Equivariant homology theory

and assembly maps by

Wolfgang Lück (Münster) July 2006 Motivation for the Baum-Connes Conjecture for torsionfree groups

The aim is to compute the topological K-theory

K_{∗}(C_{r}^{∗}(G))

of the reduced group C^{∗}-algebra C_{r}^{∗}(G) in the sense that we want to identify it with more familiar terms. The key idea comes from the observation that K_{∗}(C_{r}^{∗}(G)) has some homological properties. More precisely, if G is the amalgamated product G = G_{1} ∗_{G_{0}} G_{2} for subgroups G_{i} ⊆ G, then there is a long exact sequence due to Pimsner-Voiculescu

⋯ ⟶^{∂_{n+1}} K_{n}(C_{r}^{∗}(G_{0})) → K_{n}(C_{r}^{∗}(G_{1}))⊕K_{n}(C_{r}^{∗}(G_{2}))

→ K_{n}(C_{r}^{∗}(G)) ⟶^{∂_{n}} K_{n−1}(C_{r}^{∗}(G_{0}))

→ K_{n−1}(C_{r}^{∗}(G_{1})) ⊕ K_{n−1}(C_{r}^{∗}(G_{2})) → · · · .

If ϕ: G → G is a group automorphism and G ⋊_{ϕ} ℤ the associated semidirect product, then there is a long exact sequence due to Pimsner-Voiculescu

⋯ ⟶^{∂_{n+1}} K_{n}(C_{r}^{∗}(G)) ⟶^{ϕ_{∗}−id} K_{n}(C_{r}^{∗}(G))

→ K_{n}(C_{r}^{∗}(G ⋊_{ϕ} ℤ)) ⟶^{∂_{n}} K_{n−1}(C_{r}^{∗}(G))

⟶^{ϕ_{∗}−id} K_{n−1}(C_{r}^{∗}(G)) → · · · .

Notice that the are analogous sequences in group homology

⋯ ⟶^{∂_{n+1}} H_{n}(BG_{0}) → H_{n}(BG_{1}) ⊕ H_{n}(BG_{2})

→ H_{n}(BG) ⟶^{∂_{n}} H_{n−1}(BG_{0})

→ H_{n−1}(BG_{1}) ⊕ H_{n−1}(BG_{2})

→ H_{n−1}(BG) ⟶^{∂_{n−1}} ⋯

and

⋯ ⟶^{∂_{n+1}} H_{n}(BG) ⟶^{ϕ_{∗}−id} H_{n}(BG)

→ H_{n}(B(G ⋊_{ϕ} ℤ)) ⟶^{∂_{n}} H_{n−1}(BG)

⟶^{ϕ_{∗}−id} H_{n−1}(BG) → · · · .

The first naive guess H_{n}(BG) = K_{n}(C_{r}^{∗}(G)) fails already for the trivial group. On the other hand the guess K_{n}(BG) = K_{n}(C_{r}^{∗}(G)) works out for the trivial group. This motivates: Conjecture 1 (Baum-Connes Conjecture for torsionfree groups). There is an assembly map

K_{n}(BG) ⟶^{≅} K_{n}(C_{r}^{∗}(G))

which is bijective for all n ∈ ℤ.

Remark 2. If G is not torsionfree, the version of the Baum-Connes Conjecture above cannot hold anymore in general, Namely, for a finite group G one has

K_{n}(BG)⊗_{ℤ}ℚ ≅ K_{n}({•})⊗_{ℤ}ℚ ≅ K_{n}(C_{r}^{∗}({1}))⊗_{ℤ}ℚ

and

K_{n}(C_{r}^{∗}(G)) ≅ \{ \begin{matrix}R_{ℂ}(G) & n even , \\ \{0\} & n odd . \end{matrix}

To formulate the Baum-Connes Conjecture in general, some more input is needed.

The Farrell-Jones Conjecture for torsionfree groups

For algebraic K- and L-theory the situation is more complicated since there the Mayer-Vietoris sequence exist only modulo certain Nilterms or Unil-terms as worked out by Cappell and Waldhausen. For instance the Bass-Heller-Swan decomposition implies

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

whereas

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

If R is a regular ring, one can still hope for a torsionfree group G that there are isomorphisms

H_{n}(BG; K(R)) ⟶^{≅} K_{n}(RG).

In L-theory one can hope for an isomorphism for a torsionfree group G

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

Here the decoration ⟨−∞⟩ is forced upon us because the Shaneson splitting. It says

L_{n}^{⟨−j⟩}(Rℤ) ≅ L_{n}^{⟨−j⟩}(R) ⊕ L_{n−1}^{⟨−j−1⟩}(R), whereas

H_{n}(Bℤ; L^{⟨−j⟩}(R)) ≅ L_{n}^{⟨−j⟩}(R) ⊕ L_{n−1}^{⟨−j⟩}(R). Remark 3 (Consequence for K̃_{0}(ℤG) and Wh(G)). Let G be a torsionfree group. Then the Farrell-Jones Conjecture predicts

H_{n}(BG, K(ℤ)) ≅ K_{n}(ℤG).

Since K_{n}(ℤ) = π_{n}(K(ℤ)) = 0 for n ≤ −1 and K̃_{n}(ℤ) = 0 for n = 0, 1, an easy application of the Atiyah-Hirzebruch spectral sequence implies

K̃_{0}(ℤG) = {0};
Wh(G) = {0}.

As in the Baum-Connes Conjecture the version of the Farrell-Jones Conjecture formulated above cannot extend in general to groups with torsion.

Equivariant homology theories

Definition 4 (G-homology theory). A G-homology theory H_{∗}^{G} is a covariant functor H_{∗}^{G} from the category of G-CW -pairs to the category of ℤ-graded R-modules together with natural transformations

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

for n ∈ ℤ satisfying the following axioms:

• G-homotopy invariance;

• Long exact sequence of a pair;

• Excision;

• Disjoint union axiom.

Definition 5 (Equivariant homology theory). An equivariant homology theory H_{∗}^{?} consists of a G-homology theory H_{∗}^{G} for every group G together with the following so called induction structure: given a group homomorphism α: H → G and a H-CW -pair (X, A) there are for all n ∈ ℤ natural homomorphisms

ind_{α}: H_{n}^{H}(X, A) → H_{n}^{G}(ind_{α}(X, A))

satisfying

• Bijectivity If ker (α) acts freely on X, then ind_{α} is a bijection;

• Compatibility with the boundary homomorphisms

• Functoriality in α

• Compatibility with conjugation Example 6. Here are some examples for equivariant homology theories H_{∗}^{?}:

• Quotients Let K_{∗} be a non-equivariant homology theory. Define H_{∗}^{?} by

H_{∗}^{G}(X) := K_{∗}(G\X).

• Borel homology Let K_{∗} be a non-equivariant homology theory. Define H_{∗}^{?} by

H_{∗}^{G}(X) := K_{∗}(EG ×_{G} X).

• Equivariant bordism Let X be a proper G-CW -complex. Define the n-th G-bordism group Ω_{n}^{G}(X) by the G-bordism classes of proper cocompact smooth G-manifolds M with G-reference maps to X.

A spectrum E defines a homology theory by sending a space X to π_{∗}^{s}(X_{+} ∧ E). This generalizes to the equivariant setting as follows. Theorem 7 (Equivariant homology theories and spectra). Consider a covariant functor

E: GROUPOIDS → SPECTRA

sending equivalences of groupoids to weak equivalences of spectra.

Then there exists an equivariant homology theory H_{∗}^{?}(−; E) with the property that for every group G, subgroup H ⊆ G and n ∈ ℤ

H_{n}^{G}(G/H) = H_{n}^{H}({•}) = π_{n}^{s}(E(H)).

Proof. Given a group G and a G-set S we obtain a groupoid G^{G}(S) whose objects are elements in S and whose morphisms from s_{1} to s_{2} are the elements g ∈ G with gs_{1} = s_{2}. Composition comes from the group structure on G. Thus we obtain a covariant functor

E^{G}: Or(G) → SPECTRA , G/H ↦ E(G^{G}(G/H)).

A G-CW -complex X defines a contravariant functor

X^{?} : Or(G) → SPACES, G/H ↦ X^{H}.

We obtain a spectrum X_{+}^{?} ∧_{Or(G)} E^{G} by the balanced smash product. Its n-th stable homotopy group is H_{n}^{G}(X; E).

The next result is due to Davis-Lück. Theorem 8 (K- and L-Theory Spectra over Groupoids). Let R be a ring (with involution). There exist covariant functors

K_{R}: GROUPOIDS → SPECTRA ;
L_{R}^{⟨j⟩}: GROUPOIDS → SPECTRA ;
K^{top} : GROUPOIDS^{inj} → SPECTRA

with the following properties:

• They send equivalences of groupoids to weak equivalences of spectra;

• For every group G and all n ∈ ℤ we have

π_{n}(K_{R}(G)) ≅ K_{n}(RG);
π_{n}(L_{R}^{⟨j⟩}(G)) ≅ L_{n}^{⟨j⟩}(RG);
π_{n}(K^{top} (G)) ≅ K_{n}(C_{r}^{∗}(G)).

Classifying spaces of families of subgroups

Definition 9 (Family of subgroups). A family F of subgroups of the group G is a set of subgroups of G which is closed under conjugation and taking subgroups.

Examples for families are

{1}  trivial subgroup
FIN  finite subgroups
VCY  virtually cyclic subgroups
ALL  all subgroups

Definition 10 (Classifying space of a family). Let F be a family of subgroups of G. A model for the classifying space of the family F is a G-CW -complex E_{F}(G) such that E_{F}(G)^{H} is contractible if H ∈ F and is empty if H ∉ F.

If F is FIN or VCY, we also write EG and EG. Sometimes EG is called the classifying space for proper G-actions. Theorem 11. The G-CW -complex E_{F}(G) is characterized uniquely up to G-homotopy by the property that for every G-CW -complex X whose isotropy groups belong to F there is up to G-homotopy precisely one G-map X → E_{F}(G).

Obviously E_{{1}}(G) = EG and E_{ALL}(G) = G/G.

Remark 12 (Models for EG). The spaces EG are interesting in their own right and have often very nice geometric models which are rather small. For instance

• Rips complex for word hyperbolic groups;

• Teichmüller space for mapping class groups;

• Outer space for the group of outer automorphisms of free groups;

• L/K for a connected Lie group L, a maximal compact subgroup K ⊆ L and G ⊆ L a discrete subgroup;

• CAT(0)-spaces with proper isometric Gactions, e.g., Riemannian manifolds with non-positive sectional curvature or trees.

Formulations of the conjectures in general

Definition 13 (Assembly map). Let H_{∗}^{G} be a G-homology theory. Let F be a family of subgroups. Let pr : E_{F}(G) → G/G be the projection. The associated assembly map is

H_{n}^{G}(pr): H_{n}^{G}(E_{F}(G)) → H_{n}^{G}(G/G).

Definition 14 (Meta-Isomorphism-Conjecture). The Meta-Isomorphism Conjecture for a Ghomology theory H_{∗}^{G} and a family F says that the assembly map

H_{n}^{G}(pr): H_{n}^{G}(E_{F}(G)) → H_{n}^{G}(G/G)

is bijective for all n ∈ ℤ. Definition 15 (Baum-Connes-Conjecture). The Baum-Connes Conjecture is the Meta-Isomorphism Conjecture for H_{∗}^{G}(−, K^{top} ) = K_{∗}^{G}(−) and the family FIN, i.e., it predicts the bijectivity of

K_{n}^{G}(EG) → K_{n}(C_{r}^{∗}(G)).

Definition 16 (Farrell-Jones-Conjecture). The Farrel-Jones Conjecture is the Meta-Isomorphism Conjecture for H_{∗}^{G}(−, K_{R}) or H_{∗}^{G}(−, L_{R}^{⟨−∞⟩}) and the family VCY, i.e., it predicts the bijectivity of the assembly maps

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

Remark 17 (Assembling from subgroups). The idea behind all of these conjectures is that for a certain functor from groups to spectra its value on a group G can be assembled by its values on all subgroups occuring in F. These conjectures predict a kind of induction theorem. However, degrees are mixed in the sense that K_{n}(RG) is computable in terms of and affected by K_{m}(RH) for all H ∈ VCY and all m ∈ ℤ, m ≤ n.

Remark 18 (Status). These conjectures are known for many groups but are open in general. The methods of proofs are different depending on the groups or the conjecture considered. They mainly use techniques from controlled topology, homotopy theory, K-theory, operator theory and geometry. For a survey of the status, method of proofs, the history and the applications of the Baum-Connes Conjecture and the Farrell-Jones Conjecture we refer to the article by Lück and Reich in the handbook of K-theory (2005).

We mention the recent result of Bartels, Lück and Reich. Theorem 19. The Farrell-Jones Conjecture for algebraic K-theory K_{n}(RG) holds for all n ∈ ℤ, all coefficient rings R and all word-hyperbolic groups G.

Remark 20 (Applications). These conjectures are very deep. They give a lot of structural insight and imply a variety of classical prominent conjectures such as

• Bass Conjecture Values of Hattori-Stalling ranks.

• Borel Conjecture Topological rigidity of aspherical manifolds.

• Stable Gromov-Lawson-Rosenberg Conjecture Obstruction for positive scalar curvature metrics.

• Kadison Conjecture Idempotents in C_{r}^{∗}(G)

• Novikov Conjecture Homotopy invariance of higher signatures.

• generalized Trace Conjecture Values of the trace maps.

Remark 21 (Interpretations of the assembly maps). This implications and also the proofs of the Baum-Connes Conjecture and the Farrell-Jones Conjecture are often consequences of a good geometric interpretation of the assembly maps, e.g., in terms of index theory (Kasparov), controlled topology (Quinn) or in terms of surgery theory (Quinn, Ranicki).

These two conjectures are also the main tool in computations, where the target of the assembly map is the object of interest and the source of the assembly map is tractable for computations. The latter is due to the homotopic theoretic description of the assembly map due to Davis-Lück since well-known tools like spectral sequences and Chern characters can be generalized to the equivariant setting and then successfully applied.

We will discuss both aspects.

Remark 22 (The Meta Conjecture and other theories). The Meta Conjecture applies also to other theories. Farrell-Jones study the pseudoisotopy functor. In a project by Lück-Reich-Rognes-Varisco the version of the Meta-Con-jecture is treated for topological Hochschild homology and for topological cyclic homology, where the relevant family consists of cyclic subgroups. An important feature of the homotopy theoretic assembly map is that it is natural in the theories which are plugged in. Thus for instance construction such as the cyclotomic trace or change of rings homomorphism or the passage from algebraic to topological K-theory yield transformations between the relevant assembly maps

Index-theoretic interpretation of the Baum-Connes assembly map

The Baum-Connes assembly map

K_{0}^{G}(EG) → K_{0}(C_{r}^{∗}(G))

has the following interpretation. Elements in K_{0}^{G}(EG) are represented by pairs (M, P^{∗}) consisting of a cocompact proper smooth G-mani-fold M with G-invariant Riemannian metric and a G-equivariant elliptic complex P^{∗} of differential operators of order 1. The Baum-Connes assembly map assigns to this pair its G-index in the sense of Mishchenko-Fomenko. So the surjectivity of the Baum-Connes map says that every element in K_{0}(C_{r}^{∗}(G)) can be realized as an index of a pair (M, P^{∗}). The injectivity says that the index decides when two such cycles (M, P^{∗}) and (N, Q^{∗}) are homologous.

Example 23 (Kadison Conjecture). Let G be a torsionfree group. Then the following diagram commutes by Atiyah’s L^{2}-index theorem

[figure: K_{0}(BG) = K_{0}^{G}(EG); K_{0}(pr); K_{0}(C_{r}^{∗}(G)); tr; ℝ; K_{0}({•}); ≅; ℤ]

Hence a conclusion of the Baum-Connes Conjecture is the Trace Conjecture which predicts that the image of the trace map tr is the integers. This implies the Kadison Conjecture which says that every idempotent in C_{r}^{∗}(G) is 0 or 1.

Wall’s finiteness obstruction

Given a finitely dominated CW -complex X Wall defines its finiteness obstruction

˜0(X) ∈ K̃_{0}(ℤπ)

for π = π_{1}(X) by

˜0(X) := _{n≥0}^{∑}(−1)^{n} · [P_{n}]

where P_{∗} is any finite projective ℤπ-chain complex P_{∗} which is ℤπ-homotopy equivalent to C_{∗}(X̃). It is zero if and only X is homotopy equivalent to a finite CW -complex. For a finitely presented group G any element in K̃_{0}(ℤG) can be realized as õ(X) of a finitely dominated CW - complex for G ≅ π_{1}(X). This implies: Theorem 24 (Wall). Let G be a finitely presented group. Then K̃_{0}(ℤG) vanishes if and only if every finitely dominated CW -complex X with G ≅ π_{1}(X) is homotopy equivalent to a finite CW -complex.

s-Cobordism Theorem

We conclude from the s-Cobordism Theorem due to Barden, Kirby, Mazur, Siebemann, Smale, Stallings for every finitely presented group G and n ≥ 6 that the following statements are equivalent

• Wh(G) vanishes.

• Every n-dimensional compact s-cobordism W with π_{1}(W) ≅ G is trivial;

Since Wh({1}) vanishes, this implies the Poincaré Conjecture in dimension ≥ 5.

The Borel Conjecture

Conjecture 25 (Borel Conjecture). Let G be a finitely presented group. The Borel Conjecture for G says:

1. Let X be an aspherical finitely dominated Poincaré complex with G ≅ π_{1}(X). Then X is homotopy equivalent to a closed manifold.

2. Let M and N be two aspherical closed manifolds with G ≅ π_{1}(M) ≅ π_{1}(N). Then any homotopy equivalence M → N is homotopic to a homeomorphism. In particular M and N are homeomorphic.

The Borel Conjecture may be viewed as a topological version of Mostow rigidity.

Remark 26 (The Farrell-Jones Conjecture implies the Borel Conjecture). The Borel Conjecture 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.) It is attributed to Browder, Kirby, Novikov, Siebenmann, Sullivan and Wall. An algebraic version has been developed by Ranicki. 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. Since BG = M, the Farrell-Jones Conjecture 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 injective respectively bijective and thus by the surgery sequence that S^{top} (M) is a point and hence the Borel Conjecture holds for M.

For the question whether a Poincaré complex is homotopy equivalent to a closed manifold, is answered by the total surgery obstruction of Ranicki.

Computational aspects

Next we want to indicate how one can try to compute the source of the assembly map what is in general much easier then to compute the target.

There are basically the following general tools available:

• Analysis of nice models for EG (Leary, Lück, Nucinkis, Soulé, …)

• Splitting of Nil-terms (Bartels)

• Analysis of Nil-terms (Bass, Connolly, Grunewald, Kozniewsky, Prassidis, Weibel)

• Analysis of UNil-terms (Banagl, Brookman, Cappell, Connolly, Davis, Qayum Khan, Ranicki)

• Equivariant Atiyah-Hirzebruch spectral sequence (Davis-Lück)

• p-chain spectral sequence (Davis-Lück)

• Equivariant Chern characters (Lück)

Example 27 (Infinite dihedral group). Let D_{∞} = ℤ/2 ∗ ℤ/2 = ℤ ⋊ ℤ/2 be the infinite dihedral group. It acts on the tree ℝ by letting ℤ acting by translation and ℤ/2 by reflecting in 0. This is a model for ED_{∞}. It consists of two equivariant 0-cells D_{∞}/ℤ/2 × D^{0} and one equivariant 1-cell D_{∞} × D^{1}. If H_{∗}^{?} is an equivariant homology theory, we obtain a long exact sequence

⋯ → H_{n}^{{1}}({•}) → H_{n}^{ℤ/2}({•}) ⊕ H_{n}^{ℤ/2}({•})

→ H_{n}^{D_{∞}}(ED_{∞}) → H_{n−1}^{{1}}({•})

→ H_{n−1}^{ℤ/2}({•}) ⊕ H_{n−1}^{ℤ/2}({•}) ⋯

In the case of the Baum-Connes Conjecture which is known to be true for D_{∞} we obtain the short split exact sequence

0 → R_{ℂ}({1}) → R_{ℂ}(ℤ/2) ⊕ R_{ℂ}(ℤ/2)

→ K_{0}(C_{r}^{∗}(D_{∞})) → 0

and K_{1}(C_{r}^{∗}(D_{∞})) ≅ {0}.

Notice that the Farrell-Jones Conjecture makes no predictions since D_{∞} is virtually cyclic.

Remark 28 (Integral versus rational computations). Integral computations seem only to be possible in special cases. One cannot hope for a general formula for K_{n}(C_{r}^{∗}(G)). The computation of K_{n}(ℤG) and L_{n}(ℤG) is even harder since one has to deal with virtually cyclic subgroups instead of finite subgroups, which comes from the appearance of Nil-terms and UNil-terms.

If one is interested only in rational information, the situation improves a lot thanks to equivariant Chern characters. They predict that in all cases of interest the equivariant Atiyah-Hirzebruch spectral sequence collapses in the strongest sense.

Instead of going through its construction and the proofs we mention one consequence which follows from the Baum-Connes Conjecture, the Farrell-Jones Conjecture, the naturality of the assembly maps in the theory considered and the equivariant Chern character.

Theorem 29 (Lück). Let G be a (discrete) group. Suppose that the Baum-Connes Conjecture for G and the Farrell-Jones Conjecture for K_{n}(ℂG) are true. Let T be the set of conjugacy classes (g) of elements g ∈ G of finite order. There is a commutative diagram

⨁\begin{matrix}p+q=n \\ (g)∈T \end{matrix} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}(ℂ) ⟶ ℂ ⊗_{ℤ} K_{n}(ℂG)
↓  ↓
⨁\begin{matrix}p+q=n \\ (g)∈T \end{matrix} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}^{top} (ℂ) ⟶ ℂ ⊗_{ℤ} K_{n}^{top} (C_{r}^{∗}(G))

where C_{G}⟨g⟩ is the centralizer of the cyclic group generated by g in G and the vertical arrows come from the obvious change of ring and of K-theory maps K_{q}(ℂ) → K_{q}^{top} (ℂ) and K_{n}(ℂG) → K_{n}^{top} (C_{r}^{∗}(G)).