum,

94 The C^{∗}-Theoretic Novikov Conjecture

and Groups of Finite Asymptotic Dimension, 75

The K-and L-Theoretic Novikov Conjecture and Groups of Finite Asymptotic Dimension, 82 The Farrell-Jones Conjecture and subgroups of Lie groups, 77 The Farrell-Jones Conjecture Implies the Borel Conjecture, 22

trace standard trace of C_{r}^{∗}(G), 26 universal ℂ-trace, 50 Transitivity Principle, 40 transport groupoid, 94

uniform embedding into Hilbert space, 73

virtually cyclic, 37

weak equivalence of spectra, 93 Whitehead group, 8 generalized, 8 higher, 118