L^{2}-Topological Invariants of 3-manifolds by John Lott and Wolfgang Lück

Abstract : We give results on the L^{2}-Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain complex of Hilbert modules over a finite von Neumann algebra. We establish inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence of chain complexes. Our algebraic results, along with some analytic results on geometric 3-manifolds, are used to compute the L^{2}-Betti numbers of compact 3-manifolds which satisfy a weak form of the geometrization conjecture, and to compute or estimate their Novikov-Shubin invariants.

0. Introduction

The L^{2}-Betti numbers of a smooth closed manifold M, introduced by Atiyah [1], are invariants of M which are defined in terms of the universal cover M̃. Roughly speaking, if M is Riemannian then the p-th L^{2}-Betti number b_{p}(M) measures the size of the space of harmonic L^{2} p-forms on M̃, relative to the action of the fundamental group π on M̃. We give the precise definition later. The L^{2}-Betti numbers are homotopy invariants of M (Dodziuk [12]), and can be extended to become Γ-homotopy invariants of topological spaces upon which a countable group Γ acts (Cheeger-Gromov [10]).

By means of a Laplace transform, there is an interpretation of the L^{2}-Betti numbers in terms of the large-time asymptotics of heat flow on M̃. Let e^{−t_{Δ_{p}}}(x, y) be the Schwartz kernel of the heat operator acting on L^{2} p-forms on M̃. The von Neumann trace of the heat operator is given by

tr_{N(π)} (e^{−t_{Δ_{p}}}) = ∫_{F} tr (e^{−t_{Δ_{p}}}(x, x)) dvol(x),

where F is a fundamental domain for the π-action on M̃ and the trace on the right-hand-side is the ordinary trace on End(∧^{p}(T_{x}^{∗}M̃)). The L^{2}-Betti numbers of M can be expressed by

b_{p}(M) = lim_{t→∞} tr_{N(π)} (e^{−t_{Δ_{p}}}) .

In many examples one finds that tr_{N(π)} (e^{−t_{Δ_{p}}}) − b_{p}(M) approaches zero with an

exponential or power decay as t → ∞. Novikov and Shubin [37] introduced invariants which quantify this phenomenon. If there is an exponential decay, put α̃_{p}(M) = ∞^{+}. Otherwise, put

α̃_{p}(M) = sup {β_{p} : tr_{N(π)} (e^{−t_{Δ_{p}}}) − b_{p}(M) is O(t^{−β_{p}/2}) as t → ∞} ∈ [0, ∞].

Roughly speaking, α̃_{p}(M) measures the thickness of the spectrum of ˜△_{p} near 0; the larger α̃_{p}(M), the thinner the spectrum near 0. Novikov and Shubin stated that these invariants are independent of the choice of Riemannian metric on M, and hence are smooth invariants of M. The first author showed that they are defined for all topological manifolds and depend only on the homeomorphism type of M, and computed them in certain cases [24]. The Novikov-Shubin invariants are homotopy invariants (see Gromov-Shubin [18] and Theorems 2.6 and 5.7 of the present paper.) A combinatorial Novikov-Shubin invariant was defined by Efremov in [14] and shown to be the same as the analytically defined invariant, again under the assumption that M is closed.

In this paper we give some results on the L^{2}-Betti numbers and Novikov-Shubin invariants of compact manifolds (possibly with boundary), especially 3-manifolds. Our interest in these invariants comes from our work on related L^{2}-invariants, the L^{2}-Reidemeister and analytic torsions [6, 24, 29, 31, 32]. In particular, one wishes to know that the Novikov-Shubin invariants of a manifold are all positive, in order for the L^{2}-torsions to be defined. We make some remarks on the L^{2}-torsions in Section 7.

We define an invariant α_{p}(M) in terms of the boundary operator acting on p-chains on M̃ (compare [18, 19]). The relationship with α̃_{p}(M) is that α̃_{p}(M) = min(α_{p}(M), α_{p+1}(M)), where the left-hand-side is defined using p-forms on M̃ which satisfy absolute boundary conditions if M has boundary. Let us say that a prime 3-manifold is exceptional if it is closed and no finite cover of it is homotopy equivalent to a Haken, Seifert or hyperbolic 3manifold. No exceptional prime 3-manifolds are known, and standard conjectures (Thurston geometrization conjecture, Waldhausen conjecture) imply that there are none. The main results of this paper are given in the following theorem:

Theorem 0.1 : Let M be the connected sum M_{1}♯ … ♯M_{r} of (compact connected orientable) nonexceptional prime 3-manifolds M_{j}. Assume that π_{1}(M) is infinite. Then

1. a. The L^{2}-Betti numbers of M are given by:

b_{0}(M) = 0
b_{1}(M) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} − χ(M) + | \{C ∈ π_{0}(∂M) s.t. C ≅ S^{2}\} |
b_{2}(M) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} + | \{C ∈ π_{0}(∂M) s.t. C ≅ S^{2}\} |
b_{3}(M) = 0.

b. Equivalently, if χ(π_{1}(M)) denotes the rational-valued group Euler characteristic then b_{1}(M) = −χ(π_{1}(M)) and b_{2}(M) = χ(M) − χ(π_{1}(M)). c. In particular, M has trivial L^{2}-cohomology iff M is homotopy equivalent to RP^{3}♯RP^{3} or a prime 3-manifold with infinite fundamental group whose boundary is empty or a union of tori.

2. Let the Poincaré associate P (M) be the connected sum of the M_{j}’s which are not 3disks or homotopy 3-spheres. Then α_{p}(P (M)) = α_{p}(M) for p ≤ 2. We have α_{1}(M) = ∞^{+} except for the following cases:

(a) α_{1}(M) = 1 if P (M) is S^{1} × D^{2}, S^{1} × S^{2} or homotopy equivalent to RP^{3}♯RP^{3}.

(b) α_{1}(M) = 2 if P (M) is T^{2} × I or a twisted I-bundle over the Klein bottle K.

(c) α_{1}(M) = 3 if P (M) is a closed R^{3}-manifold.

(d) α_{1}(M) = 4 if P (M) is a closed Nil-manifold.

(e) α_{1}(M) = ∞ if P (M) is a closed Sol-manifold.

3. α_{2}(M) > 0.

4. If M is a closed hyperbolic 3-manifold then α_{2}(M) = 1. If M is a closed Seifert 3manifold then α_{2}(M) is given in terms of the Euler class e of the bundle and the Euler characteristic χ of the base orbifold by:

\underline{χ > 0} \underline{χ = 0} \underline{χ < 0}
e = 0 | ∞^{+}  3  1
e ≠ 0 | ∞^{+}  2  1

If M is a Seifert 3-manifold with boundary then α_{2}(M) is ∞^{+} if M = S^{1} × D^{2}, 2 if M is T^{2} × I or a twisted I-bundle over K, and 1 otherwise. If M is a closed Sol-manifold then α_{2}(M) ≥ 1.

5. If ∂M contains an incompressible torus then α_{2}(M) ≤ 2. If one of the M_{j}’s is closed with infinite fundamental group and does not admit an R^{3}, S^{2} × R or Sol-structure, then α_{2}(M) ≤ 2.

6. If M is closed then α_{3}(M) = α_{1}(M). If M is not closed then α_{3}(M) = ∞^{+}.

Let us briefly indicate how we prove that α_{2}(M) is positive. The important case is when M is an irreducible Haken 3-manifold with infinite fundamental group whose boundary is empty or consists of incompressible tori; the general case follows by further arguments. The Jaco-Shalen-Johannson splitting of M, together with the work of Thurston, gives a family of embedded incompressible tori which cut the manifold into pieces that are either Seifert manifolds or whose interiors admit complete finite-volume hyperbolic metrics. The α_{2}-invariants of the Seifert pieces can be computed explicitly. By analytic means we derive a lower bound for the α_{2}-invariants of the (compact) hyperbolic pieces. We then face the problem of understanding what happens to the Novikov-Shubin invariants when one glues along incompressible tori. This is done algebraically by means of inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence.

A description of the contents of the paper is as follows. The natural algebraic setting for our work is that of Hilbert A-modules, where A is a finite von Neumann algebra. In Section 1 we define the Betti numbers and Novikov-Shubin invariants of a (left-Fredholm) morphism of Hilbert A-modules, and derive some useful inequalities on the Novikov-Shubin invariants. In Section 2 we define the Betti numbers and Novikov-Shubin invariants of a Fredholm Hilbert A-chain complex. If one has a short exact sequence of Fredholm Hilbert A-chain complexes then there is an induced long weakly exact homology sequence, with which one can relate the Betti numbers of the chain complexes (Cheeger-Gromov [9]). We show that in Theorem 2.3 that the Novikov-Shubin invariants of the chain complexes are related by certain inequalities.

In Section 3 we specialize to the case of manifolds, in which A is the group von Neumann algebra N(π) of the fundamental group π. Proposition 3.2 gives the relations on the L^{2}-Betti numbers and Novikov-Shubin invariants due to Poincaré duality, and Proposition 3.7 computes the L^{2}-Betti numbers and Novikov-Shubin invariants of connected sums. In Theorem 3.8 we show that if M admits a homotopically nontrivial S^{1}-action then the L^{2}-Betti numbers vanish and the Novikov-Shubin invariants are bounded below by 1. In Corollary 3.4 we show that the Novikov-Shubin invariants of closed manifolds of dimension less than or equal to 4 depend only on the fundamental group. In Section 4 we compute the L^{2}-Betti numbers and Novikov-Shubin invariants of Seifert 3-manifolds (Theorems 4.1 and 4.4).

Section 5 first extends the results of [12, 14] on the equality of combinatorial and analytic L^{2}-topological invariants from the case of closed manifolds to that of manifolds with boundary. We then consider the Novikov-Shubin invariants of a compact 3-manifold M whose interior admits a complete finite-volume hyperbolic structure. If M is closed, the Novikov-Shubin invariants were computed in [24]. If M is not closed then we use a Mayer-Vietoris construction in the analytic setting, along with Theorem 2.3, to derive needed inequalities on the Novikov-Shubin invariants of the compact manifold, defined with absolute boundary conditions.

Theorem 0.1 is proven in Section 6. Section 7 has some remarks and gives some conjectures that are supported by the results of this paper. To understand the statements of Sections 3-7, it suffices to understand Definitions 1.3, 1.8 and 2.1.

One of us (J.L.) wishes to thank the IHES, the Max-Planck-Institut-Bonn and the Café La Chope for their hospitality while part of this work was done, and the NSF and the Humboldt Foundation for financial support.

1. L^{2}-Betti numbers and Novikov-Shubin invariants of morphisms of Hilbert A-modules

In this section we introduce the L^{2}-Betti numbers and Novikov-Shubin invariants of morphisms of Hilbert A-modules over a finite von Neumann algebra A. We study their behaviour under composition. For background material on finite von Neumann algebras and their Hilbert modules, we refer to [1, 8, 11, 31].

Let A be a von Neumann algebra with finite faithful normal trace tr_{A}. Let l^{2}(A) denote the Hilbert completion of A with respect to the inner product given by tr_{A}(a^{∗}b) for a, b ∈ A. A Hilbert A-module is a Hilbert space V with a continuous left A-module structure such that there is an isometric A-embedding of V into l^{2}(A) ⊗ H for some Hilbert space H. A morphism f : U ⟶ V of Hilbert A-modules is a bounded operator from U to V which commutes with multiplication by A. A morphism f : U ⟶ V is a weak isomorphism if its kernel is trivial and its image is dense. A sequence 0 ⟶ U ⟶^{j} V ⟶^{q} W ⟶ 0 of Hilbert A-modules is weakly exact if j is injective, clos(im(j)) = ker(q) and q has dense image.

A Hilbert A-module V is finitely generated if for some positive integer n, there is a surjective morphism ⊕_{i=1}^{n}l^{2}(A) ⟶ V. The dimension dim_{A}(V) of a finitely generated Hilbert A-module is the trace of any projection pr : ⊕_{i=1}^{n}l^{2}(A) ⟶ ⊕_{i=1}^{n}l^{2}(A) whose image is isometrically A-isomorphic to V. The notion of dimension can be extended to arbitrary Hilbert A-modules if one allows dim_{A} to take value in [0, ∞].

A morphism f : U ⟶ V has a polar decomposition f = i|f| as a product of morphisms. Here |f| : U ⟶ U is a positive operator given by |f| = \sqrt{f^{∗}f} and i : U ⟶ V is a partial isometry which restricts to an isometry between ker(f)^{⊥} and clos(im(f)). In particular, if f is a weak isomorphism then i is unitary, and so dim_{A}(U) = dim_{A}(V).

The von Neumann algebras of interest to us arise from a countable discrete group π. The group von Neumann algebra N (π) is defined to be the algebra of bounded operators on l^{2}(π) which commute with right multiplication by π. Letting e denote the identity element of π, the trace on N (π) is given by tr_{N (π)}(f) = ⟨f(e), e⟩. Then l^{2}(N (π)) is the same as l^{2}(π).

Lemma 1.1 1. dim_{A}(U) = 0 if and only if U = 0.

2. If U ⊂ V then dim_{A}(U) ≤ dim_{A}(V).

3. If U_{1} ⊃ U_{2} ⊃ … is a nested sequence of Hilbert A-submodules of U with dim_{A} U_{1} < ∞ then

dim_{A} ( ⋂_{n=1}^{∞} U_{n}) = lim_{n→∞} dim_{A}(U_{n}).

4. If 0 ⟶ U ⟶^{j} V ⟶^{q} W ⟶ 0 is weakly exact then

dim_{A}(V) = dim_{A}(U) + dim_{A}(W).

\underline{\underline{Proof}} : The first three assertions follow from the assumption that tr_{A} is a faithful normal trace. For the last assertion, we have canonical weak isomorphisms U ⟶ ker(q) and ker(q)^{⊥} ⟶ W. As V = ker(q) ⊕ ker(q)^{⊥}, the assertion follows.

Let f : U ⟶ V be a morphism of Hilbert A-modules. Let {E_{λ}^{f^{∗}f} : λ ∈ R} denote the (right-continuous) family of spectral projections of the positive operator f^{∗}f. In what follows, |x| will denote the norm of an element in a Hilbert A-module and ∥f ∥ will denote an operator norm.

Lemma 1.2 For λ ≥ 0, if x ∈ U is such that E_{λ^{2}}^{f^{∗}f}(x) = 0 and x ≠ 0 then |f(x)| > λ· |x|. If E_{λ^{2}}^{f^{∗}f}(x) = x then |f(x)| ≤ λ· |x|.

\underline{\underline{Proof}} : From the definition of the spectral family, we have

⟨f^{∗}f(x), x⟩ = ∫_{0}^{∞} λ d⟨E_{λ}^{f^{∗}f}(x), x⟩.

Since ⟨f^{∗}f(x), x⟩ =|f(x)|^{2}, the claim follows.

Definition 1.3 Define the spectral density function F : [0, ∞) ⟶ [0, ∞] of f by

F (f, λ) = dim_{A} (im(E_{λ^{2}}^{f^{∗}f})) .

We say that f is left-Fredholm if there is a λ > 0 such that F (f, λ) < ∞.

(To see the relationship with the usual notion of Fredholmness, suppose that A = C. Then f is Fredholm if and only if f and f^{∗} are left-Fredholm, and f is semi-Fredholm if and only if f or f^{∗} is left-Fredholm [3].)

Lemma 1.4 Let f : U ⟶ V be a left-Fredholm weak isomorphism. Let L ⊂ V be a Hilbert A-submodule. Then f restricts to a weak isomorphism from f^{−1}(L) to L.

\underline{\underline{Proof}} : From the polar decomposition of f, we may assume that U = V and f is positive. Clearly the restriction of f to f^{−1}(L) is 1-1, and it is enough to show that f(f^{−1}(L)) is dense in L. Now L has an orthogonal decomposition of the form L = clos(f(f^{−1}(L))) ⊕ M, where M is an A-submodule of L. As f(f^{−1}(M)) ⊂ M and f(f^{−1}(M)) ⊂ f(f^{−1}(L)), it follows that f(f^{−1}(M)) = 0. Thus M ∩ im(f) = 0. If we can show that dim_{A} M = 0 then Lemma 1.1 will imply that M = 0, and we will be done. For λ > 0, consider the map π_{λ} : M ⟶ E_{λ}^{f}(U) given by π_{λ}(m) = E_{λ}^{f}(m). If m ∈ ker(π_{λ}) then the spectral theorem shows that m ∈ im(f). Thus ker(π_{λ}) = 0, and Lemma 1.1 implies that dim_{A} M ≤ dim_{A}(E_{λ}^{f}(U)). As f is 1-1 and left-Fredholm, Lemma 1.1 implies that lim_{λ→0^{+}} dim_{A}(E_{λ}^{f}(U)) = 0. Thus dim_{A} M = 0.

Let L(f, λ) denote the set of all Hilbert A-submodules L of U with the property that if x ∈ L then |f(x)| ≤ λ· |x|.

Lemma 1.5 F (f, λ) = sup {dim_{A}(L) : L ∈ L(f, λ)} .

\underline{\underline{Proof}} : From Lemma 1.2, the image of E_{λ^{2}}^{f^{∗}f} belongs to L(f, λ). Hence

F (f, λ) ≤ sup {dim_{A}(L) : L ∈ L(f, λ)} ,

and it remains to show that for all L ∈ L(f, λ), dim_{A}(L) ≤ dim_{A}(im(E_{λ^{2}}^{f^{∗}f})). Lemma 1.2 implies that ker(E_{λ^{2}}^{f^{∗}f} |_{L}) is trivial. Hence E_{λ^{2}}^{f^{∗}f} induces a weak isomorphism from L to clos(E_{λ^{2}}^{f^{∗}f}(L)), and the claim follows from Lemma 1.1.

Lemma 1.6 Let f : U ⟶ V and g : V ⟶ W be morphisms of Hilbert A-modules. Then

1. F (f, λ) ≤ F (gf, ∥g ∥ ·λ).

2. F (g, λ) ≤ F (gf, ∥f ∥ ·λ) if f is left-Fredholm and has dense image.

3. F (gf, λ) ≤ F (g, λ^{1−r}) + F (f, λ^{r}) for all r ∈ (0, 1).

\underline{\underline{Proof}} : 1.) Consider L ∈ L(f, λ). For all x ∈ L, | gf(x) | ≤ ∥ g ∥ · | f(x) | ≤ ∥ g ∥ ·λ· | x |. This implies that L ∈ L(gf, ∥g ∥ ·λ), and the claim follows.

2.) Consider L ∈ L(g, λ). For all x ∈ f^{−1}(L), we have |gf(x)| ≤ λ· |f(x)| ≤ λ· ∥f ∥ · |x|, implying f^{−1}(L) ∈ L(gf, ∥f ∥ ·λ). Hence it remains to show dim_{A}(L) ≤ dim_{A}(f^{−1}(L)). Let p : U ⟶ U/ ker f be projection and let \overline{f} : U/ ker(f) ⟶ V be the map induced by f. Clearly \overline{f} is also left-Fredholm. Since p is surjective and \overline{f} is a weak isomorphism, Lemmas 1.1 and 1.4 imply that dim_{A} (f^{−1}(L)) ≥ dim_{A} (p(f^{−1}(L))) = dim_{A} (\overline{f}^{−1}(L)) = dim_{A}(L).

3.) Consider L ∈ L(gf, λ). Let L_{0} be the kernel of E_{λ^{2r}}^{f^{∗}f}|_{L}. We have a weakly exact sequence 0 ⟶ L_{0} ⟶ L ⟶ clos(E_{λ^{2r}}^{f^{∗}f}(L)) ⟶ 0. From Lemma 1.2, we have that | f(x) | > λ^{r}· | x | for all nonzero x ∈ L_{0}. In particular, f|_{L_{0}} : L_{0} ⟶ clos(f(L_{0})) is a weak isomorphism, and so Lemma 1.1 implies that dim_{A}(L_{0}) = dim_{A} (clos(f(L_{0}))). For x ∈ L_{0}, we have

|gf(x)| ≤ λ· |x| ≤ \frac{λ}{λ^{r}} · |f(x)| = λ^{1−r}· |f(x)| .

Hence clos(f(L_{0})) ∈ L(g, λ^{1−r}). This shows that dim_{A}(L_{0}) ≤ F (g, λ^{1−r}). From Lemma 1.1, dim_{A} (clos(E_{λ^{2r}}^{f^{∗}f}(L))) ≤ dim_{A} (im(E_{λ^{2r}}^{f^{∗}f})) = F (f, λ^{r}) and dim_{A}(L) = dim_{A}(L_{0}) +

dim_{A} (clos(E_{λ^{r}}^{f^{∗}f}(L))). This implies that dim_{A}(L) ≤ F (g, λ^{1−r}) + F (f, λ^{r}).

Definition 1.7 We say that a function F : [0, ∞) ⟶ [0, ∞] is a density function if F is monotone non-decreasing and right-continuous. If F and G are two density functions, we write F ⪯ G if there are C > 0 and ϵ > 0 such that F (λ) ≤ G(C · λ) holds for all λ ∈ [0, ϵ]. As in [18, 37], we say that F and G are dilatationally equivalent (in signs F ≃ G) if F ⪯ G and G ⪯ F. We say that F is Fredholm if there is a λ > 0 such that F (λ) < ∞.

Of course, the spectral density function F (f, λ) is a density function, and if f is left-Fredholm then F (f, λ) is a Fredholm density function.

Definition 1.8 Let F be a Fredholm density function. The Betti number of F is

b(F) = F (0).

Its Novikov-Shubin invariant is

α(F) = lim inf_{λ→0^{+}} \frac{ln(F (λ) − b(F))}{ln(λ)} ∈ [0, ∞],

provided that F (λ) > b(F) holds for all λ > 0. Otherwise, we put α(F) = ∞^{+}. If f is a left-Fredholm morphism of Hilbert A-modules, we write b(f) = b(F (f, λ)) and α(f) = α(F (f, λ)).

Here ∞^{+} is a new formal symbol which should not be confused with ∞. We have α(F) = ∞^{+} if and only if there is an ϵ > 0 such that F (λ) = b(F) for λ < ϵ. We note that any nonnegative real number, ∞ or ∞^{+} can occur as the value of the Novikov-Shubin invariant of a spectral density function. We make the following conventions:

Convention 1.9 The ordering on [0, ∞] ∪ {∞^{+}} is given by the standard ordering on R along with r < ∞ < ∞^{+} for all r ∈ R. For all α, β ∈ [0, ∞] ∪ {∞^{+}} we define

\frac{1}{α} ≤ \frac{1}{β} ⇔ α ≥ β.

Given α, β ∈ [0, ∞] ∪ {∞^{+}}, we give meaning to γ in the expression

\frac{1}{α} + \frac{1}{β} = \frac{1}{γ}

as follows: If α, β ∈ R, let γ be the real number for which this arithmetic expression of real numbers is true. If α ∈ R and β ∈ {∞, ∞^{+}}, put γ to be α. If β ∈ R and α ∈ {∞, ∞^{+}}, put γ to be β. If α and β belong to {∞, ∞^{+}} and are not both ∞^{+}, put γ = ∞. If both α and β are ∞^{+}, put γ = ∞^{+}. Given r ∈ (0, ∞) and α ∈ [0, ∞), we define rα ∈ [0, ∞) to be the ordinary product of real numbers, and we put r∞ = ∞ and r∞^{+} = ∞^{+}. For example,

\frac{1}{∞} + \frac{1}{π} = \frac{1}{π} ,  \frac{1}{∞^{+}} + \frac{1}{π} = \frac{1}{π} ,  \frac{1}{∞} + \frac{1}{∞^{+}} = \frac{1}{∞},  \frac{1}{∞^{+}} + \frac{1}{∞^{+}} = \frac{1}{∞^{+}} ,

\frac{1}{α} ≤ \frac{1}{∞} + \frac{1}{4} + \frac{1}{2} ⇔ α ≥ 4/3 and \frac{1}{α} ≤ \frac{1}{∞} + \frac{1}{∞^{+}} + \frac{1}{∞} ⇔ α ≥ ∞.

Here are the basic properties of these invariants.

Lemma 1.10 Let F and G be density functions and f : U ⟶ V be a morphism of A-Hilbert modules. Assume that G is Fredholm. Then:

1. If F ⪯ G then F is Fredholm and b(F) ≤ b(G).

2. If F ⪯ G and b(F) = b(G) then α(F) ≥ α(G).

3. If F ≃ G then b(F) = b(G) and α(F) = α(G).

4. α(G(λ^{r})) = r · α(G(λ)) for r ∈ (0, ∞).

5. α(G) = α(G − b(G)).

6. If f is left-Fredholm then b(f) = dim_{A}(ker(f^{∗}f)) = dim_{A}(ker(f)).

7. If f is zero and dim_{A}U < ∞ then f is left-Fredholm and α(f) = ∞^{+}.

8. An endomorphism f : U ⟶ U is an isomorphism if and only if f is left-Fredholm, b(f) = 0 and α(f) = ∞^{+}.

9. Assume that i : U′ → U is injective with closed image and p : V → V′ is surjective with finite-dimensional kernel. Then f is left-Fredholm if and only i ∘ f ∘ p is left-Fredholm, and in this case α(i ∘ f ∘ p) = α(f).

10. If F and G are Fredholm then α(F + G) = min {α(F), α(G)}.

\underline{\underline{Proof}} : The assertions 1.) to 5.) follow directly from the definitions.

6.) By definition, b(f) is the von Neumann dimension of im(E_{0}^{f^{∗}f}) = ker(f^{∗}f). As | f(x) |^{2}= ⟨f^{∗}f(x), x⟩, f and f^{∗}f have the same kernel.

7.) If f is zero then F (f, λ) = dim_{A} U for all nonnegative λ.

8.) If f is an isomorphism then the spectrum of f^{∗}f is bounded below by a positive number, and so F (f, λ) vanishes for small nonnegative λ. Conversely, suppose that f is left-Fredholm, b(f) = 0 and α(f) = ∞^{+}. Then the spectrum of f^{∗}f is contained in [a, b] for some positive real numbers a ≤ b, and an inverse of f^{∗}f is given by ∫_{a}^{b} λ^{−1}dE_{λ}^{f^{∗}f}. An inverse of f is given by (f^{∗}f)^{−1}f^{∗}.

9.) By the open mapping theorem, there is a positive constant C such that for all x,

C^{−1}· |x| ≤ |i(x)| ≤ C· |x| .

Hence F (f ∘ p, λ) and F (i ∘ f ∘ p, λ) are dilatationally equivalent. Assertion 3.) implies that i ∘ f ∘ p is left-Fredholm if and only if f ∘ p is left-Fredholm, and then α(i ∘ f ∘ p) = α(f ∘ p). We may write p as the composition j ∘ pr of an isomorphism j and a projection pr. Now one easily checks that F (f ∘ j, λ) and F (f, λ) are dilatationally equivalent, and that for all λ ≥ 0, we have F (f ∘ j, λ) + dim_{A}(ker(pr)) = F (f ∘ p, λ). Then assertions 3.) and 5.) prove the claim.

10.) As b(F + G) = b(F) + b(G), by assertion 5.) we may assume without loss of generality that b(F) = b(G) = b(F + G) = 0. As F, G ≤ F + G, assertion 2.) implies that α(F + G) ≤ min{α(F), α(G)}. To verify the reverse inequality, we may assume without loss of generality that α(F) ≤ α(G). The cases α(F) = 0 and α(F) = ∞^{+} are trivial, and so we assume that 0 < α(F) ≤ ∞. Consider any real number α satisfying 0 < α < α(F). Then there exists a constant K > 0 such that for small positive λ we have F (λ), G(λ) ≤ Kλ^{α}, and so F (λ) + G(λ) ≤ 2K · λ^{α}, implying that α ≤ α(F + G). The assertion follows.

Lemma 1.11 Let f : U ⟶ V and g : V ⟶ W be morphisms of Hilbert A-modules.

1. If gf is left-Fredholm then f is left-Fredholm. If in addition ker(g) ∩ im(f) = {0} then

α(f) ≥ α(gf).

2. Suppose that f is left-Fredholm and has dense image. If gf is left-Fredholm then g is

left-Fredholm and

α(g) ≥ α(gf).

3. Suppose that f and g are left-Fredholm. Then gf is left-Fredholm. If in addition ker(g) ⊂ clos(im(f)) then

\frac{1}{α(gf)} ≤ \frac{1}{α(f)} + \frac{1}{α(g)}.

\underline{\underline{Proof}} : 1.) The Fredholmness claim follows from Lemma 1.6. If in addition ker(g)∩im(f) = {0} then ker(gf) = ker(f) and hence b(gf) = b(f). Now the assertion follows from Lemma 1.6 and Lemma 1.10.2.

2. and 3.) The Fredholmness claims follow from Lemma 1.6. We can factorize f as a product of a projection p : U ⟶ U/ ker(f) and an injective morphism \overline{f} : U/ ker(f) ⟶ V. From Lemma 1.10.9, α(\overline{f}) = α(f) and α(g\overline{f}) = α(gf), so we may assume without loss of generality that f is injective. Then f induces an injection ker(gf) ⟶ ker(g), and Lemma 1.1 gives that b(gf) ≤ b(g).

If f has dense image then Lemma 1.6 gives F (g, λ) − b(g) ≤ F (gf, ∥f ∥ ·λ) − b(gf). Assertion 2.) now follows from Lemma 1.10.2. For assertion 3.), by assumption ker(g) ⊂ clos(im(f)). As f : U ⟶ clos(im(f)) is assumed to be a weak isomorphism, Lemma 1.4 implies that b(gf) = b(g) = b(f) + b(g). From Lemma 1.6, if 0 < r < 1 then

F (gf, λ) − b(gf) ≤ F (f, λ^{r}) − b(f) + F (g, λ^{1−r}) − b(g).

Parts 2, 4, 5 and 10 of Lemma 1.10 give α(gf) ≥ min {r · α(f), (1 − r) · α(g)}. Taking inverses gives

\frac{1}{α(gf)} ≤ max \{\frac{1}{r · α(f)}, \frac{1}{(1 − r) · α(g)}\} .

We only need to consider the case α(f), α(g) ∈ (0, ∞), the other cases being now trivial. Since \frac{1}{r·α(f)} (resp. \frac{1}{(1−r)·α(g)}) is a strictly monotonically decreasing (resp. increasing) function in r, the maximum on the right side, viewed as a function of r, obtains its minimum precisely when the two functions of r have the same value. One easily checks that this is the case if and only if r = \frac{α(g)}{α(f)+α(g)}, and the claim follows.

Lemma 1.12 Let ϕ : U_{1} ⟶ V_{1}, γ : U_{2} ⟶ V_{1} and ξ : U_{2} ⟶ V_{2} be morphisms of Hilbert A-modules. Then

1. ( \begin{matrix}ϕ & 0 \\ 0 & ξ \end{matrix} ) is left-Fredholm if and only if ϕ and ξ are left-Fredholm. In this case,

α ( \begin{matrix}ϕ & 0 \\ 0 & ξ \end{matrix} ) = min \{α(ϕ), α(ξ)\} .

2. Suppose that ϕ is invertible. Then ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) is left-Fredholm if and only ξ is left-

Fredholm. In this case, α ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) = α(ξ).

3. If ϕ and ξ are left-Fredholm then ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) is left-Fredholm. If ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) is left-

Fredholm then ϕ is left-Fredholm.

4. If ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) is left-Fredholm and ξ is injective then α(ϕ) ≥ α ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ).

If in addition ξ is left-Fredholm then (α ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ))^{−1} ≤ \frac{1}{α(ϕ)} + \frac{1}{α(ξ)}.

5. If ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) is left-Fredholm and ϕ has dense image then ξ is left-Fredholm and

α(ξ) ≥ α ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} )  and  (α ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ))^{−1} ≤ \frac{1}{α(ϕ)} + \frac{1}{α(ξ)}.

6. Suppose that ϕ is left-Fredholm and clos(im(ϕ))^{⊥} is finite-dimensional. Then ϕ^{∗} is left-Fredholm, F (ϕ, λ) − b(ϕ) = F (ϕ^{∗}, λ) − b(ϕ^{∗}) and α(ϕ) = α(ϕ^{∗}).

\underline{\underline{Proof}} :

1.) follows from Lemma 1.10.10., using F (( \begin{matrix}ϕ & 0 \\ 0 & ξ \end{matrix} ) , λ) = F (ϕ, λ) + F (ξ, λ).

2.) Apply Lemma 1.10.9 and assertion 1.) to ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) = ( \begin{matrix}ϕ & 0 \\ 0 & ξ \end{matrix} ) · ( \begin{matrix}1 & ϕ^{−1}γ \\ 0 & 1 \end{matrix} ).

In what follows, we write ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) = gf, where g = ( \begin{matrix}1 & γ \\ 0 & ξ \end{matrix} ) and f = ( \begin{matrix}ϕ & 0 \\ 0 & 1 \end{matrix} ).

From assertion 2.), g is left-Fredholm if and only if ξ is left-Fredholm, and in this case α(g) = α(ξ).

3.) If ϕ and ξ are left-Fredholm then Lemma 1.10.8 and assertion 1.) imply that g and f are left-Fredholm. Then Lemma 1.11.3 implies that ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} ) is left-Fredholm. If ( \begin{matrix}ϕ & γ \\ 0 & ξ \end{matrix} )

is left-Fredholm then Lemma 1.11.1 implies that f is left-Fredholm, and hence ϕ is left-Fredholm.

4.) If ξ is injective then g is injective. The first inequality now follows from Lemma 1.11.1. If ξ is left-Fredholm then g is left-Fredholm and the second inequality follows from Lemma 1.11.3.

5.) If ϕ has dense image then f has dense image and Lemma 1.11.2 implies that g is left-Fredholm. Hence ξ is left-Fredholm, and the first inequality follows from Lemma 1.11.2. The second inequality follows from Lemma 1.11.3.

6.) Write ϕ as the composition U ⟶^{p} U/ ker(ϕ) ⟶^{\overline{ϕ}} clos(im(ϕ)) ⟶^{i} V, where p is projection, i is inclusion and \overline{ϕ} is a weak isomorphism. Then ϕ^{∗} = p^{∗} ∘ \overline{ϕ}^{∗} ∘ i^{∗}. Lemma 1.10.9 shows that ϕ is left-Fredholm if and only if \overline{ϕ} is left-Fredholm, and one can check that F (ϕ, λ) − b(ϕ) = F (\overline{ϕ}, λ). As i^{∗} has finite-dimensional kernel, a similiar statement holds for ϕ^{∗} and \overline{ϕ}^{∗}. Hence we may assume that ϕ is a weak isomorphism. As ϕ(ϕ^{∗}ϕ) = (ϕϕ^{∗})ϕ and ϕ^{∗}(ϕϕ^{∗}) = (ϕ^{∗}ϕ)ϕ^{∗}, ϕ and ϕ^{∗} induce injective morphisms ϕ̂_{λ} : im(E_{λ}^{ϕ^{∗}ϕ}) ⟶ im(E_{λ}^{ϕϕ^{∗}}) and ϕ̂_{λ}^{∗} : im(E_{λ}^{ϕϕ^{∗}}) ⟶ im(E_{λ}^{ϕ^{∗}ϕ}). Using Lemma 1.1, we have that F (ϕ, λ) = F (ϕ^{∗}, λ). It follows that ϕ^{∗} is left-Fredholm.

2. L^{2}-Betti numbers and Novikov-Shubin invariants of Hilbert A-chain complexes

In this section we introduce and study the L^{2}-Betti numbers and Novikov-Shubin invariants of chain complexes, and investigate their behaviour with respect to exact sequences and homotopy equivalences.

A Hilbert A-chain complex C is a chain complex of Hilbert A-modules whose differentials are morphisms of such modules, i.e. the differentials are bounded operators compatible with the A-action. It is said to be finite if C_{n} is a finitely generated Hilbert Amodule for all integers n and is zero for all but a finite number of integers n. Letting c_{p} : C_{p} → C_{p−1} denote the p-th differential of C, the p-th homology group of C is defined by H_{p}(C) = ker(c_{p})/clos(im(c_{p+1})). Note that we have to quotient by the closure of the image of c_{p+1} if we want to ensure that H_{p}(C) is a Hilbert space. We say that C is weakly exact if its homology groups H_{p}(C) vanish. We say that C is exact if ker(c_{p}) = im(c_{p+1}) for all p.

Definition 2.1 Let C be a Hilbert A-chain complex with p-th differential c_{p}. We say that C is Fredholm at p if the induced morphism \overline{c_{p}} : C_{p}/clos(im(c_{p+1})) ⟶ C_{p−1} is left-Fredholm. We say that C is Fredholm if C is Fredholm at p for all p. Suppose that C is Fredholm at p. Its p-th Betti-number is

b_{p}(C) = dim_{A}(H_{p}(C)).

Its p-th Novikov-Shubin invariant is

α_{p}(C) = α(c_{p}).

Put

α̃_{p}(C) = min {α(c_{p+1}), α(c_{p})} .

Note that if C is Fredholm at p then H_{p}(C) = ker(\overline{c_{p}}) is finite-dimensional. The Fredholmness condition on C is automatically satisfied if C is finite. This will be the case when one deals with the cellular chain complex of a manifold. When one deals with differential forms on a manifold, the Fredholmness is not automatic and requires some extra analysis. The invariant α̃_{p}(C) corresponds to the notion of Novikov-Shubin invariants as introduced in [37]. However, it turns out to be easier and more efficient to deal with the numbers α_{p}(C).

We begin by recalling the long homology sequence associated to an exact sequence of Hilbert A-chain complexes 0 ⟶ C ⟶^{j} D ⟶^{q} E ⟶ 0. There is a sequence

… ⟶^{δ_{p+1}} H_{p}(C) ⟶^{H_{p}(j)} H_{p}(D) ⟶^{H_{p}(q)} H_{p}(E) ⟶^{δ_{p}} H_{p−1}(C) ⟶^{H_{p−1}(j)} …

where the boundary operator δ_{p} : H_{p}(E) ⟶ H_{p−1}(C) is defined as follows: Let x ∈ ker(e_{p}) be a representative of [x] in H_{p}(E). Choose y ∈ D_{p} so that q_{p}(y) = x, and z ∈ ker(c_{p−1}) so that j_{p−1}(z) = d_{p}(y). Then δ_{p}([x]) is defined to be the class [z] ∈ H_{p−1}(C). Note that the homology sequence is always defined, but is generally not weakly exact if one makes no Fredholmness assumptions. The next theorem follows from inspecting the proof of [9, Theorem 2.1].

Theorem 2.2 1. The long homology sequence is weakly exact at H_{p}(E) if C is Fredholm at p.

2. The long homology sequence is weakly exact at H_{p}(C) if D is Fredholm at p + 1.

3. The long homology sequence is weakly exact at H_{p}(D) if E is Fredholm at p+1.

The next theorem is the main result of this section. We mention that one can give examples to show that the inequalities below are sharp.

Theorem 2.3 (Additivity inequalities for the Novikov-Shubin invariants) Let 0 ⟶ C ⟶^{j} D ⟶^{q} E ⟶ 0 be an exact sequence of Hilbert A-chain complexes. Let δ : H_{p}(E) ⟶ H_{p−1}(C) be the boundary operator in the long weakly exact homology sequence.

1. Suppose that C and E are Fredholm at p. Then D is Fredholm at p, δ_{p} is Fredholm

and

\frac{1}{α_{p}(D)} ≤ \frac{1}{α_{p}(C)} + \frac{1}{α_{p}(E)} + \frac{1}{α(δ_{p})}.

2. Suppose that C is Fredholm at p − 1 and D is Fredholm at p. Then E is Fredholm at p, H_{p−1}(j) is Fredholm and

\frac{1}{α_{p}(E)} ≤ \frac{1}{α_{p−1}(C)} + \frac{1}{α_{p}(D)} + \frac{1}{α(H_{p−1}(j))}.

3. Suppose that D is Fredholm at p and E is Fredholm at p + 1. Then C is Fredholm at p, H_{p}(q) is Fredholm and

\frac{1}{α_{p}(C)} ≤ \frac{1}{α_{p}(D)} + \frac{1}{α_{p+1}(E)} + \frac{1}{α(H_{p}(q))}.

\underline{\underline{Proof}} : 1.) The exact sequence 0 ⟶ C ⟶^{j} D ⟶^{q} E ⟶ 0 induces the following commutative diagram with exact rows, where \overline{q_{p}}, \overline{d_{p}} and \overline{e_{p}} are canonical homomorphisms induced from q_{p}, d_{p} and e_{p}, and i is inclusion:

0  ⟶ ker \overline{q_{p}}  \underrightarrow{i} D_{p}/clos(im(d_{p+1}))  \underrightarrow{\overline{q_{p}}} E_{p}/ ker(e_{p})  ⟶ 0
∂_{p} ↓  _{\overline{d_{p}} ↓}  _{\overline{e_{p}} ↓}

0  ⟶ C_{p−1}  \underrightarrow{j_{p−1}}  D_{p−1}  \underrightarrow{q_{p−1}}  E_{p−1}  ⟶ 0.

To define ∂_{p} in the above diagram, let x ∈ ker(e_{p}q_{p}) represent [x] ∈ ker(\overline{q_{p}}). Then d_{p}(x) = j_{p−1}(y) for a unique y ∈ C_{p−1}. We put ∂_{p}([x]) = y. (In fact, y ∈ ker(c_{p−1}).)

Suppose for a moment that we already know that ∂_{p} is left-Fredholm. From Lemma 1.10.9, \overline{e_{p}} is left-Fredholm and α_{p}(\overline{e_{p}}) = α_{p}(E). Lemma 1.12.3 implies that \overline{d_{p}} is left-Fredholm and hence D is Fredholm at p. As \overline{e_{p}} is injective, Lemma 1.12.4 gives that

\frac{1}{α_{p}(D)} ≤ \frac{1}{α(∂_{p})} + \frac{1}{α_{p}(E)}.

Hence it remains to show that ∂_{p} and δ_{p} are left-Fredholm and that

\frac{1}{α(∂_{p})} ≤ \frac{1}{α_{p}(C)} + \frac{1}{α(δ_{p})}.

We construct a sequence which we will show to be weakly exact:

C_{p} ⟶^{\overline{j_{p}}} ker(\overline{q_{p}}) \underrightarrow{_{q_{p}}} H_{p}(E) ⟶ 0.

The map \overline{j_{p}} is induced from j_{p} in the obvious way. To define q̂_{p}, consider an x ∈ D_{p} whose class [x] ∈ D_{p}/clos(im(d_{p+1})) lies in ker(\overline{q_{p}}). Then q_{p}(x) is in the kernel of e_{p} and determines a class [q_{p}(x)] in H_{p}(E). Define q̂_{p}([x]) to be [q_{p}(x)]. One easily checks that q̂_{p} ∘ \overline{j_{p}} is zero and q̂_{p} is surjective. We will show that ker(q̂_{p}) is contained in clos(im(\overline{j_{p}})). Consider [x] ∈ ker(q̂_{p}) with representative x ∈ D_{p}. Since [q_{p}(x)] ∈ H_{p}(E) is zero, there is a sequence y_{n} ∈ E_{p+1} such that in E_{p}:

lim_{n→∞} (q_{p}(x) − e_{p+1}(y_{n})) = 0.

As q_{p+1} is surjective, there is a sequence {u_{n}}_{n=1}^{∞} in D_{p+1} such that y_{n} = q_{p+1}(u_{n}). Thus

lim_{n→∞} q_{p} (x − d_{p+1}(u_{n})) = 0.

We write x − d_{p+1}(u_{n}) = j_{p}(w_{n}) + r_{n}, where w_{n} ∈ C_{p} and r_{n} ∈ im(j_{p})^{⊥}. Then we obtain lim_{n→∞} q_{p}(r_{n}) = 0. As the restriction of q_{p} to im(j_{p})^{⊥} is an isomorphism, we conclude lim_{n→∞} r_{n} = 0. Thus

x = lim_{n→∞} (j_{p}(w_{n}) + d_{p+1}(u_{n})) ,

and hence in D_{p}/clos(im(d_{p+1}))

[x] = lim_{n→∞} \overline{j_{p}}(w_{n}).

This finishes the proof of weak exactness.

Next, we construct a commutative diagram with exact rows

0 ⟶  ker(q̂_{p})  \underrightarrow{i_{1}} ker(\overline{q_{p}}) \underrightarrow{q̂_{p}} H_{p}(E) ⟶ 0
\overline{∂_{p}} ↓  _{∂_{p} ↓}  _{δ_{p} ↓}

0 ⟶ clos(im(c_{p})) \underrightarrow{i_{2}} ker(c_{p−1}) \underrightarrow{pr} H_{p−1}(C) ⟶ 0.

The maps i_{1} and i_{2} are the canonical inclusions and the map pr is the canonical projection. The map \overline{∂_{p}} is induced from ∂_{p}, and the fact that its range lies in clos(im(c_{p})) follows from the weak exactness of the preceding sequence. One easily verifies that the diagram commutes. The rows are clearly exact.

Let ˜ \overline{j_{p}} : C_{p} ⟶ ker(q̂_{p}) be the morphism with dense image induced from \overline{j_{p}}. One easily checks that \overline{∂_{p}} ∘ ˜ \overline{j_{p}} = c_{p}. As c_{p} is left-Fredholm by assumption, Lemma 1.11.1 shows that ˜ \overline{j_{p}} is left-Fredholm. Lemma 1.11.2 implies that \overline{∂_{p}} is left-Fredholm and

α(\overline{∂_{p}}) ≥ α(c_{p}) = α_{p}(C).

As H_{p}(E) is finite-dimensional, δ_{p} is left-Fredholm. Then from Lemma 1.12.3, ∂_{p} is left-Fredholm. As \overline{∂_{p}} has dense image, Lemma 1.12.5 implies

\frac{1}{α(∂_{p})} ≤ \frac{1}{α(\overline{∂_{p}})} + \frac{1}{α(δ_{p})}.

This finishes the proof of the first assertion of Theorem 2.3.

2.) Recall that in general [27, p. 213], the n-th differential of the mapping cylinder of a chain map g : C ⟶ D is defined by

( \begin{matrix}−c_{n−1} & 0 & 0 \\ −id & c_{n} & 0 \\ g_{n−1} & 0 & d_{n} \end{matrix} ) : C_{n−1} ⊕ C_{n} ⊕ D_{n} ⟶ C_{n−2} ⊕ C_{n−1} ⊕ D_{n−1}.

There is a canonical map i : C ⟶ cyl(g) and cone(g) is defined to be the cokernel of i. That is, the n-th differential of cone(g) is

( \begin{matrix}−c_{n−1} & 0 \\ g_{n−1} & d_{n} \end{matrix} ) : C_{n−1} ⊕ D_{n} ⟶ C_{n−2} ⊕ D_{n−1}.

We define cone(C) to be the mapping cone of the identity map on C, and the suspension ΣC to be the mapping cone of the 0-map on C, i.e. (ΣC)_{n} = C_{n−1}.

In our case there is a canonical exact sequence 0 ⟶ D ⟶ cyl(q) ⟶ cone(q) ⟶ 0 and chain homotopy equivalences E ⟶ cyl(q) and ΣC ⟶ cone(q). We will show later that the numbers α(c_{p}) are homotopy invariants. So we may assume the existence of an exact sequence 0 ⟶ D ⟶ E ⟶ ΣC ⟶ 0. Moreover, the connecting map from H_{p}(ΣC) to H_{p−1}(D) agrees under these identifications with the map H_{p−1}(j) : H_{p−1}(C) ⟶ H_{p−1}(D). The claim now follows from assertion 1.).

3.) Repeat the argument in the proof of assertion 2.), yielding a short exact sequence 0 ⟶ E ⟶ ΣC ⟶ ΣD ⟶ 0.

The dual chain complex C^{∗} is the cochain complex with the same chain modules as C and the adjoints of the differentials of C as codifferentials. The definitions of the Betti numbers and the Novikov-Shubin invariants carry over directly to cochain complexes. The Laplace operator Δ_{p} : C_{p} ⟶ C_{p} is defined to be c_{p+1}c_{p+1}^{∗} + c_{p}^{∗}c_{p}.

Lemma 2.4 Let C and D be Hilbert A-chain complexes.

1. Δ_{p} is left-Fredholm if and only if C is Fredholm at p and p + 1. In this case,

2 · α̃_{p}(C) = α(Δ_{p}) and b_{p}(C) = b_{p}(Δ_{p}).

2. C is Fredholm at p if and only if C^{∗} is Fredholm at p. In this case,

α_{p}(C) = α_{p}(C^{∗}) and b_{p}(C) = b_{p}(C^{∗}).

3. C ⊕ D is Fredholm at p if and only if C and D are Fredholm at p. In this case,

α_{p}(C ⊕ D) = min {α_{p}(C), α_{p}(D)} and b_{p}(C ⊕ D) = b_{p}(C) + b_{p}(D).

1.) The Hodge decomposition theorem (see e.g. [31, Theorem 3.7], the proof of which extends to the Fredholm case) gives the claim for the Betti numbers. Moreover, we have the following commutative square with isomorphisms as horizontal morphisms:

ker(c_{p})^{⊥} ⊕ ker(c_{p+1}^{∗})^{⊥} ⊕ ker(Δ_{p}) \underrightarrow{≅} C_{p}
c_{p}^{∗}c_{p} ⊕ c_{p+1}c_{p+1}^{∗} ⊕ 0 ↓  _{↓ Δ_{p}}

ker(c_{p})^{⊥} ⊕ ker(c_{p+1}^{∗})^{⊥} ⊕ ker(Δ_{p}) \underrightarrow{≅} C_{p}

In what follows, we consider c_{p} to be an operator from ker(c_{p})^{⊥} to ker(c_{p}^{∗})^{⊥}, and similarly for c_{p+1}, c_{p}^{∗} and c_{p+1}^{∗}. Lemmas 1.10.7 and 1.12.1 imply that Δ_{p} is left-Fredholm if and only if both c_{p}^{∗}c_{p} and c_{p+1}c_{p+1}^{∗} are left-Fredholm, and in this case, α(Δ_{p}) = min {α(c_{p}^{∗}c_{p}), α(c_{p+1}c_{p+1}^{∗})}.

In general, as E_{λ}^{f^{∗}f} = E_{λ^{2}}^{(f^{∗}f)^{2}}, Lemma 1.10.4 implies that f is left-Fredholm if and only if f^{∗}f is, and in this case α(f^{∗}f) = 2 · α(f). We have shown in Lemma 1.12.6 that if f is left-Fredholm and its cokernel is finite-dimensional then f^{∗} is left-Fredholm and α(f) = α(f^{∗}). This implies that if c_{p} is left-Fredholm then c_{p}^{∗}c_{p} is left-Fredholm, and in this case 2 · α(c_{p}) = α(c_{p}^{∗}c_{p}). Moreover, c_{p+1}c_{p+1}^{∗} is left-Fredholm if and only if c_{p+1} is left-Fredholm, and in this case 2 · α(c_{p+1}) = α(c_{p+1}c_{p+1}^{∗}). Now the claim follows.

2.) follows from assertion 1.)

3.) is a consequence of Lemma 1.12.1.

We recall that C is said to be contractible if C has a chain contraction γ, i.e. a collection of morphisms γ_{p} : C_{p} ⟶ C_{p+1} such that γ_{p−1}c_{p} + c_{p+1}γ_{p} = id. for all p.

Lemma 2.5 The following assertions are equivalent for a Hilbert A-chain complex C:

1. C is contractible.

2. Δ_{p} is invertible for all p.

3. C is Fredholm and for all p, b_{p}(C) = 0 and α_{p}(C) = ∞^{+}.

\underline{\underline{Proof}} : 1.) ⇒ 3.) Using c_{p} and γ_{p−1}, we can construct morphisms \overline{c_{p}} : C_{p}/clos(im(c_{p+1})) ⟶ C_{p−1} and \overline{γ_{p−1}} : C_{p−1} ⟶ C_{p}/clos(im(c_{p+1})) such that \overline{γ_{p−1}} ∘ \overline{c_{p}} = id. Hence \overline{c_{p}} induces an invertible operator onto its image. Lemma 1.10.8-9 implies that \overline{c_{p}} is left-Fredholm, b_{p}(\overline{c_{p}}) = 0 and α(\overline{c_{p}}) = ∞^{+}.

3.) ⇒ 2.) From Lemma 2.4, Δ_{p} is left-Fredholm, b(Δ_{p}) = 0 and α(Δ_{p}) = ∞^{+} for all p. Now apply Lemma 1.10.8.

2.) ⇒ 1.) Suppose that Δ_{p} is invertible for all p. Then Δ_{p+1}^{−1} ∘ c_{p+1}^{∗} is a chain contraction of C.

We now reprove the homotopy invariance of the L^{2}-Betti numbers and the Novikov-Shubin invariants [12, 14, 18].

Theorem 2.6 (Homotopy invariance) If f : C ⟶ D is a chain homotopy equivalence then for all p ∈ Z we have

F (c_{p}) ≃ F (d_{p}), b_{p}(C) = b_{p}(D), α_{p}(C) = α_{p}(D) and α̃_{p}(C) = α̃_{p}(D).

\underline{\underline{Proof}} : There are exact sequences of chain complexes 0 ⟶ C ⟶ cyl(f) ⟶ cone(f) ⟶ 0 and 0 ⟶ D ⟶ cyl(f) ⟶ cone(C) ⟶ 0 with cone(f) and cone(C) being contractible. We obtain chain isomorphisms C ⊕ cone(f) ⟶ cyl(f) and D ⊕ cone(C) ⟶ cyl(f) by the following general construction for an exact sequence 0 ⟶ C ⟶^{j} D ⟶^{q} E ⟶ 0 with contractible E: Choose a chain contraction ϵ for E and for each p a morphism t_{p} : E_{p} ⟶ D_{p} such that q_{p} ∘ t_{p} = id. Put

s_{p} = d_{p+1} ∘ t_{p+1} ∘ ϵ_{p} + t_{p} ∘ ϵ_{p−1} ∘ e_{p}.

This defines a chain map s : E ⟶ D such that q ∘ s = id. Define a chain map u : D → C by saying that for x ∈ D_{p}, u_{p}(x) is the unique y ∈ C_{p} such that x = s_{p}q_{p}(x) + j_{p}(y). Then j + s is a chain isomorphism C ⊕ E ⟶ D, with inverse u ⊕ q. Since C ⊕ cone(f) and D ⊕ cone(C) are isomorphic and cone(f) and cone(C) are contractible, Lemma 2.5 implies that F (c_{p}) ≃ F (d_{p}), from which the other assertions follow.

3. L^{2}-Betti numbers and Novikov-Shubin invariants of manifolds

In this section we analyse the L^{2}-Betti numbers and the Novikov-Shubin invariants of compact manifolds.

Throughout this section we will use the following setup: Let M be a compact connected orientable smooth manifold of dimension m with fundamental group π and universal cover M̃. Suppose that ∂M is the union of two submanifolds ∂_{0}M and ∂_{1}M such that ∂(∂_{0}M) = ∂_{0}M ∩ ∂_{1}M = ∂(∂_{1}M). We allow that ∂_{0}M or ∂_{1}M are empty. Let ˜∂_{0}M denote the preimage of ∂_{0}M under the projection M̃ ⟶ M. Let A be a finite von Neumann algebra, V be a finitely generated Hilbert A-module and μ : π ⟶ Iso_{A}(V)^{op} be a right unitary representation of π. In most applications A will be the von Neumann algebra N(π) of π, V will be l^{2}(π) and μ will be the right regular representation.

Let C(M̃, ˜∂_{0}M) be the cellular Zπ-chain complex coming from the lift of any CW - decomposition of (M, ∂_{0}M) to a π-equivariant CW -decomposition of (M̃, ˜∂_{0}M). Note that π acts on the left on C(M̃, ˜∂_{0}M), and on the right on V. Let C(M, ∂_{0}M; V) denote the Hilbert A-chain complex V ⊗_{Zπ} C(M̃, ˜∂_{0}M). If c denotes the differential of C(M, ∂_{0}M; V), define the L^{2}-homology H_{p}(M, ∂_{0}M; V) with coefficients in V to be the Hilbert A-module ker(c_{p})/clos(im(c_{p})). In this section we will only deal with homology. We note that the corresponding cohomology groups are isometrically isomorphic to the homology groups. Recall that we have defined the L^{2}-Betti numbers and Novikov-Shubin invariants for chain complexes in Definition 2.1. Since they are homotopy invariants (see Theorem 2.6), the following definition is independent of the choice of the CW -decomposition:

Definition 3.1 Define the p-th L^{2}-Betti-number of (M, ∂_{0}M), with coefficients in V, to be

b_{p}(M, ∂_{0}M; V) = b_{p}(C(M, ∂_{0}M; V)) = dim_{A}(H_{p}(M, ∂_{0}M; V)).

Define the p-th Novikov-Shubin invariant to be

α_{p}(M, ∂_{0}M; V) = α_{p}(C(M, ∂_{0}M; V))

and put

α̃_{p}(M, ∂_{0}M; V) = α̃_{p}(C(M, ∂_{0}M; V)).

If V = l^{2}(π) then we abbreviate:

b_{p}(M, ∂_{0}M) = b_{p}(M, ∂_{0}M; l^{2}(π));
α_{p}(M, ∂_{0}M) = α_{p}(M, ∂_{0}M; l^{2}(π));
α̃_{p}(M, ∂_{0}M) = α̃_{p}(M, ∂_{0}M; l^{2}(π)).

We abbreviate b_{p}(M, ∅) by b_{p}(M), α_{p}(M, ∅) by α_{p}(M) and α̃_{p}(M, ∅) by α̃_{p}(M).

We refer to α_{p}(M, ∂_{0}M; V) as the Novikov-Shubin invariant, whereas in the previous literature α̃_{p}(M, ∂_{0}M; V) is called the Novikov-Shubin invariant. Also, in previous articles the values ∞ and ∞^{+} are not distinguished. Moreover, we use the normalization of [24], which differs by a factor of 2 from that used in [14, 18, 37].

We start with Poincaré duality. It gives a Zπ-chain homotopy equivalence

∩ [M] : C^{m−∗}(M̃, ˜∂_{1}M) ⟶ C_{∗}(M̃, ˜∂_{0}M).

Tensoring over Zπ with V then gives a chain homotopy equivalence of Hilbert A-chain complexes. From Theorem 2.6 and Lemma 2.4 we derive

Proposition 3.2 (Poincaré duality) 1. b_{m−p}(M, ∂_{1}M; V) = b_{p}(M, ∂_{0}M; V).

2. α_{m+1−p}(M, ∂_{1}M; V) = α_{p}(M, ∂_{0}M; V).

3. α̃_{m−p}(M, ∂_{1}M; V) = α̃_{p}(M, ∂_{0}M; V).

Lemma 3.3 Let (f, f_{0}) : (M, ∂_{0}M) ⟶ (N, ∂_{0}N) be a map between pairs such that f and f_{0} are n-connected for some n ≥ 2. Then

1. b_{p}(M, ∂_{0}M; V) = b_{p}(N, ∂_{0}N; V) for p ≤ n − 1 and b_{n}(M, ∂_{0}M; V) ≥ b_{n}(N, ∂_{0}N; V).

2. α_{p}(M, ∂_{0}M; V) = α_{p}(N, ∂_{0}N; V) for p ≤ n.

\underline{\underline{Proof}} : Let C(f̃) : C(M̃, ˜∂_{0}M) ⟶ C(Ñ, ˜∂_{0}N) be the Zπ-chain map induced by f. We will abbreviate cyl(C(f̃)) by cyl and cone(C(f̃)) by cone. We have the exact sequence

0 ⟶ C(M̃, ˜∂_{0}M) \underrightarrow{i} cyl \underrightarrow{pr} cone ⟶ 0

Let P be the subcomplex of cone such that P_{i} = {0} for i ≤ n, P_{n+1} is the kernel of the (n + 1)-differential of cone and P_{i} = cone_{i} for i > n + 1. As cone is n-connected by the Hurewicz theorem, P_{n+1} is finitely-generated stably free, and the inclusion of P into cone is a homotopy equivalence. A chain complex C is elementary if it is concentrated in two adjacent dimensions n and n+1 and is given there by the same module C_{n+1} = C_{n}, with the identity as the n + 1-th differential. By possibly adding a finitely-generated free elementary chain complex concentrated in dimensions n + 1 and n + 2 to P, we obtain a finite free Zπ-chain complex Q together with a chain homotopy equivalence g : Q ⟶ cone. Let D be the pullback chain complex of g : Q ⟶ cone and the canonical projection cyl ⟶ cone, i.e. the kernel of g ⊕ pr : Q ⊕ cyl ⟶ cone. Then we obtain a short exact sequence

0 ⟶ C(M̃, ˜∂_{0}M) ⟶ D ⟶ Q ⟶ 0

of finitely-generated free Zπ-chain complexes such that D is chain homotopy equivalent to C(Ñ, ˜∂_{0}N) and Q_{i} = {0} for i ≤ n. By Theorem 2.6, it suffices to prove the claim for l^{2}(π) ⊗_{Zπ} C(M̃, ˜∂_{0}M) and l^{2}(π) ⊗_{Zπ} D. Since these chain complexes have the same chain modules and differentials in dimensions less than or equal to n, the claim follows.

Corollary 3.4 1. The L^{2}-Betti numbers b_{p}(M) (respectively the Novikov-Shubin invariants α_{p}(M)) of a compact connected manifold depend only on the fundamental group provided that p ≤ 1 (respectively p ≤ 2).

2. The L^{2}-Betti numbers b_{p}(M) and the Novikov-Shubin invariants α_{p}(M) of a closed connected 3-manifold depend only on the fundamental group.

3. The Novikov-Shubin invariants α_{p}(M) of a closed connected 4-manifold depend only on the fundamental group.

\underline{\underline{Proof}} : The classifying map M ⟶ Bπ for π = π_{1}(M) is 2-connected, and Bπ can be chosen to be a CW -complex whose 2-skeleton Bπ^{2} is finite. Hence Lemma 3.3 implies that α_{p}(M) = α_{p}(Bπ^{2}) (respectively b_{p}(M) = b_{p}(Bπ^{2})) depends only on π provided that p ≤ 2 (respectively p ≤ 1). (Note that in the proof of Lemma 3.3, one only needs that C_{p}(Ñ, ∂_{0}Ñ) be a finitely generated Zπ_{1}(N)-module for p ≤ n.) The other claims follow from Theorem 3.2 on Poincaré duality.

Note that the second L^{2}-Betti number of a closed 4-manifold depends on more than just the fundamental group. For example, by taking repeated connected sums with CP^{2} one can increase b_{2} by any positive integer.

In the top and bottom dimensions the invariants can be computed completely. We recall that a finitely generated group Γ is said to be amenable if there is a π-invariant bounded linear operator μ : L^{∞}(Γ) ⟶ R such that

inf{f(γ) : γ ∈ Γ} ≤ μ(f) ≤ sup{f(γ) : γ ∈ Γ}.

Note that any finitely generated abelian group is amenable and any finite group is amenable. A subgroup and a quotient group of an amenable group are amenable. An extension of an amenable group by an amenable group is amenable. A group containing a free group on two generators is not amenable. A finitely generated group Γ is nilpotent if Γ possesses a finite lower central series

Γ = Γ_{1} ⊃ Γ_{2} ⊃ … ⊃ Γ_{s} = {1} Γ_{k+1} = [Γ, Γ_{k}].

If \overline{Γ} contains a nilpotent subgroup Γ of finite index then \overline{Γ} is said to be virtually nilpotent. Let d_{i} be the rank of the quotient Γ_{i}/Γ_{i+1} and let d be the integer ∑_{i≥1} id_{i}. Then \overline{Γ} has polynomial growth of degree d [2]. Note that a group has polynomial growth if and only if it is virtually nilpotent [16].

Lemma 3.5

1. α_{1}(M) = α̃_{0}(M) is finite if and only if π is infinite and virtually nilpotent. In this case, α_{1}(M) is the growth rate of π.

2. α_{1}(M) = α̃_{0}(M) is ∞^{+} if and only if π is finite or nonamenable.

3. α_{1}(M) = α̃_{0}(M) is ∞ if and only if π is amenable and not virtually nilpotent.

4. b_{0}(M) = 0 if π is infinite and 1/ |π | otherwise.

5. If ∂_{0}M is not empty then α_{1}(M, ∂_{0}M; V) and α_{m}(M, ∂_{1}M; V) are equal to ∞^{+} and b_{0}(M, ∂_{0}M; V) and b_{m}(M, ∂_{1}M; V) are zero.

6. If ∂_{0}M is empty then α_{m}(M; V) = α_{1}(M; V) and b_{m}(M; V) = b_{0}(M; V).

\underline{\underline{Proof}} : 1.) to 3.) Since α_{1}(M) depends only on the fundamental group and there is a closed manifold with π as its fundamental group, we may assume that M is closed. Efremov [14] shows that α_{1}(M) equals its analytic counterpart. For the analytic counterpart, assertion 1.) is proven in [45] and assertion 2.) is proven in [4]. Assertion 3.) is a direct consequence of 1.) and 2.)

4.) is proven in [10, Proposition 2.4].

5.) and 6.) If ∂_{0}M is nonempty then the pair (M, ∂_{0}M) is homotopy equivalent to a pair of finite CW -complexes (X, A) such that all of the 0-cells of X lie in A. Hence the cellular Zπ_{1}(M)-chain complex C(M̃, ˜∂_{0}M; V) is Zπ_{1}(M)-chain homotopy equivalent to a Zπ_{1}(M)- chain complex which is trivial in dimension 0. Now apply Theorems 2.6 and 3.2.

For later purposes we will need the following result:

Lemma 3.6 Let j : π_{1}(M) ⟶ Γ be an inclusion of discrete groups. Let j^{∗}l^{2}(Γ) be the unitary representation π_{1}(M) ⟶ Iso_{N(Γ)}(l^{2}(Γ))^{op} obtained from the right regular representation of Γ by composing with j. Then for all p, we have

1. b_{p}(M, ∂_{0}M) = b_{p}(M, ∂_{0}M; j^{∗}l^{2}(Γ)).

2. α_{p}(M, ∂_{0}M) = α_{p}(M, ∂_{0}M; j^{∗}l^{2}(Γ)).

\underline{\underline{Proof}} : Let f : ⊕_{i=1}^{n}Zπ_{1}(M) ⟶ ⊕_{i=1}^{n}Zπ_{1}(M) be a Zπ_{1}(M)-linear map. By tensoring with l^{2}(π_{1}(M)) (resp. j^{∗}l^{2}(Γ)), we get a morphism of Hilbert N(π_{1}(M)) (resp. N(Γ))-modules denoted by f_{1} (resp. f_{2}). Let {E_{λ}^{f_{2}^{∗}f_{2}} : λ ∈ R} denote the spectral family of the self-adjoint operator f_{2}^{∗}f_{2} : ⊕_{i=1}^{n}l^{2}(Γ) ⟶ ⊕_{i=1}^{n}l^{2}(Γ) and {E_{λ}^{f_{1}^{∗}f_{1}} : λ ∈ R} denote the spectral family of f_{1}^{∗}f_{1} : ⊕_{i=1}^{n}l^{2}(π_{1}(M)) ⟶ ⊕_{i=1}^{n}l^{2}(π_{1}(M)). Then E_{λ}^{f_{2}^{∗}f_{2}} maps ⊕_{i=1}^{n}l^{2}(π_{1}(M)) into itself and the restriction of E_{λ}^{f_{2}^{∗}f_{2}} to ⊕_{i=1}^{n}l^{2}(π_{1}(M)) is just E_{λ}^{f_{1}^{∗}f_{1}}. By [11, Theorem 1, p. 97], this implies

F (f_{1}, λ) = tr_{N(π_{1}(M))} (E_{λ^{2}}^{f_{1}^{∗}f_{1}}) = ⟨E_{λ^{2}}^{f_{1}^{∗}f_{1}}(1), 1⟩_{l^{2}(π_{1}(M))} =
⟨E_{λ^{2}}^{f_{2}^{∗}f_{2}}(1), 1⟩_{l^{2}(Γ)} = tr_{N(Γ)} (E_{λ^{2}}^{f_{2}^{∗}f_{2}}) = F (f_{2}, λ),

and the claim follows.

We now investigate the behaviour with respect to connected sums.

Proposition 3.7 Let M_{1}, M_{2}, … M_{r} be compact connected m-dimensional manifolds, with m ≥ 3. Let M be their connected sum M_{1}♯ … ♯M_{r}. Then

1. b_{1}(M) − b_{0}(M) = r − 1 + ∑_{j=1}^{r} (b_{1}(M_{j}) − b_{0}(M_{j})).

2. b_{p}(M) = ∑_{j=1}^{r} b_{p}(M_{j}) for 2 ≤ p ≤ m − 2.

3. α_{p}(M) = min {α_{p}(M_{j}) : 1 ≤ j ≤ r} for 2 ≤ p ≤ m − 1.

4. If π_{1}(M_{i}) is trivial for all i except for i = i_{0} then α_{1}(M) = α_{1}(M_{i_{0}}). Suppose π_{1}(M_{i}) is trivial for all i except for i ∈ {i_{0}, i_{1}}, i_{0} ≠ i_{1}, and that π_{1}(M_{i_{0}}) = π_{1}(M_{i_{1}}) = Z/2. Then α_{1}(M) = 1. In all other cases α_{1}(M) = ∞^{+}.

\underline{\underline{Proof}} : We may assume without loss of generality that r = 2. The connected sum M_{1}♯M_{2} is obtained by glueing M_{1}\int(D^{m}) and M_{2}\int(D^{m}) together along ∂D^{m}. Since ∂D^{m} ⟶ D^{m} is (m − 1)-connected, the inclusion of M_{j}\int(D^{m}) into M_{j} is (m − 1)-connected. Hence the inclusion

M_{1}\int(D^{m}) ∪_{∂D^{m}} M_{2}\int(D^{m}) ⟶ M_{1} ∪_{D^{m}} M_{2}

is (m−1)-connected. Since M_{1} ∪_{D^{m}} M_{2} is homotopy equivalent to the wedge M_{1} ⋁ M_{2}, from Lemma 3.3 it suffices to prove the claims for M_{1} ⋁ M_{2}.

1.) to 3.) Let π denote π_{1}(M_{1} ⋁ M_{2}) = π_{1}(M_{1}) ∗ π_{1}(M_{2}). We obtain an exact sequence 0 ⟶ C(∗; l^{2}(π)) ⟶ C(M_{1}; l^{2}(π)) ⊕ C(M_{2}; l^{2}(π)) ⟶ C(M_{1} ⋁ M_{2}; l^{2}(π)) ⟶ 0, where ∗ denotes the base point. The long weakly exact Mayer-Vietoris sequence reduces to weak isomorphisms

H_{p}(M_{1}; l^{2}(π)) ⊕ H_{p}(M_{2}; l^{2}(π)) ⟶ H_{p}(M_{1} ⋁ M_{2}; l^{2}(π)) , p ≥ 2,

and the weakly exact sequence

0 ⟶ H_{1}(M_{1}; l^{2}(π)) ⊕ H_{1}(M_{2}; l^{2}(π)) ⟶ H_{1}(M_{1} ⋁ M_{2}; l^{2}(π)) ⟶ l^{2}(π)

⟶ H_{0}(M_{1}; l^{2}(π)) ⊕ H_{0}(M_{2}; l^{2}(π)) ⟶ H_{0}(M_{1} ⋁ M_{2}; l^{2}(π)) ⟶ 0

We conclude from Lemmas 1.4 and 3.6 that

b_{1}(M_{1}) + b_{1}(M_{2}) − b_{1}(M_{1} ⋁ M_{2}) + 1 − b_{0}(M_{1}) − b_{0}(M_{0}) + b_{0}(M_{1} ⋁ M_{2}) = 0
b_{p}(M_{1}) + b_{p}(M_{2}) = b_{p}(M_{1} ⋁ M_{2}) for p ≥ 2,

from which assertions 1.) and 2.) follow. We obtain assertion 3.) from Theorem 2.3.

4.) Since α_{1}(M) only depends on the fundamental group and π_{1}(M) = π_{1}(M_{1}) if π_{1}(M_{2}) is trivial, the first part of the assertion follows. It remains to consider the case when π_{1}(M_{1}) and π_{1}(M_{2}) are nontrivial. From Lemma 3.5.2, α_{1}(M) is ∞^{+} if and only if π_{1}(M) is nonamenable. We claim that π_{1}(M) is amenable if and only if π_{1}(M_{1}) = π_{1}(M_{2}) = Z/2, in which case α_{1}(M) = 1. Namely, suppose that π_{1}(M) is amenable. Then it follows from [10, Theorem 0.2] that b_{1}(M) = b_{0}(M) = 0. But then assertion 1.) and Lemma 3.5 imply that | π_{1}(M_{i}) |= 2 for i = 1, 2. As Z/2 ∗ Z/2 is an extension of Z by Z/2, it is amenable. Also, there is a two-fold covering of M with the fundamental group of a circle. Hence α_{1}(M) = α_{1}(S^{1}), which is 1 by a simple calculation.

Next we study manifolds with an S^{1}-action. Let (M; ∂_{0}M) be as above. Suppose that S^{1} acts smoothly on M. Let ϕ : π_{1}(M) ⟶ Γ be an homomorphism such that for one orbit (and hence all orbits) S^{1}/H in M, the composition of ϕ with the map induced by the inclusion π_{1}(S^{1}/H) ⟶ π_{1}(M) has infinite image. In particular, the S^{1}-action has no fixed points. Choose A to be N(Γ) and the representation ϕ^{∗}l^{2}(Γ) to be the composition of the regular representation Γ ⟶ Iso_{N(Γ)}(l^{2}(Γ)) with ϕ. In other words, we are looking at the cover \overline{M} ⟶ M of M associated with ϕ.

Theorem 3.8 (S^{1}-manifolds) With the above conditions on the S^{1}-manifold M, for all p ≥ 0 we have:

1. b_{p}(M, ∂_{0}M; ϕ^{∗}l^{2}(Γ)) = 0.

2. α_{p}(M, ∂_{0}M; ϕ^{∗}l^{2}(Γ)) ≥ 1.

\underline{\underline{Proof}} : The first assertion was proven in [31, Theorem 3.20].

In what follows we will write l^{2}(Γ) instead of ϕ^{∗}l^{2}(Γ), or j^{∗}ϕ^{∗}l^{2}(Γ) for j an inclusion. Since we have a smooth S^{1}-action, M carries a S^{1}-equivariant CW -structure. This means that we have a filtration

∅ = M_{−1} ⊂ M_{0} ⊂ M_{1} ⊂ … M_{m−1} = M

such that M_{i} is obtained from M_{i−1} by attaching a finite number of S^{1}-equivariant cells S^{1}/H × D^{i} with attaching maps S^{1}/H × S^{i−1} ⟶ M_{i−1}. Since the S^{1}-action has no fixed points, the subgroups H ⊂ S^{1} are all finite cyclic groups. We will show that

α_{p}(M_{i}, ∂_{0}M ∩ M_{i}; l^{2}(Γ)) ≥ 1 for p ≤ i + 1
α_{p}(M_{i}, ∂_{0}M ∩ M_{i}; l^{2}(Γ)) = ∞^{+} for p > i + 1

by induction over i, where the representation of π_{1}(M_{i}) is induced from the inclusion π_{1}(M_{i}) ⟶ π_{1}(M). The initial step i = −1 is trivial. The induction step from i − 1 to i is done as follows:

There is an exact sequence of chain complexes

0 ⟶ C(M_{i−1}, ∂_{0}M ∩ M_{i−1}; l^{2}(Γ)) ⟶ C(M_{i}, ∂_{0}M ∩ M_{i}; l^{2}(Γ)) ⟶ C(M_{i}, M_{i−1} ∪ (∂_{0}M ∩ M_{i}); l^{2}(Γ)) ⟶ 0.

The last chain complex is isomorphic to a direct sum of chain complexes of the form C(S^{1}/H × D^{i}, S^{1}/H × S^{i−1}; l^{2}(Γ)). Since all isotropy groups H must be finite, such a chain complex looks like Σ^{i}C(S^{1}; l^{2}(Γ)), where l^{2}(Γ) is viewed as a representation space of π_{1}(S^{1}) by means of an injection π_{1}(S^{1}) ⟶ Γ. Lemma 3.6 and a simple calculation of α_{1}(S^{1}) show that α_{p}(Σ^{i}C(S^{1}; l^{2}(Γ))) = α(S^{1}) is 1 if p = i + 1 and ∞^{+} otherwise. Lemma 2.4.3 implies that α_{p}(C(M_{i}, M_{i−1} ∪ (∂_{0}M ∩ M_{i}); l^{2}(Γ)) is also 1 for p = i + 1 and ∞^{+} otherwise. Upon applying Theorem 2.3.1 to the short exact sequence of weakly acyclic chain complexes above and using the induction hypothesis on M_{i−1}, the claim follows.

Remark 3.9 If g : (M, ∂_{0}M) ⟶ (N, ∂_{0}N) is an n-fold finite covering then b_{p}(M, ∂_{0}M) = n · b_{p}(N, ∂_{0}N) and α_{p}(M, ∂_{0}M) = α_{p}(N, ∂_{0}N) for all p ≥ 0. Note that the ordinary Betti numbers of a manifold are generally not multiplicative under finite coverings.

Example 3.10 We state the values of the L^{2}-Betti numbers and Novikov-Shubin invariants for all compact connected 1- and 2-manifolds. In dimension 1 there are only S^{1} and the unit interval I. One easily checks that b_{0}(S^{1}) = b_{1}(S^{1}) = 0 and α_{1}(S^{1}) = 1. As I is contractible, we have that b_{0}(I) = 1, b_{1}(I) = 0 and α_{1}(I) = ∞^{+}.

Let F_{g}^{d} be the orientable closed surface of genus g with d embedded 2-disks removed. (As any nonorientable compact surface is finitely-covered by an orientable surface, Remark 3.9 shows that it is enough to handle the orientable case.) Using the general formula for the Euler characteristic in terms of L^{2}-Betti numbers [8] :

χ(M) = ∑_{p}(−1)^{p}b_{p}(M),

Lemma 3.5 and the fact that a compact surface with boundary is homotopy-equivalent to a bouquet of circles, one derives:

b_{0}(F_{g}^{d}) = \{ \begin{matrix}1 & g = 0, d = 0, 1 \\ 0 & otherwise. \end{matrix}

b_{1}(F_{g}^{d}) = \{ \begin{matrix}0 & g = 0, d = 0, 1 \\ d + 2(g − 1) & otherwise. \end{matrix}

b_{2}(F_{g}^{d}) = \{ \begin{matrix}1 & g = 0, d = 0 \\ 0 & otherwise. \end{matrix}

α_{1}(F_{g}^{d}) = α̃_{0}(F_{g}^{d}) = α̃_{1}(F_{g}^{d}) = \{ \begin{matrix}1 & g = 0, d = 2 \\ 2 & g = 1, d = 0 \\ ∞^{+} & otherwise. \end{matrix}

α_{2}(F_{g}^{d}) = α̃_{2}(F_{g}^{d}) = \{ \begin{matrix}2 & g = 1, d = 0 \\ ∞^{+} & otherwise. \end{matrix}

Example 3.11 Suppose that M is a compact connected orientable 3-manifold with finite fundamental group π. We have that α_{p}(M) = ∞^{+} for all p. If M is closed then M̃ is a homotopy sphere, and Remark 3.9 implies that b_{0}(M) = b_{3}(M) = \frac{1}{|π|} and b_{1}(M) = b_{2}(M) = 0. If ∂M is nonempty then M̃ is a connected sum of a homotopy sphere and k 3-disks, for some positive integer k [20]. Then b_{0}(M) = \frac{1}{|π|}, b_{2}(M) = \frac{k−1}{|π|} and b_{1}(M) = b_{3}(M) = 0.

4. Seifert 3-Manifolds

In this section we compute the L^{2}-Betti numbers and Novikov-Shubin invariants of Seifert 3-manifolds. We also discuss Sol manifolds. We use the definition of Seifert fibred 3-manifold , or briefly Seifert manifold , given in [41], which we will use as a reference on Seifert manifolds. Recall that a geometry on a 3-manifold M is a complete locally homogeneous Riemanian metric on its interior. The universal cover of the interior has a complete homogeneous Riemannian metric, meaning that the isometry group acts transitively [42]. Thurston has shown that there are precisely eight maximal simply-connected 3-dimensional geometries having compact quotients, namely S^{3}, R^{3}, S^{2} × R, H^{2} × R, Nil, S̃L_{2}(R), Sol and H^{3}. If a closed 3-manifold admits a geometric structure modelled on one of these eight geometries then the geometry involved is unique. In the case of the L^{2}-Betti numbers, the following result was already given in [6].

Theorem 4.1 Let M be a closed Seifert 3-manifold. If its fundamental group is infinite then it has vanishing L^{2}-cohomology. In terms of the Euler class e of the bundle and the Euler characteristic χ of the base orbifold, α_{1}(M) = α_{3}(M) is given by

\underline{χ > 0} \underline{χ = 0} \underline{χ < 0}
e = 0 | 1  3  ∞^{+}
e ≠ 0 | ∞^{+}  4  ∞^{+}

and α_{2}(M) is given by

\underline{χ > 0} \underline{χ = 0} \underline{χ < 0}
e = 0 | ∞^{+}  3  1
e ≠ 0 | ∞^{+}  2  1.

\underline{\underline{Proof}} : The geometric structure of M is determined as follows: [41, Theorem 5.3]:

\underline{χ > 0}  \underline{χ = 0}  \underline{χ < 0}
e = 0 |  S^{2} × R  R^{3}  H^{2} × R
e ≠ 0 |  S^{3}  Nil  S̃L_{2}(R).

If M has a S^{3}-structure then π_{1}(M) is finite and we can apply Example 3.11.

In all other cases M is finitely covered by the total space \overline{M} of an S^{1}-principal bundle over an orientable closed surface F. Moreover, e(M) = 0 iff e(\overline{M}) = 0, and the Euler characteristic χ of the orbifold base of M is negative, zero or positive according to the same condition for χ(\overline{M}/S^{1}) [41, p. 426, 427 and 436]. From Remark 3.9, in what follows we may assume without loss of generality that M is \overline{M}. Theorem 3.8 implies that b_{p}(M) = 0. If χ(F) is negative then π_{1}(F) is non-amenable since it contains a free subgroup of rank 2. As π_{1}(F) is a quotient of π_{1}(M), π_{1}(M) is also non-amenable and so α_{1}(M) = ∞^{+} by Lemma 3.5. Next, we verify the remaining claims for α_{1} and α_{2}.

R^{3}: We may assume that M = T^{3}. A direct computation by Fourier analysis gives that α_{p}(T^{3}) = 3 for all 1 ≤ p ≤ 3.

S^{2} × R: We may assume that M = S^{1} × S^{2}. Now apply Lemma 4.2.

H^{2} × R: We may assume that M = S^{1} × F_{g} for g ≥ 2. Now apply Lemma 4.2.

Nil: From [24] we have that α̃_{0}(M) = 4 and α̃_{1}(M) = 2, and so the claim for α_{1} and α_{2} follows.

S̃L_{2}(R): A computation using harmonic analysis on S̃L_{2}(R) and the results of [38] gives α_{2}(M) = 1. We will not reproduce the computation here.

The next lemma will finish the proof of Theorem 4.1.

Lemma 4.2 Let F_{g}^{d} be the (orientable compact connected) surface of genus g with d boundary components. Then

1. b_{p}(S^{1} × F_{g}^{d}) = 0 for all p.

2. α_{1}(S^{1} × F_{g}^{d}) = \{ \begin{matrix}1 & g = 0, d = 0, 1 \\ 2 & g = 0, d = 2 \\ 3 & g = 1, d = 0 \\ ∞^{+} & otherwise \end{matrix}

3. α_{2}(S^{1} × F_{g}^{d}) = \{ \begin{matrix}∞^{+} & g = 0, d = 0, 1 \\ 3 & g = 1, d = 0 \\ 2 & g = 0, d = 2 \\ 1 & otherwise \end{matrix}

4. α_{3}(S^{1} × F_{g}^{d}) = \{ \begin{matrix}1 & g = 0, d = 0 \\ 3 & g = 1, d = 0 \\ ∞^{+} & otherwise \end{matrix}

\underline{\underline{Proof}} : The claim for the L^{2}-Betti numbers follows from Theorem 3.8. In the cases g = 0, d = 0, 1, 2 and g = 1, d = 0, i.e. S^{1} × S^{2}, S^{1} × D^{2}, S^{1} × S^{1} × I and T^{3}, the claim follows from earlier computations for S^{1}, T^{2} and T^{3} (see Example 3.10 and Theorem 4.1). In the remaining cases Example 3.10 gives that α_{p}(F_{g}^{d}) = ∞^{+} for all p and b_{p}(F_{g}^{d}) = 0 for p ≠ 1. We abbreviate F = F_{g}^{d}. Let H be the Hilbert chain complex over the von Neumann algebra of π_{1}(F) which is concentrated in dimension 1, and is given there by ker(Δ_{1}), where Δ_{1} : C_{1}(F; l^{2}(π_{1}(F))) ⟶ C_{1}(F; l^{2}(π_{1}(F))) is the Laplace operator. There is a natural split inclusion i : H ⟶ C(F; l^{2}(π_{1}(F))). From Lemma 2.5, i is a homotopy equivalence. We have that C(S^{1} × F, l^{2}(π_{1}(S^{1} × F))) is the Hilbert tensor product of C(F; l^{2}(π_{1}(F))) and C(S^{1}; l^{2}(π_{1}(S^{1}))), and so is homotopy equivalent to the Hilbert tensor product of H and C(S^{1}; l^{2}(π_{1}(S^{1}))). As the part of H in dimension one is isomorphic to ⊕_{i=1}^{−χ(F)}l^{2}(π_{1}(F)), this Hilbert tensor product is isometrically isomorphic to the suspension of the direct sum of −χ(F) copies of C(S^{1}; l^{2}(π_{1}(S^{1} × F))). From Lemma 2.4, Theorem 2.6 and Lemma 3.6, the Novikov-Shubin invariants of M are the same as those of the suspension of C(S^{1}; l^{2}(π_{1}(S^{1}))). The claim now follows from Example 3.10.

Remark 4.3 The fact that the Novikov-Shubin invariants are the same for closed H^{2} × R-manifolds and S̃L_{2}(R)-manifolds is probably related to the fact that they are K(π, 1) manifolds whose universal covers are quasi-isometric [15].

Theorem 4.4 Let M be a Seifert manifold with nonempty boundary. Then all L^{2}-Betti numbers vanish. We have that α_{3}(M) = ∞^{+}, and the other Novikov-Shubin invariants are given by:

\underline{α_{1}} \underline{α_{2}} 1 ∞^{+} M is a solid torus or Klein bottle 2 2 M is an I-bundle over T^{2} or over a Klein bottle K ∞^{+} 1 otherwise. .

\underline{\underline{Proof}} : We have that the boundary of M is compressible iff M is homeomorphic to a solid torus or Klein bottle [41, Corollary 3.3]. The theorem follows in this case from Remark 3.9 and Lemma 4.2, and so we may assume that M has incompressible boundary. As any 2-dimensional orbifold with boundary is finitely covered by a 2-dimensional surface with boundary, we can find a finite cover \overline{M} of M which is homeomorphic to some S^{1} × F_{g}^{d}, with d ≥ 1. From Remark 3.9 and Lemma 4.2, we have to know that M is an I-bundle over T^{2} or K iff F_{g}^{d} = S^{1} × I. This follows from [20, Theorem 10.5].

Proposition 4.5 If M is a closed Sol-manifold then M has vanishing L^{2}-Betti numbers, α_{1}(M) = ∞ and α_{2}(M) ≥ 1.

\underline{\underline{Proof}} : By taking a finite cover, we may assume that our Sol-manifold is a torus bundle over S^{1} with hyperbolic glueing map ϕ [41, Theorem 5.3]. Hence π_{1}(M) is a semi-direct product of Z^{2} and Z where the action of Z on Z^{2} is given by a hyperbolic automorphism of Z^{2}. Then π_{1}(M) is amenable, as it is an extension of amenable groups. It is easy to see that π_{1}(M) is not virtually nilpotent. Lemma 3.5.3 implies that α_{1}(M) = ∞.

By Example 3.10, b_{p}(T^{2}) = 0 for all p and α_{p}(T^{2}) = 2 for p ∈ {1, 2}. From [30], the L^{2}-Betti numbers of M vanish. We have a short exact (Wang) sequence of Hilbert chain complexes:

0 → C(T^{2}; l^{2}(π_{1}(M))) →^{j} C(T^{2} × I; l^{2}(π_{1}(M))) →^{q} C(M; l^{2}(π_{1}(M))) → 0.

Theorem 2.3.2 gives

\frac{1}{α_{2}(M)} ≤ \frac{1}{2} + \frac{1}{2}.

5. Analytic L^{2}-Betti numbers and Novikov-Shubin invariants for manifolds with boundary, and hyperbolic 3-manifolds

In this section we define analytic Novikov-Shubin invariants and L^{2}-Betti numbers for manifolds with boundary, and show the equivalence between the analytic invariants and the combinatorial invariants of the previous section. As an application, we give a lower bound for the Novikov-Shubin invariants of a compact 3-manifold whose interior admits a complete finite-volume hyperbolic metric.

For closed manifolds, the facts that the analytic L^{2}-Betti numbers and Novikov-Shubin invariants equal their combinatorial counterparts were proven in [12] and [14]. In order to make the comparisons between the analytic and combinatorial invariants for a compact manifold M with boundary, it will be convenient for us to think of the combinatorial invariants as defined by cellular cochains, instead of cellular chains. In this section, except where otherwise stated, the Novikov-Shubin invariants will be those of the coboundary operator. The smooth forms on M̃ will be denoted by C^{∞}(∧^{∗}(M̃)). Those with compact support will be denoted by C_{0}^{∞}(∧^{∗}(M̃)). Note that the elements of C_{0}^{∞}(∧^{∗}(M̃)) do not necessarily vanish on ˜∂M.

We assume that M has a smooth Riemannian metric and corresponding Levi-Civita connection. We give M̃ the induced Riemannian metric and Levi-Civita connection. Let d denote the exterior differentiation and let b : ∂M̃ → M̃ denote the boundary inclusion of ∂M̃ into M̃. Let ∇ be covariant differentiation on the smooth tensors on M̃. As before, π denotes the fundamental group of M.

Definition 5.1 Define norms ∥ • ∥_{s} on the smooth compactly-supported tensors on M̃ for nonnegative integers s inductively by saying that ∥ • ∥_{0} is the L^{2}-norm and

∥ ω ∥_{s+1}^{2}=∥ ω ∥_{s}^{2} + ∥ ∇ω ∥_{s}^{2} .

Let the Sobolev space H_{s}^{∗}(M; l^{2}(π)) be the Hilbert space completion of C_{0}^{∞}(∧^{∗}(M̃)) under the norm ∥ • ∥_{s}.

Put A = N(π). There is a Hilbert A-cochain complex concentrated in dimensions p − 1, p and p + 1 given by

… ⟶ 0 ⟶ H_{2}^{p−1}(M; l^{2}(π)) →^{d_{2}^{p−1}} H_{1}^{p}(M; l^{2}(π)) →^{d_{1}^{p}} H_{0}^{p+1}(M; l^{2}(π)) ⟶ 0 ⟶ …. (1)

In Definition 2.1 we introduced the condition of Fredholmness at p, the p-th L^{2}-Betti number and the p-th Novikov-Shubin invariant of such a Hilbert A-complex. We will show in Theorem 5.7 that the complex (1) is indeed Fredholm at p.

Definition 5.2 The analytic p-th L^{2}-cohomology of M is the p-th cohomology group of the Hilbert A-cochain complex (1). The analytic p-th L^{2}-Betti number of M and analytic p-th Novikov-Shubin invariant of M are defined similarly.

If we put α̃_{p}(M) = min(α_{p}(M), α_{p−1}(M)) then the application of a Laplace transform to the spectral density function shows that the analytic invariants of the introduction, defined using heat kernels, are the same as those defined here [18, Appendix].

As a topological vector space, H_{s}^{∗}(M; l^{2}(π)) is independent of the choice of Riemannian metric on M. Given two different Riemannian metrics on M, the identity map on C_{0}^{∞}(∧^{∗}(M̃)) induces a bounded invertible mapping between the corresponding complexes (1), and in particular a cochain homotopy equivalence. Theorem 2.6 then implies that the analytic L^{2}-Betti numbers and Novikov-Shubin invariants are independent of the Riemannian metric on M.

Definition 5.3 Define Sobolev spaces of differential forms with absolute boundary conditions by

H_{2,abs}^{p}(M; l^{2}(π)) = {ω ∈ H_{2}^{p}(M; l^{2}(π)) : b^{∗}(∗ω) = b^{∗}(∗dω) = 0},
H_{1,abs}^{p}(M; l^{2}(π)) = {ω ∈ H_{1}^{p}(M; l^{2}(π)) : b^{∗}(∗ω) = 0},
H_{0,abs}^{p}(M; l^{2}(π)) = H_{0}^{p}(M; l^{2}(π)).  (2)

There is a Hilbert A-cochain complex concentrated in dimensions p − 1, p and p + 1 given by

… → 0 → H_{2,abs}^{p−1}(M; l^{2}(π)) →^{d_{2,abs}^{p−1}} H_{1,abs}^{p}(M; l^{2}(π)) →^{d_{1,abs}^{p}} H_{0,abs}^{p+1}(M; l^{2}(π)) → 0 → …. (3)

Let ϵ > 0 be small enough that there is a coordinate function t ∈ [0, 2ϵ] near ∂M̃ such that ∂_{t} is a unit length vector field whose flow generates unit speed geodesics which are normal to the boundary, and ∂M̃ corresponds to t = 0. Then a tubular neighborhood of ∂M̃ is diffeomorphic to [0, 2ϵ] × ∂M̃. A differential form ω on M̃ can be decomposed near the boundary as

ω = ω_{1}(t) + dt ∧ ω_{2}(t),

where ω_{1}(t) and ω_{2}(t) are smooth 1-parameter families of forms on ∂M̃. We can write the condition for ω to be in H_{1,abs}^{p}(M; l^{2}(π)) as ω_{2}(0, ·) = 0, and the condition for ω to be in H_{2,abs}^{p}(M; l^{2}(π)) as ω_{2}(0, ·) = ∂_{t}ω_{1}(0, ·) = 0.

Let ρ : [0, 2ϵ] → R be a smooth bump function which is identically one near t = 0 and identically zero for t ≥ ϵ. Let ˆ△ denote the Laplacian acting on forms on ∂M̃. For u ≥ 0, define the operator e^{−(ue^{1+_{△}})} by the spectral theorem.

Definition 5.4 For ω a form on M̃, restrict ω to [0, 2ϵ] × ∂M̃ and put

(Kω)(t, ·) = ρ(t) ∫_{0}^{t} e^{−(ue^{1+_{△}})}ω_{2}(u, ·)du.  (4)

We can extend Kω by zero to become a form on M̃.

Lemma 5.5 The map K induces bounded π-equivariant operators

K_{s}^{p} : H_{s}^{p}(M; l^{2}(π)) → H_{s+1}^{p−1}(M; l^{2}(π))

for s ∈ {0, 1, 2} and

K_{s,abs}^{p} : H_{s,abs}^{p}(M; l^{2}(π)) → H_{s+1,abs}^{p−1}(M; l^{2}(π))

for s ∈ {0, 1}.

\underline{\underline{Proof}} : Whether or not a given distributional form lies in H_{s}^{p}(M; l^{2}(π)) is independent of the choice of Riemannian metric. Moreover, the condition for a form to lie in H_{∗,abs}^{p}(M; l^{2}(π)) is the same whether one uses the given metric g_{M} on M or a metric which becomes the product metric dt^{2} + g_{∂M} near the boundary. Thus we may assume without loss of generality that [0, 2ϵ]×∂M̃ has a product metric. Let ω be an element of H_{0}^{p}(M; l^{2}(π)). From Definition 5.4, we may assume that ω has support on [0, 2ϵ]×∂M̃. Clearly ω_{2} lies in H_{0}^{p−1}([0, 2ϵ]×∂M; l^{2}(π)). From (4),

(∂_{t}Kω)(t, ·) = ρ′(t) ∫_{0}^{t} e^{−(ue^{1+_{△}})}ω_{2}(u, ·)du + ρ(t)e^{−(te^{1+_{△}})}ω_{2}(t, ·),

and so ∂_{t}K gives a bounded linear map from H_{0}^{p−1}([0, 2ϵ]×∂M; l^{2}(π)) to itself. Let ˆ∇ denote covariant differentiation in the ∂M̃ directions. Then from (4),

(ˆ∇Kω)(t, ·) = ρ(t) ∫_{0}^{t} ˆ∇e^{−(ue^{1+_{△}})}ω_{2}(u, ·)du.

Thus ˆ∇K is an integral operator with kernel function

(ˆ∇K)(t, u) = Θ(t − u)ρ(t)ˆ∇e^{−(ue^{1+_{△}})},

where Θ is the Heaviside step function. In order to show that ˆ∇K is L^{2}-bounded, it suffices to show that ∫_{[0,2ϵ]} ∫_{[0,2ϵ]} Tr ((ˆ∇K)(t, u)^{∗}(ˆ∇K)(t, u)) dtdu < ∞, as that would imply that ˆ∇K is actually Hilbert-Schmidt. We have

∫_{0}^{2ϵ} ∫_{0}^{2ϵ} Tr ((ˆ∇K)(t, u)^{∗}(ˆ∇K)(t, u)) dtdu = ∫_{0}^{2ϵ} ∫_{0}^{2ϵ} Θ(t − u)ρ^{2}(t)

Tr (e^{−(ue^{1+_{△}})}ˆ∇^{∗}ˆ∇e^{−(ue^{1+_{△}})}) dtdu

≤ const. ∫_{0}^{2ϵ} Tr (ˆ∇^{∗}ˆ∇e^{−(2ue^{1+_{△}})}) du
≤ const. ∫_{0}^{∞} Tr (ˆ∇^{∗}ˆ∇e^{−(2ue^{1+_{△}})}) du
= const. Tr (\frac{1}{2} ˆ∇^{∗}ˆ∇e^{−(1+_{△})}) < ∞.

Thus ∇K gives an L^{2}-bounded linear map on H_{0}^{p−1}(M; l^{2}(π)). It follows that K gives a bounded linear map from H_{0}^{p−1}(M; l^{2}(π)) to H_{1}^{p−1}(M; l^{2}(π)). A similar argument shows that K gives a bounded linear map from H_{s}^{p}(M; l^{2}(π)) to H_{s+1}^{p−1}(M; l^{2}(π)) for s = 1, 2.

As (Kω)_{2} = 0, it follows that K maps H_{0,abs}^{p}(M; l^{2}(π)) to H_{1,abs}^{p−1}(M; l^{2}(π)). Given ω ∈ H_{1,abs}^{p}(M; l^{2}(π)), we have

∂_{t}(Kω)_{1}(0, ·) = ∂_{t}(Kω)(0, ·) = ω_{2}(0, ·) = 0.

Thus K maps H_{1,abs}^{p}(M; l^{2}(π)) to H_{2,abs}^{p−1}(M; l^{2}(π)).

Proposition 5.6 The Hilbert A-cochain complex (1) is Fredholm at p if and only if (3) is Fredholm at p. In this case, they have the same p-th L^{2}-Betti number and the same p-th Novikov-Shubin invariant.

\underline{\underline{Proof}} : The inclusion i induces a chain map from (3) to (1). We will show that there is a chain map j from the cochain complex (1) to the cochain complex (3) of the form

H_{2}^{p−1}(M; l^{2}(π))  \underrightarrow{d_{2}^{p−1}} H_{1}^{p}(M; l^{2}(π))  \underrightarrow{d_{1}^{p}} H_{0}^{p+1}(M; l^{2}(π))
j_{2}^{p−1} ↓  _{j_{1}^{p} ↓}  _{j_{0}^{p+1} ↓}

H_{2,abs}^{p−1}(M; l^{2}(π))  \underrightarrow{d_{2,abs}^{p−1}} H_{1,abs}^{p}(M; l^{2}(π))  \underrightarrow{d_{1,abs}^{p}} H_{0,abs}^{p+1}(M; l^{2}(π)),

where

j_{2}^{p−1} = 1 − d_{3}^{p−2}K_{2}^{p−1} − K_{1}^{p−1}d_{2}^{p−1},
j_{1}^{p} = 1 − d_{2}^{p−1}K_{1}^{p} − K_{0}^{p+1}d_{1}^{p},
j_{0}^{p+1} = 1 − d_{1}^{p}K_{0}^{p+1}.

Lemma 5.5 implies that that the vertical operators j do indeed map H_{s}^{p+1−s}(M; l^{2}(π)) to H_{s}^{p+1−s}(M; l^{2}(π)) for s ∈ {0, 1, 2}. It remains to check that the images actually lie in H_{s,abs}^{p+1−s}(M; l^{2}(π)).

Given ω ∈ H_{0}^{p+1}(M; l^{2}(π)), it is trivial that j_{0}^{p+1}ω lies in H_{0,abs}^{p+1}(M; l^{2}(π)). Given ω ∈ H_{1}^{p}(M; l^{2}(π)), we have

(d_{2}^{p−1}K_{1}^{p}ω)(0, ·) = dt ∧ ω_{2}(0, ·)

and

(K_{0}^{p+1}d_{1}^{p}ω)(0, ·) = 0.

Then

(ω − (d_{2}^{p−1}K_{1}^{p} + K_{0}^{p+1}d_{1}^{p})ω) (0, ·) = ω_{1}(0, ·),

and so j_{1}^{p}ω lies in H_{1,abs}^{p}(M; l^{2}(π)).

Given ω ∈ H_{2}^{p−1}(M; l^{2}(π)), in order to show that j_{2}^{p−1}ω lies in H_{2,abs}^{p−1}(M; l^{2}(π)) it suffices to show that j_{2}^{p−1}ω lies in H_{1,abs}^{p−1}(M; l^{2}(π)) and that d_{2}^{p−1}j_{2}^{p−1}ω lies in H_{1,abs}^{p}(M; l^{2}(π)). As ω also lies in H_{1}^{p−1}(M; l^{2}(π)), the preceding argument gives that j_{2}^{p−1}ω lies in H_{1,abs}^{p−1}(M; l^{2}(π)). As d_{2}^{p−1}j_{2}^{p−1}ω = j_{1}^{p}d_{2}^{p−1}ω and d_{2}^{p−1}ω lies in H_{1}^{p}(M; l^{2}(π)), the preceding argument also gives that d_{2}^{p−1}j_{2}^{p−1}ω lies in H_{1,abs}^{p−1}(M; l^{2}(π)).

We now have chain maps i and j. Consider the cochain complexes, concentrated in dimensions p and p + 1, given by

… → 0 → H_{1}^{p}(M; l^{2}(π))/clos(im(d_{2}^{p−1})) →^{d_{1}^{p}} H_{0}^{p+1}(M; l^{2}(π)) → 0 → ….  (5)

and

… → 0 → H_{1,abs}^{p}(M; l^{2}(π))/clos(im(d_{2,abs}^{p−1})) →^{d_{1,abs}^{p}} H_{0,abs}^{p+1}(M; l^{2}(π)) → 0 → ….  (6)

Clearly (1) is Fredholm at p if and only if (5) is Fredholm at p, and in this case they have the same p-th Betti number and p-th Novikov-Shubin invariant. A similar statement holds for (3) and (6). Now i and j induce chain maps between (5) and (6), and K and K_{abs} induce chain homotopies between the two compositions and the identity. Thus by Theorem 2.6, the cochain complex (5) is Fredholm if and only if (6) is Fredholm, and in this case they have the same p-th Betti number and p-th Novikov-Shubin invariant.

Theorem 5.7 The Hilbert A-cochain complexes (1) and (3) are Fredholm at p. Moreover, the analytic L^{2}-Betti numbers and Novikov-Shubin invariants of Definition 5.2 are equal to the combinatorial invariants of Section 3, with ∂_{0}M = ∅.

\underline{\underline{Proof}} : The choice of Riemannian metric on M does not affect whether or not (1) is Fredholm. If (1) is Fredholm, its p-th L^{2}-Betti number and p-th Novikov-Shubin invariant are independent of the choice of Riemannian metric on M. From Proposition 5.6, it suffices to show that (3) is Fredholm at p and that if the metric is a product near the boundary then the p-th L^{2}-Betti number and p-th Novikov-Shubin invariant agree with the combinatorially defined invariants.

There is an induced Riemannian metric on the double DM, upon which Z_{2} acts by isometries. With π still denoting π_{1}(M), there is a π-normal cover of DM, namely the double DM̃ of M̃, and it is easy to see that H_{s,abs}^{p}(M; l^{2}(π)) is isomorphic to (H_{s}^{p}(DM; l^{2}(π)))^{Z_{2}}, the subspace of H_{s}^{p}(DM; l^{2}(π)) which is invariant under the induced Z_{2}-action, for s ∈ {0, 1, 2}. In particular, (3) is isomorphic to the Z_{2}-invariant part of the cochain complex H_{p+1−∗}^{∗}(DM; l^{2}(π)), restricted to the dimensions p − 1, p and p + 1.

It follows from [18], [12] and [14] that H_{p+1−∗}^{∗}(DM; l^{2}(π)) is Fredholm at p and that one has equality of the analytic and combinatorial invariants on DM, defined using the π-cover. One can go through the proofs making everything equivariant with respect to the Z_{2} action, in order to show that the same is true when one restricts to the Z_{2}-invariant subspaces. (As in [12] and [14], one first deals with Sobolev spaces of a high enough order that the de Rham map is well-defined. One then shows the analytic invariants are independent of the order of the Sobolev space. In our case, we are finally interested in the Sobolev space H_{1}^{p}. All of these steps will go through equivariantly.) Putting all this together, we have shown Theorem 5.7.

Now let M be a compact 3-manifold whose interior admits a complete finite-volume hyperbolic metric. If M is closed then we have that b_{∗}(M; l^{2}(π)) = 0 [13] and the Novikov-Shubin invariants of the exterior derivative operator are computed in [24] as

α_{0}(M; l^{2}(π)) = α_{2}(M; l^{2}(π)) = ∞^{+}, α_{1}(M; l^{2}(π)) = 1.

Suppose that M is not closed. Then it has incompressible torus boundary and the interior M′ of M is the union of a compact core and a finite number of hyperbolic cusps (see [44] or [35, p. 52, 54]). Let i : M → M′ be an embedding of M in M′ obtained by smoothly truncating the cusps of M′ and let M have the induced Riemannian metric. Let i_{1} : M_{1} → M′ be the embedding of a submanifold (with boundary) M_{1} of M′ obtained by attaching a collar to M, and let i_{2} : M_{2} → M′ be the embedding of a submanifold (with boundary) M_{2} of M′ obtained by attaching a collar to M′ −M. Then M_{3} = M_{1} ∩M_{2} is diffeomorphic to a disjoint union of I × T^{2}’s (where we take {0} × T^{2} to be contained in the interior of M_{1} and {1} × T^{2} to be contained in the interior of M_{2}) and is embedded in M′ by a map i_{3} : M_{3} → M′. Let i_{4} : M_{3} → M_{1} and i_{5} : M_{3} → M_{2} be the obvious embeddings. Put π = π_{1}(M).

For each p ∈ {0, 1, 2, 3}, consider the following Hilbert cochain complexes concentrated in dimensions p − 1, p and p + 1:

C_{(p)}^{∗} = H_{p+1−∗}^{∗}(M′; l^{2}(π))
D_{(p)}^{∗} = H_{p+1−∗}^{∗}(M_{1}; l^{2}(π)) ⊕ H_{p+1−∗}^{∗}(M_{2}; i_{2}^{∗}l^{2}(π))
E_{(p)}^{∗} = H_{p+1−∗}^{∗}(M_{3}; i_{3}^{∗}l^{2}(π)),

with differentials c, d and e given by exterior differentiation. Although M′ is noncompact, the Sobolev space H_{s}^{∗}(M′; l^{2}(π)) can be defined as in Definition 5.1, and is in fact a Sobolev space of differential forms on the hyperbolic 3-space H^{3}. The complexes C_{(p)} and E_{(p)} are Fredholm at p.

Lemma 5.8 There is an exact sequence of Hilbert cochain complexes

0 → C_{(p)} →^{j} D_{(p)} →^{k} E_{(p)} → 0,  (7)

with j(ω) = i_{1}^{∗}(ω) ⊕ i_{2}^{∗}(ω) and k(ω_{1} ⊕ ω_{2}) = i_{4}^{∗}(ω_{1}) − i_{5}^{∗}(ω_{2}).

\underline{\underline{Proof}} : It follows from the definitions that ker(j) = 0, and it is easy to check that ker(k) = im(j). To see that k is onto, let ϕ : I → R be a bump function which is identically zero near 0 and identically one near 1. Let ϕ̃ : M̃_{3} → R denote the composition of the pullbacks of ϕ to M_{3} and then to M̃_{3}, the preimage of M_{3} in H^{3}. We can think of an element η of E_{(p)}^{∗} as a differential form η̃ on M̃_{3}. Then ϕ̃η̃ extends by zero to a differential form on M̃_{1}, which comes from an element ω_{1} of H_{p+1−∗}^{∗}(M_{1}; l^{2}(π)). Similarly, we can extend (ϕ̃ − 1)η̃ by zero to a differential form on M̃_{2}, which comes from an element ω_{2} of H_{s}^{∗}(M_{2}; i_{2}^{∗}l^{2}(π)). Then k(ω_{1} ⊕ ω_{2}) = η.

It follows from Theorem 2.3 that the complex D_{(p)} is Fredholm at p.

Proposition 5.9 b_{p}(E_{(p)}) = 0, α_{0}(E_{(0)}) = α_{1}(E_{(1)}) = 2 and α_{2}(E_{(2)}) = ∞^{+}.

\underline{\underline{Proof}} : As the map Z^{2} = π_{1}(M_{3}) → π is an inclusion, the proof of Lemma 3.6 goes through for the analytic invariants to give that b_{p}(E_{(p)}) = b_{p}(I × T^{2}; l^{2}(Z^{2})) and α_{p}(E_{(p)}) = α_{p}(I × T^{2}; l^{2}(Z^{2})), where the right-hand-sides are defined by Definition 5.2. By the equivalence of the analytic and combinatorial invariants and the homotopy invariance of the combinatorial invariants (Theorem 2.6), these are the same as the invariants of T^{2}, which were given in Example 3.10.

Proposition 5.10 b_{p}(C_{(p)}) = 0, α_{0}(C_{(0)}) = α_{2}(C_{(2)}) = ∞^{+} and α_{1}(C_{(1)}) = 1.

\underline{\underline{Proof}} : As the universal cover of M′ is isometrically H^{3}, this follows from the same calculation in [24] as was cited above for the case of closed hyperbolic 3-manifolds.

Theorem 5.11 α_{1}(M; l^{2}(π)) ≥ 2/3.

\underline{\underline{Proof}} : We apply Theorem 2.3 to the exact sequence (7) with p = 1. As H_{1}(E_{(1)}) = 0, α(δ_{1}) = ∞^{+}. From Proposition 5.10, α_{1}(C_{(1)}) = 1 and from Proposition 5.9, α_{1}(E_{(1)}) = 2. Then Theorem 2.3 gives α_{1}(D_{(1)}) ≥ 2/3. From Lemma 1.10,

α_{1}(D_{(1)}) = min(α_{1}(M_{1}; l^{2}(π)), α_{1}(M_{2}; i_{2}^{∗}l^{2}(π)),

from which the assertion of the theorem follows.

Theorem 5.12 b_{p}(M; l^{2}(π)) = 0 for all p.

\underline{\underline{Proof}} : We can exhaust M′ = int(M) by a sequence of compact manifolds (with boundary) {M_{k}} which are all diffeomorphic to M. From [9, Theorem 1.1], b_{p}(M; l^{2}(π)) = b_{p}(M_{k}; l^{2}(π)) is the von Neumann dimension of the space of L^{2} harmonic p-forms on M̃′. As M̃′ is H^{3}, such forms vanish [13].

We now revert to letting the α_{p}(M)-invariants refer to boundaries, as in the previous sections, as opposed to coboundaries. The translation is that α_{p}(M), defined using coboundaries, equals α_{p+1}(M), defined using boundaries.

Theorem 5.13 α_{1}(M) = α_{3}(M) = ∞^{+}.

\underline{\underline{Proof}} : It follows from [47, Proposition 4.1.11] that π_{1}(M) is nonamenable. We derive from Lemma 3.5.2 that α_{1}(M) = ∞^{+}. As M has nonempty boundary, Lemma 3.5.5 gives that α_{3}(M) = ∞^{+}.

In summary, we have shown

Theorem 5.14 If M is a compact 3-manifold whose interior admits a complete finitevolume hyperbolic structure then M has vanishing L^{2}-cohomology and α_{1}(M) = α_{3}(M) = ∞^{+}. If M is closed then α_{2}(M) = 1 and if M is not closed then α_{2}(M) ≥ 2/3.

It will follow from Theorem 0.1.5 that if M is not closed then α_{2}(M) ≤ 2.

6. L^{2}-Betti numbers and Novikov-Shubin invariants of 3-manifolds

In this section we analyse the L^{2}-Betti numbers and Novikov-Shubin invariants of compact connected orientable 3-manifolds. It is easy to extend the results to the nonorientable case by means of the orientation covering.

We recall some basic facts about (compact connected orientable) 3-manifolds [20, 41]. A 3-manifold M is prime if for any decomposition of M as a connected sum M_{1}♯M_{2}, M_{1} or M_{2} is homeomorphic to S^{3}. It is irreducible if every embedded 2-sphere bounds an embedded 3-disk. Any prime 3-manifold is irreducible or is homeomorphic to S^{1} ×S^{2} [20, Lemma 3.13]. One can write M as a connected sum

M = M_{1}♯M_{2}♯ … ♯M_{r}

where each M_{j} is prime, and this prime decomposition is unique up to renumbering [20, Theorems 3.15, 3.21]. By the sphere theorem [20, Theorem 4.3], an irreducible 3-manifold is a K(π, 1) Eilenberg-MacLane space if and only if it is a 3-disk or has infinite fundamental group.

A properly-embedded orientable connected surface in a 3-manifold is incompressible if it is not a 2-sphere and the inclusion induces a injection on the fundamental groups. One says that ∂M is incompressible in M if and only if ∂M is empty or any component C of ∂M is incompressible in the sense above. An irreducible 3-manifold is Haken if it contains an embedded orientable incompressible surface. If M is irreducible and in addition H_{1}(M) is infinite, which is implied if ∂M contains a surface other than S^{2}, then M is Haken [20, Lemma 6.6 and 6.7]. (With our definitions, any properly embedded 2-disk is incompressible, and the 3-disk is Haken.)

Before we prove the main theorem of this paper, we must mention what is known about Thurston’s geometrization conjecture for irreducible 3-manifolds with infinite fundamental groups. (Again, our 3-manifolds are understood to be compact, connected and orientable.) Johannson [22] and Jaco and Shalen [21] have shown that given an irreducible 3-manifold M with incompressible boundary, there is a finite family of disjoint, pairwise-nonisotopic incompressible tori in M which are not isotopic to boundary components and which split M into pieces that are Seifert manifolds or are geometrically atoroidal, meaning that they admit no embedded incompressible torus (except possibly parallel to the boundary). A minimal family of such tori is unique up to isotopy, and we will say that it gives a toral splitting of M. We will say that the toral splitting is a geometric toral splitting if the geometrically atoroidal pieces which do not admit a Seifert structure have complete hyperbolic metrics on their interiors. Thurston’s geometrization conjecture for irreducible 3-manifolds with infinite fundamental groups states that such manifolds have geometric toral splittings.

Suppose that M is Haken. The pieces in its toral splitting are certainly Haken. Let N be a geometrically atoroidal piece. The torus theorem says that N is a special Seifert manifold or is homotopically atoroidal i.e. any subgroup of π_{1}(N) which is isomorphic to Z × Z is conjugate into the fundamental group of a boundary component. Thurston has shown that a homotopically atoroidal Haken manifold is a twisted I-bundle over the Klein bottle (which is Seifert), or admits a complete hyperbolic metric on its interior.

Thus the case in which Thurston’s geometrization conjecture for an irreducible 3manifold M with infinite fundamental group is still open is when M is a closed non-Haken irreducible 3-manifold with infinite fundamental group which is not Seifert. The conjecture states that such a manifold is hyperbolic.

Our goal is to make general statements about the L^{2}-Betti numbers and Novikov-Shubin invariants of a 3-manifold. We have already treated the case when the fundamental group is finite in Example 3.11. We will confine ourselves in the sequel to the case when π_{1}(M) is infinite. We will compute the invariants using the putative geometric decomposition of M. As we are studying homotopy invariants which have a simple behaviour with respect to finite coverings, it is enough to assume a weaker condition than that M have a geometric decomposition. Recall from the introduction that we say that a prime 3-manifold is exceptional if it is closed and no finite cover of it is homotopy-equivalent to a Haken, Seifert or hyperbolic 3-manifold.

Part 4 of Theorem 0.1 has been proven in Sections 4 and 5. We now prove parts 1a, 2 and 3 of Theorem 0.1. The proof will be by a succession of lemmas. In order to prove the statement about α_{1}(M), we will show that if α_{1}(M) < ∞^{+} then M is one of the special cases listed in the statement of the theorem. The values of α_{1}(M) in these special cases follow from previous calculations.

Lemma 6.1 If M is an irreducible Haken manifold with incompressible torus boundary then M has vanishing L^{2}-cohomology and α_{2}(M) > 0. If α_{1}(M) < ∞^{+} then M is one of the special cases listed in Theorem 0.1.2.

\underline{\underline{Proof}} : We know that M has a geometric toral splitting. As a compact connected orientable 3-manifold with torus boundary whose interior has a complete hyperbolic metric is either T^{2} × I or has a complete finite-volume hyperbolic metric [35, p. 52], the pieces in the toral splitting either admit a Seifert structure or have a complete finite-volume hyperbolic metric on their interior. Let s be the number of tori in such a minimal splitting. We will use induction over s. To begin the induction, if s = 0 then M is Seifert or hyperbolic and the claim follows from Theorems 4.1, 4.4 and 5.14. The induction step from s − 1 to s is done as follows:

Let T^{2} be a torus in a minimal family of splitting tori. Depending on whether T^{2} is separating or not, we get decompositions M = M_{1}∪_{T^{2}} M_{2} or M = M_{1}∪_{T^{2}×∂I} T^{2}×I by cutting M open along T^{2}, and an exact sequence 0 ⟶ C(T^{2}) ⟶ C(M_{1}) ⊕ C(M_{2}) ⟶ C(M) ⟶ 0 or 0 ⟶ C(T^{2} × ∂I) ⟶ C(M_{1}) ⊕ C(T^{2} × I) ⟶ C(M) ⟶ 0 with coefficients in l^{2}(π_{1}(M)). Note that each M_{j} satisfies the induction hypothesis. Hence b_{p}(M_{j}) = 0 for all p and α_{2}(M_{j}) > 0. From Lemma 3.6 and Example 3.10 we have that b_{p}(T^{2}) = 0 for all p and α_{p}(T^{2}) = 2 for p ∈ {1, 2}. The weakly exact Mayer-Vietoris sequence gives that M has vanishing L^{2}-cohomology, and Theorem 2.3.2 and Lemma 2.4.3 give the inequalities

\frac{1}{α_{2}(M)} ≤ \frac{1}{α_{1}(T^{2})} + \frac{1}{min\{α_{2}(M_{1}), α_{2}(M_{2})\}} or

\frac{1}{α_{2}(M)} ≤ \frac{1}{α_{1}(T^{2} × ∂I)} + \frac{1}{min\{α_{2}(M_{1}), α_{2}(T^{2} × I)\}}.

Thus α_{2}(M) > 0.

We also have the exact sequences 0 ⟶ C(M_{1}) ⟶ C(M) ⟶ C(M_{2}, T^{2}) ⟶ 0 or 0 ⟶ C(M_{1}) ⟶ C(M) ⟶ C(T^{2} × I, T^{2} × ∂I) ⟶ 0 with l^{2}(π_{1}(M)) as coefficients. As M_{1} has vanishing L^{2}-cohomology, Theorem 2.3.1 gives that

\frac{1}{α_{1}(M)} ≤ \frac{1}{α_{1}(M_{1})} + \frac{1}{α_{1}(M_{2}, T^{2})}  or  \frac{1}{α_{1}(M)} ≤ \frac{1}{α_{1}(M_{1})} + \frac{1}{α_{1}(T^{2} × I, T^{2} × ∂I)} .

From Lemma 3.5 we have that α_{1}(M_{2}, T^{2}) = α_{1}(T^{2} × I, T^{2} × ∂I) = ∞^{+}. This implies in both cases that α_{1}(M_{1}) ≤ α_{1}(M). Hence α_{1}(M_{1}) < ∞^{+}, and by symmetry α_{1}(M_{2}) < ∞^{+} in the first case. By the induction hypothesis, M_{j} must be T^{2} × I or a twisted I-bundle over K. Thus M is either the gluing of two twisted I-bundles over K along their boundaries, or a T^{2}-bundle over S^{1}. If M is the gluing of two twisted I-bundles over K over their boundaries then M is double-covered by a T^{2}-bundle over S^{1}. In either case, Lemma 6.2 will give that M has the geometric type of some T^{2}-bundle over S^{1}. (For later purposes, Lemma 6.2 is stated in greater generality than is needed here.) Then [41, Theorem 5.5] implies that M has a Sol, Nil or R^{3}-structure, and is one of the special cases listed.

Lemma 6.2 Let \overline{M} be a finite cover of an irreducible closed oriented 3-manifold M with infinite fundamental group. If \overline{M} is homotopy-equivalent to a closed 3-manifold N with a Seifert or Sol-structure then M has the same geometric type as N.

\underline{\underline{Proof}} : From [33, Theorem 3] we have that \overline{M} is irreducible. If N has a Seifert structure then [40, pages 35 and 36] gives that \overline{M} is homeomorphic to N and that M is also a Seifert manifold of the same geometric type. If N has a Sol-structure then \overline{M} and N are Haken, and so \overline{M} is homeomorphic to N [20, Theorem 13.6]. It follows from [41, Theorem 5.3] that M has a Sol-structure.

Lemma 6.3 If M is an irreducible Haken manifold with incompressible boundary then b_{p}(M) = 0 for p ≠ 1, b_{1}(M) = −χ(M) and α_{2}(M) > 0. If α_{1}(M) < ∞^{+} then M is one of the special cases listed in Theorem 0.1.2.

\underline{\underline{Proof}} : Because of Lemma 6.1, we may assume that ∂M is nonempty. Let N be M ∪_{∂M} M. Then [46, Satz 1.8] implies that N is irreducible. Clearly N is a closed Haken manifold. From Lemma 6.1 we have that N has vanishing L^{2}-cohomology and α_{2}(N) > 0. We have the exact sequence 0 ⟶ C(∂M) ⟶ C(M) ⊕ C(M) ⟶ C(N) ⟶ 0 with coefficients in l^{2}(π_{1}(N)). From Example 3.10 we have that b_{p}(∂M) = 0 for p ≠ 1 and α_{p}(∂M) > 0 for all p. Then we get from the weakly exact Mayer-Vietoris sequence that b_{p}(M) = 0 for p ≠ 1. From the Euler characteristic formula we derive that b_{1}(M) = −χ(M). Theorem 2.3.1 and Lemma 2.4.3 imply that

\frac{1}{α_{2}(M)} ≤ \frac{1}{α_{2}(∂M)} + \frac{1}{α_{2}(N)}

and hence α_{2}(M) > 0. Next we prove the claim for α_{1}(M). Suppose that M does not have a toral boundary. Then ∂M contains a component F_{g} for g ≥ 2. As π_{1}(F_{g}) is nonamenable and is a subgroup of π_{1}(M), π_{1}(M) is nonamenable and Lemma 3.5.2 implies that α_{1}(M) = ∞^{+}. Hence the claim follows already from Lemma 6.1.

Lemma 6.4 If M is an irreducible Haken manifold and is not a 3-disk, then b_{p}(M) = 0 for p ≠ 1, b_{1}(M) = −χ(M) and α_{2}(M) > 0. If α_{1}(M) < ∞^{+} then M is one of the special cases listed in Theorem 0.1.2.

\underline{\underline{Proof}} : Because of Lemma 6.3, we may assume that ∂M is compressible. The loop theorem [20, Theorem 4.2] gives an embedded disk D^{2} in M such that D^{2} meets ∂M transversally, and ∂D^{2} = D^{2} ∩ ∂M is an essential curve on ∂M. Depending on whether the disk D^{2} is separating or not, we get the following two cases:

If D^{2} is separating then there are 3-manifolds M_{1} and M_{2} and embedded disks D^{2} ⊂ ∂M_{1} and D^{2} ⊂ ∂M_{2} such that M = M_{1} ∪_{D^{2}} M_{2}. In particular, M is homotopy equivalent to M_{1} ⋁ M_{2}. Since M is prime, M_{1} and M_{2} are prime. As M_{1} and M_{2} have nonempty boundary, they are not S^{1} × S^{2}, and so are irreducible. As M is irreducible with infinite fundamental group, it is a K(π, 1) Eilenberg-MacLane space. Then the same must be true for M_{1} and M_{2}. If M_{i} were a 3-disk then the boundary of the embedded 2-disk would not be an essential curve on ∂M. Thus M_{1} and M_{2} have infinite fundamental groups.

If D^{2} is nonseparating then there is a 3-manifold M_{1} with embedded S^{0} × D^{2} ⊂ ∂M_{1} such that M = M_{1} ∪_{S^{0}×D^{2}} D^{1} × D^{2}. The same argument as above shows that M_{1} is an irreducible 3-manifold which is a 3-disk or has infinite fundamental group. If it were a 3-disk then M would be S^{1} × D^{2}, which satisfies the claim of the Lemma. So we may assume that M_{1} has infinite fundamental group.

We will prove the Lemma using the fact that M is homotopy equivalent to M_{1} ⋁ M_{2} (respectively M_{1} ⋁ S^{1}). It suffices to verify the claim for M_{1} and M_{2} (respectively M_{1}), since the claim for M then follows from the proof of Proposition 3.7. If M_{1} and M_{2} (respectively M_{1}) have incompressible boundary then we are done by Lemma 6.3. Otherwise, we repeat the process of cutting along 2-disks described above. This process must stop after finitely many steps.

Proof of Parts 1a, 2 and 3 of Theorem 0.1 : We have the prime decomposition

M = M_{1}♯M_{2}♯ … ♯M_{r}.

By assumption, each M_{j} in the decomposition is nonexceptional. We claim first that if π_{1}(M_{j}) is finite then b_{1}(M_{j}) = 0, if π_{1}(M_{j}) is infinite then b_{1}(M_{j}) = −χ(M_{j}), and that α_{2}(M_{j}) > 0. The case of finite fundamental group follows from Example 3.11. From Theorem 2.6 and Remark 3.9 we may assume that if M_{j} is closed then M_{j} is Seifert, hyperbolic or Haken. If M_{j} is closed and Seifert then the result follows from Theorem 4.1. If M_{j} is closed and hyperbolic then the result follows from Theorem 5.14. If M_{j} is closed and Haken then the result follows from Lemma 6.1. If M_{j} has a boundary component which is a 2-sphere then M_{j} is a 3-disk and the result follows from Example 3.11. If M_{j} has a nonempty boundary with no 2-spheres then it is Haken and the result follows from Lemma 6.4.

From Lemma 3.5 we have that b_{0}(M) = b_{3}(M) = 0. From Proposition 3.7.1 we have that

b_{1}(M) = r − 1 + ∑_{j=1}^{r} (b_{1}(M_{j}) − \frac{1}{| π_{1}(M_{j}) |}) .

As we have shown that b_{1}(M_{j}) = −χ(M_{j}) + {1 if M_{j} ≅ D^{3}}, the claim of Theorem 0.1.1 for b_{1}(M) follows. The claim for b_{2}(M) now follows from the Euler characteristic equation. From Proposition 3.7.3 we have α_{2}(M) = min{α_{2}(M_{j}) : j = 1, … r} > 0.

From Corollary 3.4.1 we have that α_{1}(M) = α_{1}(P (M)). Thus, by removing the simplyconnected factors, we may assume that M = P (M). Suppose that α_{1}(M) < ∞^{+}. From Proposition 3.7, we have the possibilities that r = 1, or that r = 2 and π_{1}(M_{1}) = π_{1}(M_{2}) = Z/2. If r = 1 then M ≅ S^{1} × S^{2} and is one of the special cases listed, or M is irreducible. If M is not closed then it is Haken and Lemma 6.4 implies that it is one of the special cases listed. If M is closed then by assumption a finite cover \overline{M} of M is homotopy equivalent to a Seifert, hyperbolic or Haken manifold N, which must also be closed and orientable. If N is Seifert or hyperbolic then Theorems 4.1, 4.4 and 5.14 imply that N is a closed S^{2} × R, R^{3}, or Nil manifold. If N is Haken then Lemma 6.4 implies that N is a closed S^{2} × R, R^{3}, Nil or Sol manifold. Lemma 6.2 gives that M is of the same geometric type as N, and so is one of the special cases listed.

If r = 2, it remains to show that an irreducible (compact connected orientable) 3manifold M with π_{1}(M) = Z/2 is homotopy equivalent to RP^{3}. This follows from [43, Theorem 1.8].

Proof of Theorem 0.1.1b : First, for the group Euler characteristic [5, Section IX.7] to be defined we must show that π_{1}(M) is virtually torsion-free and of finite homological type. Let {M_{j}}_{j=1}^{s} be the prime factors of M with finite fundamental group. Put Γ_{1} = π_{1}(M_{1}) ∗ … ∗ π_{1}(M_{s}) and Γ_{2} = π_{1}(M_{s+1}) ∗ … ∗ π_{1}(M_{r}). It is known that Γ_{1} has a finiteindex free subgroup F and that Γ_{2} is torsion-free. Let ϕ : Γ_{1} ∗ Γ_{2} → Γ_{1} be the natural homomorphism. Then ϕ^{−1}(F) is finite-index in π_{1}(M), and the Kurosh subgroup theorem [20, Theorem 8.3] implies that it is torsion-free. As Γ_{1} and Γ_{2} have finite homological type, [5, Proposition IX.7.3.e] implies that π_{1}(M) is of finite homological type and that:

χ(π_{1}(M)) = r − 1 + ∑_{j=1}^{r} χ(π_{1}(M_{j})).

Thus in order to show that b_{1}(M) = −χ(π_{1}(M)), it is enough to verify that for each j,

− \frac{1}{| π_{1}(M_{j}) |} − χ(M_{j}) + \{1 if M_{j} ≅ D^{3}\} = −χ(π_{1}(M_{j})).

As M_{j} is either a K(π, 1) Eilenberg-MacLane space with π infinite, a 3-disk or a closed manifold with finite fundamental group, the equation is easy to verify. The statement for b_{2}(M) now follows from the Euler characteristic equation.

We now prove a slightly stronger version of Theorem 0.1.1c.

Proposition 6.5 Let M be a (compact connected orientable) 3-manifold. If all L^{2}-Betti numbers of M vanish then M satisfies one of the following conditions:

1. M is homotopy equivalent to an irreducible 3-manifold N with infinite fundamental group whose boundary is empty or a disjoint union of tori.

2. M is homotopy equivalent to S^{1} × S^{2} or RP^{3}♯RP^{3}.

If condition 2.) holds, or if condition 1.) holds and N is nonexceptional, then all of the L^{2}-Betti numbers of M vanish.

\underline{\underline{Proof}} : Suppose that M has vanishing L^{2}-cohomology. From Example 3.11, π_{1}(M) must be infinite. From Proposition 3.7.1 we have that

r − 1 + ∑_{j=1}^{r} (b_{1}(M_{j}) − \frac{1}{| π(M_{j}) |}) = 0.

Equivalently,

∑_{j=1}^{r} (b_{1}(M_{j}) − \frac{1}{| π(M_{j}) |} + 1) = 1.

It follows that the prime decomposition of M must consist of homotopy 3-spheres, 3-disks and either A. A prime manifold M′ with infinite fundamental group and vanishing b_{1} or B. Two prime manifolds M^{1} and M^{2} with fundamental group Z/2.

In case A, M′ is S^{1} ×S^{2} or is irreducible. If M′ is irreducible and has nonempty boundary then Lemma 6.4 implies that its boundary components must be tori. From the Euler characteristic equation we have that χ(M) = 0, and so no 3-disks can occur in the prime decomposition of M. In case B, we have already shown that M^{1} and M^{2} are homotopyequivalent to RP^{3}. Again, because χ(M) = 0, no 3-disks can occur in the prime decomposition of M. Thus we have shown that if M has vanishing L^{2}-cohomology then M satisfies one of the two conditions of the corollary.

If M satisfies condition 2. of the corollary then Theorems 2.6 and 4.1 imply that M has vanishing L^{2}-cohomology. If M satisfies condition 1. of the corollary, from Theorem 2.6 we may assume without loss of generality that M = N. We have that its Euler characteristic vanishes. If M has nonempty boundary then Lemma 6.4 implies that it has vanishing L^{2}- cohomology. If M is closed and nonexceptional then by passing to a finite cover and using Theorem 2.6, we may assume that M is Seifert, hyperbolic or Haken. Theorems 4.1, 4.4 and 5.14 imply that M has vanishing L^{2}-cohomology.

We now prove Theorem 0.1.5. Again, we build up to the proof by a sequence of lemmas.

Lemma 6.6 If M is irreducible and ∂M contains an incompressible torus then α_{2}(M) ≤ 2.

\underline{\underline{Proof}} : From Lemma 6.3 we get b_{2}(M) = 0. As T^{2} has vanishing L^{2}-cohomology, the long weakly exact homology sequence of the pair (M, T^{2}) implies that H_{2}(M, T^{2}; l^{2}(π_{1}(M))) vanishes. We have a short exact sequence 0 ⟶ C(T^{2}) ⟶ C(M) ⟶ C(M, T^{2}) ⟶ 0 and so from Theorem 2.3.3,

\frac{1}{α_{2}(T^{2})} ≤ \frac{1}{α_{2}(M)} + \frac{1}{α_{3}(M, T^{2})} .

Proposition 3.2 implies that α_{3}(M, T^{2}) = α_{1}(M, ∂M − T^{2}). If this is ∞^{+} then α_{2}(M) ≤ α_{2}(T^{2}) = 2 and we are done. If ∂M − T^{2} ≠ ∅ then Lemma 3.5.5 implies that α_{1}(M, ∂M − T^{2}) = ∞^{+}. If ∂M −T^{2} = ∅ then Theorem 0.1.2 gives the possible cases in which α_{1}(M, ∂M − T^{2}) < ∞^{+}. The only case in which ∂M is a single incompressible torus is when M is a twisted I-bundle over K, and in this case Theorem 4.4 gives that α_{2}(M) = 2.

Lemma 6.7 If M is a closed Haken manifold and does not admit an R^{3} or Sol structure then α_{2}(M) ≤ 2.

\underline{\underline{Proof}} : If M is Seifert or hyperbolic then the proposition follows from Theorems 4.1 and 5.14. Otherwise, consider the nonempty minimal family of splitting tori. Let T^{2} be a member of the minimal family. Cutting M open along T^{2} yields decompositions M = M_{1} ∪_{T^{2}} M_{2} or M = M_{1} ∪_{T^{2}×∂T^{2}} T^{2} × I, depending on whether T^{2} is separating or not. We get the exact sequences 0 ⟶ C(M_{1}) ⟶ C(M) ⟶ C(M_{2}, T^{2}) ⟶ 0 or 0 ⟶ C(M_{1}) ⟶ C(M) ⟶ C(T^{2} × I, T^{2} × ∂I) ⟶ 0 with coefficients in l^{2}(π_{1}(M)). Since b_{1}(M) = 0 (Lemma 6.1), we derive from Theorem 2.3.2 that

\frac{1}{α_{2}(M_{2}, T^{2})} ≤ \frac{1}{α_{1}(M_{1})} + \frac{1}{α_{2}(M)}  or  \frac{1}{α_{2}(T^{2} × I, T^{2} × ∂I)} ≤ \frac{1}{α_{1}(M_{1})} + \frac{1}{α_{2}(M)}.

Suppose that α_{1}(M_{1}) ≥ ∞. Then we have that α_{2}(M) ≤ α_{2}(M_{2}, T^{2}) (respectively α_{2}(M) ≤ α_{2}(T^{2} × I, T^{2} × ∂I) = 2). Proposition 3.2 gives that α_{2}(M_{2}, T^{2}) = α_{2}(M_{2}), and we have already proven that this is less than or equal to two. By symmetry, it remains to treat the case when α_{1}(M_{1}), α_{1}(M_{2}) < ∞, (respectively α_{1}(M_{1}) < ∞). From Theorem 0.1.2, M_{1} and M_{2} must be I-bundles over K (respectively M_{1} must be I × T^{2}). As before, in either case M carries a Sol, Nil or R^{3}-structure. Since α_{2}(M) = 2 in the Nil case (Theorem 4.1), the lemma follows.

Proof of Theorem 0.1.5 : From Proposition 3.7.3 we have that

α_{2}(M) = min{α_{2}(M_{j}) : j = 1, … r}.

Clearly, it is enough to verify the theorem under the assumption that M is prime. As S^{1} × S^{2} has an S^{2} × R-structure, we may assume that M is irreducible. If ∂M contains an incompressible torus then we are done by Lemma 6.6. Suppose that M is closed, has infinite fundamental group and is nonexceptional. Then a finite cover \overline{M}, which is closed, orientable and irreducible, is homotopy equivalent to a manifold N which is Seifert, hyperbolic or Haken. If α_{2}(M) > 2 then Theorems 4.1 and 5.14 and Lemma 6.7 imply that N has an R^{3}, S^{2} × R or Sol structure. By Lemma 6.2, M also has such a structure.

Finally, Theorem 0.1.6 follows from Propositions 3.2 and 3.5.5.

7. Remarks and Conjectures

Conjecture 7.1 Let M be a compact connected manifold, possibly with boundary. Then

1. The L^{2}-Betti numbers of M are rational. If π_{1}(M) is torsion-free then the L^{2}-Betti numbers of M are integers.

2. The Novikov-Shubin invariants of M are positive and rational.

In the case of the L^{2}-Betti numbers, this seems to be a well-known conjecture. The question of the rationality of the L^{2}-Betti numbers, for closed manifolds, appears in [1]. Theorem 0.1 shows that Conjecture 7.1.1 is true for the class of 3-manifolds considered there. By Lemma 3.5.1, Conjecture 7.1.2 is trivially true for α_{1}(M). Theorems 4.1 and 4.4 give that it is true for α_{2}(M) if M is a Seifert 3-manifold. Note that for any positive integer k there are examples of closed manifolds in higher dimensions with π_{1}(M) = Z such that α_{3}(M) = \frac{1}{k} [24]. Conjecture 7.1 is equivalent to the following purely algebraic conjecture:

Conjecture 7.2 Let π be a finitely presented group and let f : ⊕_{i=1}^{r}Zπ ⟶ ⊕_{i=1}^{r}Zπ be a Zπ-homomorphism. We get a bounded π-equivariant operator \overline{f} : ⊕_{i=1}^{r}l^{2}(π) ⟶ ⊕_{i=1}^{r}l^{2}(π) by tensoring by l^{2}(π). Then

1. The von Neumann dimension of ker(\overline{f}) is rational. If π is torsion-free then it is an integer.

2. The Novikov-Shubin invariant of \overline{f} is a positive rational number.

To see the equivalence between Conjectures 7.1 and 7.2, suppose first that we are given a compact manifold M. Let K be a finite CW -complex which is homotopy equivalent to M. Taking f in Conjecture 7.2 to be the combinatorial Laplacian Δ_{p} coming from K and using Lemma 2.4, we see that the validity of Conjecture 7.2.1 would imply that Conjecture 7.1.1 holds for M. Taking f to be the differential c_{p} of the cellular chain complex of K̃, we see that the validity of Conjecture 7.2.2 would imply that Conjecture 7.1.2 holds for M. It remains to show that Conjecture 7.1 implies Conjecture 7.2. Let X be a finite CW -complex with fundamental group π. Let f : ⊕_{i=1}^{r}Zπ ⟶ ⊕_{i=1}^{r}Zπ be any Zπ-module homomorphism. For any given n ≥ 2, one can attach cells to X in dimensions n and n + 1 in such a way that the resulting finite CW -complex Y has the same fundamental group as X, and the relative chain complex C(Ỹ , X̃) is concentrated in dimensions n and n+1 and given there by f [26, Theorem 2.1]. If we choose n > dim(X) then b_{n+1}(Y) = b(\overline{f}) and α_{n+1}(Y) = α(\overline{f}). Moreover, there is a compact manifold M, possibly with boundary, which is homotopy equivalent to Y. Since the L^{2}-Betti numbers and the Novikov-Shubin invariants are homotopy invariants, we get b_{n+1}(M) = b(\overline{f}) and α_{n+1}(M) = α(\overline{f}). Hence Conjecture 7.1 is equivalent to Conjecture 7.2.

Conjecture 7.2.1 is proven for a large class of groups, which includes elementary amenable groups and free groups, in [23]. It is not hard to see that Conjecture 7.2.2 is true if π is abelian. (A proof of the equivalent Conjecture 7.1.2 in this case appears in [24].) D. Voiculescu informs us that Conjecture 7.2.2 is true when π is a free group. Conjecture 7.2.1 implies a well-known conjecture of algebra.

Conjecture 7.3 Let π be a finitely-presented group. Then the group ring Qπ has no zerodivisors if and only if π is torsion-free.

The only-if statement is trivial. The if statement would follow from the second conjecture as follows: Let u ∈ Qπ be a zero-divisor. We want to show that u = 0. We may assume that u lies in Zπ. Let f : Zπ ⟶ Zπ be given by right multiplication with u. Since u is a zero-divisor, \overline{f} has a non-trivial kernel. Since the dimension of the kernel of \overline{f} must be a positive number less or equal to the dimension of l^{2}(π), which is 1, it can only be an integer if it is 1. Hence the kernel of \overline{f} is l^{2}(π). This implies that u = 0.

Conjecture 7.4 The second L^{2}-Betti number of a compact prime 3-manifold vanishes.

We have shown in Example 3.11 and Theorem 0.1 that the second L^{2}-Betti number of a nonexceptional compact prime 3-manifold vanishes. However, there may be a reason why it should vanish which is independent of any geometric decomposition theorem.

Conjecture 7.5 If M is a closed K(π, 1) manifold then its L^{2}-Betti numbers vanish outside of the middle dimension.

Proposition 6.5 implies that a closed K(π, 1) 3-manifold of the type considered there has vanishing L^{2}-Betti numbers, thereby verifying the conjecture. Conjecture 7.5 includes the unproven conjecture of Singer which states the same for nonpositively-curved manifolds. If π_{1}(M) contains an infinite normal amenable subgroup then the truth of the conjecture follows immediately from [10, Theorem 0.2]. Conjecture 7.5 was emphasized in the case of 4-manifolds in [17, p. 154]. A consequence would be that if dim(M) = 4k+2 then χ(M) ≤ 0, and if dim(M) = 4k then χ(M) ≥ | σ(M) |.

As mentioned in the introduction, our motivation to study L^{2}-Betti numbers and Novikov-Shubin invariants comes from the L^{2}-torsion invariants [6, 24, 29, 31, 32]. Let M be a compact Riemannian manifold whose boundary is decomposed as ∂M = ∂_{0}M ∐ ∂_{1}M, where ∂_{0}M and ∂_{1}M are disjoint unions of components of ∂M. One can try to define L^{2}- analogs of the usual Reidemeister and analytic torsions of the pair (M, ∂_{0}M). However, one would encounter difficulties in the definitions if the spectrum of the combinatorial or differential-form Laplacian were too thick near zero. A sufficient condition for the L^{2}-torsions to be well-defined is that the Novikov-Shubin invariants α_{∗}(M, ∂_{0}M) are all positive. In this case, we denote the corresponding L^{2}-torsion invariants by ρ_{comb}(M, ∂_{0}M) and ρ_{an}(M, ∂_{0}M), respectively. Conjecture 7.1.2 would imply that the L^{2}-torsions are always well-defined. If in addition the L^{2}-cohomology of (M, ∂_{0}M) vanishes then the L^{2}-Reidemeister torsion is a simple homotopy invariant (and in particular a homeomorphism invariant) and the L^{2}-analytic torsion is a diffeomorphism invariant.

Conjecture 7.6 The L^{2}-Reidemeister and analytic torsions of (M, ∂_{0}M) are related by

ρ_{comb}(M, ∂_{0}M) = ρ_{an}(M, ∂_{0}M) + \frac{ln(2)}{2} · χ(∂M).

This is the L^{2}-analog of the Cheeger-Müller theorem for the ordinary Reidemeister and analytic torsions [7, 36], as extended to manifolds with boundary in [25, 28].

Our results show that if M is a 3-manifold of the type considered in Theorem 0.1 then the L^{2}-torsions are well-defined. Sufficient conditions for the vanishing of the L^{2}-cohomology are given in Proposition 6.5. If M is a Seifert 3-manifold with infinite fundamental group then its L^{2}-Reidemeister torsion vanishes [31]. If M is a closed 3-manifold which admits a hyperbolic structure then its L^{2}-analytic torsion is − \frac{1}{3π}V ol(M, g_{hyp}), where g_{hyp} is the unique (up to isometry) hyperbolic metric on M [24, 32].

Conjecture 7.7 If M is a compact connected 3-manifold with a Thurston geometric decomposition which satisfies one of the conditions of Proposition 6.5 then its L^{2}-torsion is − \frac{1}{3π} times the sum of the (finite) volumes of its hyperbolic pieces.

As one has a formula for the relationship between the L^{2}-Reidemeister torsions of the terms in a short exact sequence [31], Conjecture 7.7 would follow from Conjecture 7.6 if one knew that the L^{2}-torsion of a compact 3-manifold whose interior admitted a complete finitevolume hyperbolic metric were equal to − \frac{1}{3π} times the hyperbolic volume of the interior. We note that Conjecture 7.7 would imply that for the manifolds it considers, the L^{2}-torsion is a universal constant times the simplicial volume discussed in [44].

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Current address

John Lott^{∗} Wolfgang Lück^{∗∗} Department of Mathematics Fachbereich Mathematik University of Michigan Johannes Gutenberg-Universität Ann Arbor, MI 48109-1003 Staudingerweg 9 USA 6500 Mainz 1 lott@math.lsa.umich.edu Germany

lueck@topologie.mathematik.uni-mainz.de

* Partially supported by NSF-grant DMS 9101920 ** Partially supported by NSF-grant DMS 9103327