Lectures on L^{2}-Betti

numbers

Wolfgang Lück^{∗}

Fachbereich Mathematik und Informatik

Westfälische Wilhelms-Universität

Münster Einsteinstr. 62 48149 Münster

Germany

lueck@math.uni-muenster.de http://www.math.unimuenster.de/u/lueck

October 2, 2001

1. Introduction to L^{2}-Betti Numbers

Let G be a discrete group.

Definition 1.1 Denote by ℤG, ℚG and ℂG the integral, rational and complex group ring. An element in ℂG is a (formal) sum ∑_{g∈G} λ_{g} · g such that λ_{g} ∈ ℂ and λ_{g} ≠ 0 for only finitely many elements g ∈ G.

Denote by l^{2}(G) the Hilbert space of (formal) sums ∑_{g∈G} λ_{g} ·g such that λ_{g} ∈ ℂ and ∑_{g∈G} |λ_{g}|^{2} < ∞. This is the Hilbert space completion of ℂG with respect to the inner product for which G is an orthonormal basis.

Define the group von Neumann algebra

N (G) := B(l^{2}(G))^{G}

to be the algebra of bounded G-equivariant operators l^{2}(G) → l^{2}(G).

The von Neumann trace is defined by

tr_{N (G)} : N (G) → ℂ, f ↦ ⟨f(e), e⟩_{l^{2}(G)}.

Example 1.2 If G is finite, then ℂG = l^{2}(G) = N (G). The trace tr_{N (G)} assigns to ∑_{g∈G} λ_{g} · g the coefficient λ_{e}.

Example 1.3 Let G be ℤ^{n}. Let L^{2}(T^{n}) be the Hilbert space of L^{2}-integrable functions T^{n} → ℂ. Let L^{∞}(T^{n}) be the Banach space of essentially bounded functions f : T^{n} → ℂ ∐{∞}. An element (k_{1}, …, k_{n}) in ℤ^{n} acts isometrically on L^{2}(T^{n}) by pointwise multiplication with the function T^{n} → ℂ which maps (z_{1}, z_{2}, …, z_{n}) to z_{1}^{k_{1}} · . . . · z_{n}^{k_{n}}. Fourier transform yields an isometric ℤ^{n}-equivariant isomorphism l^{2}(ℤ^{n}) ⟶^{≅} L^{2}(T^{n}). We obtain an isomorphism

L^{∞}(T^{n}) ⟶^{≅} N (ℤ^{n}) = B(L^{2}(T^{n}))^{ℤ^{n}}

by sending f ∈ L^{∞}(T^{n}) to the ℤ^{n}-operator M_{f} : L^{2}(T^{n}) → L^{2}(T^{n}) g ↦ g ·f Under this identification the trace becomes

tr_{N (ℤ^{n})} : L^{∞}(T^{n}) → ℂ f ↦ ∫_{T^{n}} fdμ.

Definition 1.4 A Hilbert N (G)-module V is a Hilbert space V together with a linear isometric G-action such that there exists a Hilbert space H and an isometric linear G-embedding of V into the tensor product of Hilbert spaces H ⊗ l^{2}(G) with the obvious G-action.

A map of Hilbert N (G)-modules f : V → W is a bounded G-equivariant operator.

We call a Hilbert N (G)-module V finitely generated if there is a non-negative integer n and a surjective map ⊕_{i=1}^{n} l^{2}(G) → V of Hilbert N (G)-modules.

Definition 1.5 Let V be a finitely generated Hilbert N (G)-module. Choose a Gequivariant projection p : l^{2}(G)^{n} → l^{2}(G)^{n} with im(p) ≅_{N (G)} V. Define the von Neumann dimension of V by

dim_{N (G)}(V) := tr_{N (G)}(p)

:= n ; ∑ ; i=1 tr_{N (G)}(p_{i,i}) ∈ [0, ∞).

This notion extends to arbitrary Hilbert N (G)-modules if we allow the value ∞.

Definition 1.6 We call a sequence of Hilbert N (G)-modules U ⟶^{i} V ⟶^{p} W weakly exact at V if the kernel ker (p) of p and the closure clos (im(i)) of the image im(i) of i agree.

A map of Hilbert N (G)-modules f : V → W is a weak isomorphism if it is injective and has dense image.

Example 1.7 The morphism of N (ℤ)-Hilbert modules

M_{z−1} : l^{2}(ℤ) → l^{2}(ℤ), u ↦ (z − 1) · u

is a weak isomorphism but not an isomorphism.

Theorem 1.8 1. We have for a Hilbert N (G)-module V

V = 0 ⟺ dim_{N (G)}(V) = 0;

2. If 0 → U → V → W → 0 is a weakly exact sequence of Hilbert N (G)-modules, then

dim_{N (G)}(U) + dim_{N (G)}(W) = dim_{N (G)}(V);

3. Let {V_{i} | i ∈ I} be a directed system of Hilbert N (G)- submodules of V, directed by ⊂. Then

dim_{N (G)}(clos (∪_{i∈I}V_{i}))

= sup {dim_{N (G)}(V_{i}) | i ∈ I};

Example 1.9 If G is finite, then a finitely generated Hilbert N (G)-module is the same a unitary G-representation and

dim_{N (G)}(V) = \frac{1}{|G|} · dim_{ℂ}(V).

Example 1.10 Let G be ℤ^{n}. Let X ⊂ T^{n} be any measurable set with characteristic function χ_{X} ∈ L^{∞}(T^{n}). Let M_{χ_{X}} : L^{2}(T^{n}) → L^{2}(T^{n}) be the ℤ^{n}-equivariant unitary projection given by multiplication with χ_{X}. Its image V is a Hilbert N (ℤ^{n})- module with

dim_{N (ℤ^{n})}(V) = vol(X).

Definition 1.11 A G-CW -complex X is a G-space with a G-invariant filtration

∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{n} ⊂ … ∪_{n≥0} X_{n} = X

such that X carries the colimit topology and X_{n} is obtained from X_{n−1} by attaching equivariant n-dimensional cells, i.e. there exists a G-pushout

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

We call X finite if it is built by finitely many equivariant cells. We call X of finite type if each skeleton X_{n} is finite.

• A G-CW -complex is finite if and only if G\X is compact;

• It is a proper G-space if and only if each isotropy group is finite;

• Let X be a simplicial complex with simplicial G-action. Then its barycentric division X′ carries a G-CW -structure;

• If the smooth manifold M carries a smooth proper cocompact group action G, then it admits an equivariant triangulation and hence a G-CW -structure;

• If X → Y is a regular G-covering, then a CW -structure on Y induces a G-CW - structure on X and vice versa; Definition 1.12 Let X be a free G-CW - complex of finite type. Denote by C_{∗}(X) its cellular ℤ-chain complex. Define its cellular L^{2}-chain complex C_{∗}^{(2)} (X) to be the Hilbert N (G)-chain complex

C_{∗}^{(2)} (X) := l^{2}(G) ⊗_{ℤG} C_{∗}(X).

Define its p-th L^{2}-homology to be the finitely generated Hilbert N (G)-module

H_{p}^{(2)} (X; N (G)) := ker (C_{p}^{(2)} )/\overline{im(c_{p+1}^{(2)} )}.

Define its p-th L^{2}-Betti number

b_{p}^{(2)} (X; N (G)) = dim_{N (G)} (H_{p}^{(2)} (X; N (G))) .

Notice that C_{p}(X) = ⊕_{I_{p}}ℤG. Hence C_{p}^{(2)} (X) =

⊕_{I_{p}}l^{2}(G). Each differential c_{p}^{(2)} is a morphism of finitely generated Hilbert N (G)- modules since each I_{p} is finite by assumption.

The p-th Laplace operator is defined by

Δ_{p} := c_{p+1}^{(2)} ∘ (c_{p+1}^{(2)} )^{∗} + (c_{p}^{(2)} )^{∗} ∘ c_{p}^{(2)} :

C_{p}^{(2)} (X) → C_{p}^{(2)} (X).

Each H_{p}^{(2)} (X; N (G)) carries the structure of a finitely generated Hilbert N (G)-module since the natural map

ker (Δ_{p}) = ker (c_{p}^{(2)} ) ∩ ker (c_{p+1}^{(2)} )

⟶^{≅} H_{p}^{(2)} (X; N (G))

is an isometric G-equivariant isomorphism and ker (Δ_{p}) ⊂ C_{p}^{(2)} (X) = ⊕_{I_{p}}l^{2}(G).

Theorem 1.13 (Cellular L^{2}-Betti numbers).

1. Homotopy invariance Let f : X → Y be a G-map of free G-CW -complexes of finite type. If f is a weak homotopy equivalence (after forgetting the G-action), then

b_{p}^{(2)} (X) = b_{p}^{(2)} (Y);

2. Euler-Poincaré formula (Atiyah) Let X be free finite G-CW -complex. Let χ(G\X) be the Euler characteristic of the finite CW -complex. Then

χ(G\X) = _{p≥0}^{∑}(−1)^{p} · b_{p}^{(2)} (X);

3. Poincaré duality Let M be a cocompact free proper Gmanifold of dimension n which is orientable. Then

b_{p}^{(2)} (M) = b_{n−p}^{(2)}(M, ∂M);

4. Künneth formula (Zucker)

Let X be a free G-CW -complex of finite type and Y be a free H-CW -complex of finite type. Then X × Y is a free G × H-CW -complex of finite type and we get for all n ≥ 0

b_{n}^{(2)} (X × Y) = _{p+q=n}^{∑} b_{p}^{(2)} (X) · b_{q}^{(2)} (Y);

5. Wedges Let X_{1}, X_{2}, …, X_{r} be connected CW - complexes of finite type and X = ∨_{i=1}^{r} X_{i} be their wedge. Then

b_{1}^{(2)} (X̃) − b_{0}^{(2)} (X̃)

= r − 1 + r ; ∑ ; j=1 (b_{1}^{(2)} (X̃_{j}) − b_{0}^{(2)} (X̃_{j})) ;

and for 2 ≤ p

b_{p}^{(2)} (X̃) = r ; ∑ ; j=1 b_{p}^{(2)} (X̃_{j});

6. Morse inequalities (Novikov-Shubin) Let X be a free G-CW -complex of finite type. Let β_{p}(G\X) be the number of p-cells in G\X. Then we get for n ≥ 0

n ; ∑ ; p=0 (−1)^{n−p} · b_{p}^{(2)} (X)

≤ n ; ∑ ; p=0 (−1)^{n−p} · β_{p}(G\X);

7. Zero-th L^{2}-Betti number Let X be a connected free G-CW -complex of finite type. Then

b_{0}^{(2)} (X) = \frac{1}{|G|},

where \frac{1}{|G|} is to be understood as zero if the order |G| of G is infinite;

8. Restriction Let X be a free G-CW -complex of finite type and let H ⊂ G be a subgroup of finite index [G : H]. Then res _{G}^{H} X is a free H-CW -complex of finite type and

[G : H] · b_{p}^{(2)} (X; N (G))

= b_{p}^{(2)} (res _{G}^{H} X; N (H));

9. Induction Let H be a subgroup of G and let X be a free H-CW -complex of finite type. Then G ×_{H} X is a G-CW -complex of finite type and

b_{p}^{(2)} (G ×_{H} X; N (G)) = b_{p}^{(2)} (X; N (H)).

Example 1.14 If G is finite and X is a free G-CW -complex of finite type, then b_{p}^{(2)} (X) is the classical p-th Betti number of X multiplied with \frac{1}{|G|}.

Lemma 1.15 Let X be a free ℤ^{n}-CW -complex of finite type. Then

b_{p}^{(2)} (X) = dim_{ℂ[ℤ^{n}]^{(0)}} (ℂ[ℤ^{n}]^{(0)} ⊗_{ℤ[ℤ^{n}]} H_{p}(X)) ,

where ℂ[ℤ^{n}]^{(0)} is the quotient field of ℂ[ℤ^{n}].

Example 1.16 Let X → Y be a finite covering with d-sheets of connected CW -complexes of finite type. Then Theorem 1.13 (8) implies

b_{p}^{(2)} (Ỹ) = d · b_{p}^{(2)} (X̃).

In particular we get for a connected CW - complex X of finite type for which there is a selfcovering X → X with d-sheets for some integer d ≥ 2 that b_{p}^{(2)} (X̃) = 0 for all p ≥ 0. This implies for any finite CW - complex X of finite type

b_{p}^{(2)} ({ \atop S^{1} × X}) = 0.

Theorem 1.17 (Long weakly exact L^{2}- homology sequence, Cheeger-Gromov).

Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{p_{∗}} E_{∗} → 0 be an exact

sequence of Hilbert N (G)-chain complexes whose chain modules have finite dimension. Then there is a long weakly exact homology sequence

… ⟶^{H_{n+1}^{(2)} (p_{∗})} H_{n+1}^{(2)} (E_{∗}) ⟶^{∂_{n+1}} H_{n}^{(2)} (C_{∗})

⟶^{H_{n}^{(2)} (i_{∗})} H_{n}^{(2)} (D_{∗}) ⟶^{H_{n}^{(2)} (p_{∗})} H_{n}^{(2)} (E_{∗}) ⟶^{∂_{n}} …

Theorem 1.18 (L^{2}-Betti numbers and S^{1}-actions).

Let X be a connected S^{1}-CW -complex of finite type, for instance a connected compact manifold with S^{1}-action. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.) Then we get for all p ≥ 0

b_{p}^{(2)} (X̃) = 0.

Theorem 1.19 (L^{2}-Betti numbers and aspherical S^{1}-manifolds). Let M be an aspherical closed manifold with non-trivial S^{1}-action. Then the action has no fixed points and the inclusion of any orbit into X induces an injection on the fundamental groups. All L^{2}-Betti numbers b_{p}^{(2)} (M̃) are trivial and χ(M) = 0.

Example 1.20 Let F_{g} be the orientable closed surface. Since F_{0} = S^{2} is simplyconnected, we get

b_{p}^{(2)} (F̃_{0}) = b_{p}(S^{2}) = 1 if p = 0, 2.
b_{p}^{(2)} (F̃_{0}) = b_{p}(S^{2}) = 0 if p ≠ 0, 2;

If g ≥ 0, then π_{1}(F_{g}) is infinite and hence b_{0}^{(2)} (F̃_{g}) = 0. By Poincaré duality b_{2}^{(2)} (F̃_{g}) = 0. Since dim (F_{g}) = 2, we get b_{p}^{(2)} (F̃_{g}) = 0 for p ≥ 3. Using the Euler-Poincaré formula we get

b_{1}^{(2)} (F̃_{g}) = −χ(F_{g}) = 2g − 2;
b_{p}^{(2)} (F̃_{0}) = 0 for p ≠ 1.

Theorem 1.21 (L^{2}-Betti numbers of 3manifolds, Lott-Lück). Let M be the connected sum M_{1}♯ … ♯M_{r} of (compact connected orientable) prime 3-manifolds M_{j} which are non-exceptional. Assume that π_{1}(M) is infinite. Then the L^{2}-Betti numbers of the universal covering M̃ are given by

b_{0}^{(2)} (M̃) = 0;
b_{1}^{(2)} (M̃) = (r − 1) − r ; ∑ ; j=1 \frac{1}{| π_{1}(M_{j}) |} − χ(M)

+ |{C ∈ π_{0}(∂M) | C ≅ S^{2}}| ;

b_{2}^{(2)} (M̃) = (r − 1) − r ; ∑ ; j=1 \frac{1}{| π_{1}(M_{j}) |}

+ |{C ∈ π_{0}(∂M) | C ≅ S^{2}}| ;

b_{3}^{(2)} (M̃) = 0.

Theorem 1.22 L^{2}-Hodge-de Rham Theorem, Dodziuk).

Let M be a cocompact free proper Gmanifold with G-invariant Riemannian metric and K an equivariant smooth triangulation of M. Suppose that M has no boundary. Let

H_{(2)}^{p} (M) = {ω ∈ Ω^{p}(M̃) | Δ_{p}(M) = 0}

be the space of harmonic L^{2}-forms on M̃. Then integration defines an isomorphism of finitely generated Hilbert N (G)-modules

H_{(2)}^{p} (M) ⟶^{≅} H_{(2)}^{p} (K).

Corollary 1.23

b_{p}^{(2)} (M) = lim_{t→∞} ∫_{F} tr_{ℂ}(e^{−tΔ_{p}} (x, x)) dvol.

where F is a fundamental domain for the G-action and e^{−tΔ_{p}} (x, y) is the heat kernel on M̃.

Theorem 1.24 Let M be a hyperbolic closed Riemannian manifold of dimension n. Then:

b_{p}^{(2)} (M̃)  \{ \begin{matrix}= 0 & , if 2p ≠ n \\ > 0 & , if 2p = n \end{matrix} .

Proof: A direct computation shows that H_{(2)}^{p} (H^{n}) is not zero if and only if 2p = n. Notice that M is hyperbolic if and only if M̃ is isometrically diffeomorphic to the standard hyperbolic space H^{n}.

Theorem 1.25 Let M be a hyperbolic closed manifold of dimension n. Then

1. If n = 2m is even, then

(−1)^{m} · χ(M) > 0;

2. M carries no non-trivial S^{1}-action.

Proof: (1) We get from the Euler-Poincaré formula and Theorem 1.24

(−1)^{m} · χ(M) = b_{m}^{(2)} (M̃) > 0.

(2) We give the proof only for n = 2m even. Then b_{m}^{(2)} (M̃) > 0. Since M̃ = H^{n} is contractible, M is aspherical. Now apply Theorem 1.19.

Theorem 1.26 Vanishing of L^{2}-Betti numbers of mapping tori). Let f : X → X be a cellular selfhomotopy equivalence of a connected CW -complex X of finite type. Then we get for all p ≥ 0

b_{p}^{(2)} (T̃_{f}) = 0.

Proof: There is a d-sheeted covering T_{f^{d}} → T_{f}. Hence

b_{p}^{(2)} (T̃_{f}) = \frac{b_{p}^{(2)} (T̃_{f^{d}})}{d} .

If β_{p}(X) is the number of p-cells, then there is up to homotopy equivalence a CW - structure on T_{f^{d}} with β(T_{f^{d}}) = β_{p}(X) + β_{p−1}(X). We have

b_{p}^{(2)} (T̃_{f^{d}}) = dim_{N (G)}(H_{p}^{(2)} (C_{p}^{(2)} (T̃_{f^{d}}))

≤ dim_{N (G)} (C_{p}^{(2)} (T̃_{f^{d}})) = β_{p}(T_{f^{d}}).

This implies for all d ≥ 1

b_{p}^{(2)} (T̃_{f}) ≤ \frac{β_{p}(X) + β_{p−1}(X)}{d} .

Taking the limit for d → ∞ yields the claim.

Example 1.27 The following examples show that in general there are hardly any relations between the ordinary Betti numbers b_{p}(X) and the L^{2}-Betti numbers b_{p}^{(2)} (X̃) for a connected CW -complex X of finite type.

Given a group G such that BG is of finite type, define its p-th L^{2}-Betti number and its p-th Betti number by

b_{p}^{(2)} (G) := b_{p}^{(2)} (EG; N (G));
b_{p}(G) := b_{p}(BG).

We get from Theorem 1.13 (4), (5) and

(8) for r ≥ 2 and non-trivial groups G_{1}, G_{2}, …, G_{r} whose classifying spaces BG_{i} are of finite type

b_{1}^{(2)} (∗_{i=1}^{r} G_{i}) = r − 1 + r ; ∑ ; i=1 (b_{1}^{(2)} (G_{i}) − \frac{1}{|G_{i}|}) ;
b_{0}^{(2)} (∗_{i=1}^{r} G_{i}) = 0;
b_{p}^{(2)} (∗_{i=1}^{r} G_{i}) = r ; ∑ ; i=1 b_{p}^{(2)} (G_{i}) for p ≥ 2;
b_{p}(∗_{i=1}^{r} G_{i}) = r ; ∑ ; i=1 b_{p}(G_{i}) for p ≥ 1;
b_{0}^{(2)} (ℤ/n) = \frac{1}{n};
b_{p}^{(2)} (ℤ/n) = 0 for p ≥ 1;
b_{p}(ℤ/n) = 0 for p ≥ 1;
b_{p}^{(2)} (G_{1} × G_{2}) = p ; ∑ ; i=0 b_{i}^{(2)} (G_{1}) · b_{p−i}^{(2)}(G_{2});
b_{p}(G_{1} × G_{2}) = p ; ∑ ; i=0 b_{i}(G_{1}) · b_{p−i}(G_{2}).

From this one easily verifies for any integers m ≥ 0, n ≥ 1 and i ≥ 1 that for the group

G_{i}(m, n) = ℤ/n×(∗_{k=1}^{2m+2} ℤ/2)×( i−1 ; ∏ ; j=1 ∗_{l=1}^{4} ℤ/2)

its classifying space BG_{i}(m, n) is of finite type and

b_{i}^{(2)} (G_{i}(m, n)) = \frac{m}{n};
b_{p}^{(2)} (G_{i}(m, n)) = 0 for p ≠ i;
b_{p}(G_{i}(m, n)) = 0 for p ≥ 1.

Given an integer l ≥ 1 and a sequence r_{1}, r_{2}, …, r_{l} of non-negative rational numbers, we can construct a group G such that BG is of finite type and

b_{p}^{(2)} (G) = r_{p} for 1 ≤ p ≤ l;
b_{p}^{(2)} (G) = 0 for l + 1 ≤ p;
b_{p}(G) = 0 for p ≥ 1.

Namely, take

G = ℤ/n × ∗_{i=2}^{k} G_{i}(m_{i}, n_{i}).

On the other hand we can construct for any sequence n_{1}, n_{2}, … of non-negative integers a CW -complex X of finite type such that b_{p}(X) = n_{p} and b_{p}^{(2)} (X̃) = 0 holds for p ≥ 1, namely take

X = B(ℤ/2 ∗ ℤ/2) × ∨_{p=1}^{∞} (∨_{i=1}^{n_{p}} S^{p}) .

Theorem 1.28 Proportionality Principle for L^{2}-invariants Let M be a simply connected Riemannian manifold. Then there are constants B_{p}^{(2)} (M) for p ≥ 0 depending only on the Riemannian manifold M such that for any discrete group G with a free proper cocompact action on M by isometries the following holds

b_{p}^{(2)} (M; N (G)) = B_{p}^{(2)} (M) · vol(G\M).

2. The Generalized Dimension function

Remark 2.1 Recall that by definition

N (G) := B(l^{2}(G), l^{2}(G))^{G} = mor _{N (G)}(l^{2}(G), l^{2}(G)).

This induces a bijection of ℂ-vector spaces

M(m, n, N (G)) ⟶^{≅} mor _{N (G)}(l^{2}(G)^{m}, l^{2}(G)^{n}).

It is compatible with multiplication of matrices and composition of morphisms. This extends to finitely generated Hilbert N (G)- modules and finitely projective N (G)-modules.

Theorem 2.2 (Modules over N (G) and Hilbert N (G)-modules). We obtain an equivalence of ℂ-categories

ν: {fin. gen. proj. N (G)-mod. } → {fin. gen. Hilb. N (G)-mod. }.

Definition 2.3 Let R be a ring. Let M be a R-submodule of N. Define the closure of M in N to be the R-submodule of N

\overline{M} = \{x ∈ N | f(x) = 0 for all

f ∈ N^{∗} with M ⊂ ker (f)}.

For a R-module M define the R-submodule TM and the R-quotient module PM by:

TM := {x ∈ M | f(x) = 0

for all f ∈ M^{∗}};

PM := M/TM.

We call a sequence of R-modules L ⟶^{i} M ⟶^{q} N weakly exact if \overline{im(i)} = ker (q).

Notice that TM is the closure of the trivial submodule in M. It can also be described as the kernel of the canonical map

i(M) : M → (M^{∗})^{∗}

which sends x ∈ M to the map M^{∗} → R f ↦ f(x)^{∗}. Notice that TPM = 0 and that PM = 0 is equivalent to M^{∗} = 0.

Example 2.4 Let R = ℤ. Let M be a finitely generated ℤ-module and K ⊂ M. Then

\overline{K} = \{x ∈ M | n · x ∈ K for some n ∈ ℤ\};
TM := tors (M);
PM = M/ tors (M).

A sequence M_{0} → M_{1} → M_{2} of finitely generated ℤ-modules is weakly exact if and only if it is exact after applying ℚ ⊗_{ℤ} −.

Definition 2.5 Let P be a finitely generated projective N (G)-module. Choose a matrix A ∈ M_{n}(N (G)) with A^{2} = A such that the image of r_{A} : N (G)^{n} → N (G)^{n} is N (G)-isomorphic to P. Define

dim_{N (G)}(P) := tr_{N (G)}(A) [0, ∞).

Theorem 2.6 1. The functors ν and ν^{−1} preserve exact sequences and weakly exact sequences;

2. If P is a finitely generated projective N (G)-module, then

dim_{N (G)}(P) = dim_{N (G)}(ν(P)).

Remark 2.7 N (G) is Noetherian if and only if G is finite. It contains zero-divisors if G is non-trivial.

Definition 2.8 A ring R is called semihereditary if any finitely generated submodule of a projective module is projective.

Lemma 2.9 N (G) is semihereditary.

Proof: It suffices to prove for a finitely generated N (G)-submodule M ⊂ N (G)^{n} that it is projective. Choose a N (G)-map f : N (G)^{m} → N (G)^{n} whose image is M. Let ν(f) : l^{2}(G)^{m} → l^{2}(G)^{n} be the morphism corresponding to f under ν. Choose a projection pr : l^{2}(G)^{m} → l^{2}(G)^{m} with image ker (ν(f)). Then

l^{2}(G)^{m} ⟶^{pr} l^{2}(G)^{m} ⟶^{ν^{−1}(f)} l^{2}(G)^{n}

is exact. Hence

N (G)^{m} ⟶^{ν^{−1}(pr)} N (G)^{m} ⟶^{f} N (G)^{n}

is exact and ν^{−1}(pr)^{2} = ν^{−1}(pr). Hence ker (f) ⊂ N (G)^{m} is a direct summand and M = im(f) is projective.

Remark 2.10 The following results and definitions can be understood by the slogan that N (G) behaves like ℤ if one forgets that ℤ is Noetherian and has no-zero-divisors. In this sense all properties of ℤ carry over to N (G).

Lemma 2.11 Let M be a finitely generated N (G)-module. Then

1. Let K ⊂ M be a submodule. Then \overline{K} ⊂ M is a direct summand and M/K is finitely generated projective;

2. PM is a finitely generated projective N (G)-module and we get a splitting

M ≅ TM ⊕ PM;

3. If M is finitely presented, then there is an exact sequence

0 → N (G)^{n} → N (G)^{n} → TM → 0.

Theorem 2.12 (Dimension function for arbitrary N (G)-modules).

There is precisely one dimension function

dim : {N (G) − modules } → [0, ∞]

which has the following properties;

1. Extension Property

If M is a finitely generated projective R-module, then dim (M) agrees with the previously defined notion;

2. Additivity

If 0 → M_{0} ⟶^{i} M_{1} ⟶^{p} M_{2} → 0 is an exact sequence of R-modules, then

dim (M_{1}) = dim (M_{0}) + dim (M_{2});

3. Cofinality

Let {M_{i} | i ∈ I} be a cofinal system of submodules of M, i.e. M = ∪_{i∈I}M_{i} and for two indices i and j there is an index k in I satisfying M_{i}, M_{j} ⊂ M_{k}. Then

dim (M) = sup {dim (M_{i}) | i ∈ I};

4. Continuity

If K ⊂ M is a submodule of the finitely generated R-module M, then

dim (K) = dim (\overline{K});

5. If M is a finitely generated R-module, then

dim (M) = dim (PM);
dim (TM) = 0;

Proof: We give the proof of uniqueness which leads to the definition of dim . Any N (G)-module M is the colimit over the directed system of its finitely generated submodules {M_{i} | i ∈ I}. Hence by Cofinality

dim (M) = sup {dim (M_{i}) | i ∈ I}.

We get for each M_{i} from Additivity

dim (M_{i}) = dim (PM_{i}).

Hence we get

dim (M) = sup {dim (P) | P ⊂ M

finitely generated projective }.

The hard part is now to show that with this definition all the properties are satisfied.

Theorem 2.13 Let {M_{i} | i ∈ I} be a direct system of R-modules over the directed set I. For i ≤ j let ϕ_{i,j} : M_{i} → M_{j} be the associated morphism of R-modules. Suppose for each i ∈ I that there is i_{0} ∈ I with i ≤ i_{0} such that dim (im(ϕ_{i,i_{0}})) < ∞ holds. Then

dim (colim _{i∈I} M_{i})
= sup {inf {dim (im(ϕ_{i,j} : M_{i} → M_{j})) |
j ∈ I, i ≤ j} | i ∈ I} .

Remark 2.14 The results above are motivated by the following observations for R = ℤ. If M is a finitely generated ℤmodule, then M/ tors (M) is finitely generated free and

M = tors (M) ⊕ M/ tors (M).

We get a dimension function for all ℤmodules by

dim (M) := dim_{ℚ}(ℚ ⊗_{ℤ} M).

The difference between ℤ and N (G) is that for a projective ℤ-module P we have dim (P) < ∞ if and only if P is finitely generated. This is not true for N (G).

Definition 2.15 Let X be a (left) G-space. Its homology with coefficients in N (G) is

H_{p}^{G}(X; N (G)) = H_{p} (N (G) ⊗_{ℤG} C_{∗}^{sing} (X)) .

Define the p-th L^{2}-Betti number of X by

b_{p}^{(2)} (X; N (G)) := dim_{N (G)}(H_{p}^{G}(X; N (G)))
∈ [0, ∞].

Lemma 2.16 Let X be a G-CW -complex of finite type. Then Definition 2.15 of L^{2}- Betti numbers b_{p}^{(2)} (X; N (G)) agrees with the previous one.

Definition 2.17 The p-th L^{2}-Betti number of a group G is

b_{p}^{(2)} (G) := b_{p}^{(2)} (EG, N (G)).

Remark 2.18 Notice that we work with homology. This is very convenient since homology transforms colimits into colimits in general whereas cohomology tranforms colimits into exact sequences involving inverse limits and higher inverse limits. Moreover, the dimension function behaves well under colimits but its behaviour under inverse limits is much more complicated.

Theorem 2.19 L^{2}-Betti numbers for arbitrary spaces).

1. Homotopy invariance Let f : X → Y be a G-map. Suppose such that for each subgroup H ⊂ G the induced map f^{H} : X^{H} → Y^{H} is a ℂhomology equivalence, i.e. H_{p}^{sing} (f^{H}; ℂ) : H_{p}^{sing} (X^{H}; ℂ) → H_{p}^{sing} (Y^{H}; ℂ) is bijective for p ≥ 0. Then for all p ≥ 0 the induced map f_{∗} : H_{p}^{G}(X; N (G)) → H_{p}^{G}(Y; N (G)) is bijective and we get

b_{p}^{(2)} (X) = b_{p}^{(2)} (Y) for p ≥ 0;

2. Comparison with the Borel construction Let X be a G-CW -complex. Suppose that for all x ∈ X the isotropy group G_{x} is finite or satisfies b_{p}^{(2)} (G_{x}) = 0 for all p ≥ 0. Then for p ≥ 0

b_{p}^{(2)} (X; N (G)) = b_{p}^{(2)} (EG × X; N (G));

3. Independence of equivariant cells with infinite isotropy Let X be a G-CW -complex. Let X[∞] be the G-CW -subcomplex consisting of those points whose isotropy subgroups are infinite. Then we get for all p ≥ 0

b_{p}^{(2)} (X; N (G)) = b_{p}^{(2)} (X, X[∞]; N (G));

4. Künneth formula Let X be a G-space and Y be a Hspace. Then X × Y is a G × H-space and we get for all n ≥ 0

b_{n}^{(2)} (X × Y) = _{p+q=n}^{∑} b_{p}^{(2)} (X) · b_{q}^{(2)} (Y),

where we use the convention that 0 · ∞ = 0, r · ∞ = ∞ for r ∈ (0, ∞] and r + ∞ = ∞ for r ∈ [0, ∞];

5. Induction Let i : H → G be an inclusion of groups and let X be a H-space. Let i : N (H) → N (G) be the induced ring homomorphism. Then

H_{p}^{G}(G ×_{H} X; N (G)) = i_{∗}H_{p}^{H}(X; N (H));
b_{p}^{(2)} (G ×_{H} X; N (G)) = b_{p}^{(2)} (X; N (H));

6. Restriction Let H ⊂ G be a subgroup of finite index [G : H]. Let X be a G-space and let res (X) be the H-space obtained from X by restriction. Then

b_{p}^{(2)} (res (X); N (H)) = [G : H] · b_{p}^{(2)} (X; N (G));

7. Zero-th homology and L^{2}-Betti number Let X be a path-connected G-space. Then

b_{0}^{(2)} (X; N (G)) = |G|^{−1}.

Moreover H_{0}^{G}(X; N (G)) is trivial if and only if G is non-amenable.

Definition 2.20 A group G is called amenable if there is a (left) G-invariant linear operator μ : l^{∞}(G, ℝ) → ℝ with μ(1) = 1 which satisfies

inf {f(g) | g ∈ G} ≤ μ(f) ≤ sup {f(g) | g ∈ G} for all f ∈ l^{∞}(G, ℝ).

The class of elementary amenable groups is defined as the smallest class of groups which has the following properties:

1. It contains all finite and all abelian groups;

2. It is closed under taking subgroups;

3. It is closed under taking quotient groups;

4. It is closed under extensions;

5. It is closed under directed unions, Remark 2.21 The class of amenable groups is closed under the operations above. Hence it contains the class of elementary amenable groups. A group which contains ℤ∗ℤ is not amenable.

Corollary 2.22 (Brooks). Let M be a closed Riemannian manifold. Then the Laplace operator acting on functions on M̃ has zero in its spectrum if and only if π_{1}(M) is amenable.

Theorem 2.23 (Dimension-flatness of N (G) over ℂG for amenable G).

Let G be amenable and M be a ℂGmodule. Then for p ≥ 1

dim_{N (G)} (Tor _{p}^{ℂG}(N (G), M)) = 0.

Theorem 2.24 Let G be an amenable group and X be a G-space. Then

b_{p}^{(2)} (X; N (G)) = dim_{N (G)} (N (G) ⊗_{ℂG} H_{p}^{sing} (X; ℂ)) .

Corollary 2.25 (Cheeger-Gromov). Let G be a group which contains an infinite normal amenable subgroup. Then for p ≥ 0

b_{p}^{(2)} (G; N (G)) = 0.

If there is a finite model for BG, then

χ(G) := χ(BG) = 0.

Proof: If G is amenable, this follows from H_{p}(EG; ℂ) = 0 for p ≥ 1. In the general case one uses a spectral sequence argument.

Definition 2.26 Let R be an (associative) ring (with unit). Define its projective class group K_{0}(R) to be the abelian group whose generators are isomorphism classes [P] of finitely generated projective R-modules P and whose relations are [P_{0}] + [P_{2}] = [P_{1}] for any exact sequence 0 → P_{0} → P_{1} → P_{2} → 0 of finitely generated projective Rmodules. Define G_{0}(R) analogously but replacing finitely generated projective by finitely generated.

Theorem 2.27 Let G be an amenable group. Then we get a well-defined map

dim : G_{0}(ℂG) → ℝ, [M] ↦ dim_{N (G)}(N (G) ⊗_{ℂG} M).

In particular [ℂG] generates an infinite cyclic subgroup in G_{0}(ℂG).

Lemma 2.28 If G contains ℤ ∗ ℤ as subgroup, then

[ℂG] = 0 ∈ G_{0}(ℂG).

Conjecture 2.29 G is amenable if and only if

[ℂG] ≠ 0 ∈ G_{0}(ℂG).

Theorem 2.30 (L^{2}-Betti numbers and S^{1}-actions). Let X be a connected S^{1}-CW -complex. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.) Let X̃ be the universal covering of X with the canonical π_{1}(X)-action. Then we get for all p ≥ 0

b_{p}^{(2)} (X̃) = 0.

Theorem 2.31 L^{2}-Betti numbers and fibrations Let F ⟶^{i} E ⟶^{p} B be a fibration of connected CW -complexes. Suppose that π_{1}(F) → π_{1}(E) is injective. Suppose for a given integer d ≥ 1 that b_{p}^{(2)} (F̃) = 0 for p ≤ d−1 and b_{d}^{(2)} (F̃) < ∞ holds. Suppose that π_{1}(B) contains an element of infinite order or finite subgroups of arbitrary large order. Then b_{p}^{(2)} (Ẽ) = 0 for p ≤ d.

Definition 2.32 Let G be a finitely presented group. Define its deficiency def (G) to be the maximum g(P) − r(P), where P runs over all presentations P of G and g(P) is the number of generators and r(P) is the number of relations of a presentation P.

Lemma 2.33 Let G be a finitely presented group. Then

def (G) ≤ 1 − b_{0}^{(2)} (G) + b_{1}^{(2)} (G) − b_{2}^{(2)} (G).

Proof We have to show for any presentation P that

g(P)−r(P) ≤ 1−b_{0}^{(2)} (G)+b_{1}^{(2)} (G)−b_{2}^{(2)} (G).

Let X be a CW -complex realizing P. Then

χ(X) = 1 − g(P) + r(P) = b_{0}^{(2)} (X̃) + b_{1}^{(2)} (X̃) − b_{2}^{(2)} (X̃).

Since the classifying map X → BG is 2connected, we get

b_{p}^{(2)} (X̃) = b_{p}^{(2)} (G) for p = 0, 1;
b_{2}^{(2)} (X̃) ≥ b_{2}^{(2)} (G).

Example 2.34 The free group F_{g} has the obvious presentation ⟨s_{1}, s_{2}, … s_{g} | ∅⟩ and its deficiency is realized by this presentation, namely def (F_{g}) = g.

If G is a finite group, def (G) ≤ 0 by Lemma 2.33 because of b_{0}^{(2)} (G) = |G|^{−1} and b_{1}^{(2)} (G) = 0.

The deficiency of a cyclic group ℤ/n is 0, the obvious presentation ⟨s | s^{n}⟩ realizes the deficiency.

The deficiency of ℤ/n × ℤ/n is −1, the obvious presentation ⟨s, t | s^{n}, t^{n}, [s, t]⟩ realizes the deficiency.

Example 2.35 One may expect that the deficiency is additive under free products. This is not true by the following example due to Hog, Lustig and Metzler(1985). The group (ℤ/2 × ℤ/2) ∗ (ℤ/3 × ℤ/3) has the obvious presentation

⟨s_{0}, t_{0}, s_{1}, t_{1} | s_{0}^{2} = t_{0}^{2} = [s_{0}, t_{0}] = s_{1}^{3} = t_{1}^{3} = [s_{1}, t_{1}] = 1⟩

and one may think that its deficiency is −2. However, it turns out that its deficiency is −1. For instance, there is the following presentation, which looks on the first glance to be the presentation above with one relation missing

⟨s_{0}, t_{0}, s_{1}, t_{1} | s_{0}^{2} = 1, [s_{0}, t_{0}] = t_{0}^{2}, s_{1}^{3} = 1, [s_{1}, t_{1}] = t_{1}^{3}, t_{0}^{2} = t_{1}^{3}⟩.

The following calculation shows that, from the five relations appearing in the presentation above, the relation t_{0}^{2} = 1 follows which shows that the presentation above indeed one of of (ℤ/2 × ℤ/2) ∗ (ℤ/3 × ℤ/3) .

We start by proving inductively for k = 1, 2, … the relation s_{i}^{k}t_{i}s_{i}^{−k} = t_{i}^{r_{i}^{k}} for i = 0, 1 where r_{0} = 3 and r_{1} = 4. The beginning of the induction is obvious, the induction step follows from the calculation

s_{i}^{k+1} t_{i}s_{i}^{−(k+1)} = s_{i}s_{i}^{k}t_{i}s_{i}^{−k}s_{i}^{−1}

= s_{i}t_{i}^{r_{i}^{k}}s_{i}^{−1} = (s_{i}t_{i}s_{i}^{−1} )^{r_{i}^{k}} = (t_{i}^{r_{i}})^{r_{i}^{k}} = t_{i}^{r_{i}^{k+1}} .

This implies, for k = 2, i = 0 and k = 3 ,i = 1

t_{0} = t_{0}^{3^{2}} ;
t_{1} = t_{1}^{4^{3}} .

Since t_{0}^{2} = t_{1}^{3}, we conclude

(t_{0}^{2})^{4} = 1;
(t_{0}^{2})^{21} = 1.

As 4 and 21 are prime, we get t_{0}^{2} = 1 and the claim follows.

Theorem 2.36 Let 1 → H ⟶^{i} G ⟶^{q} K → 1 be an exact sequence of infinite groups. Suppose that G is finitely presented and one of the following conditions is satisfied.

1. b_{1}^{(2)} (H) < ∞;

2. The ordinary first Betti number of H satisfies b_{1}(H) < ∞ and b_{1}^{(2)} (K) = 0;

Then:

(i) def (G) ≤ 1;

(ii) Let M be a closed oriented 4-manifold with G as fundamental group. Then

| sign (M)| ≤ χ(M);

Remark 2.37 Next we compare our approach with the one of Cheeger and Gromov. We only consider the case of a countable simplicial complex X with free simplicial G-action. Then for any exhaustion X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … ⊂ X by G-equivariant simplicial subcomplexes for which G\X is compact, the p-th L^{2}-Betti number in the sense and notation of Cheeger-Gromov is given by

b_{p}^{(2)} (X : G) = lim_{j→∞} lim_{k→∞} dim_{N (G)}

(im (\overline{H}_{(2)}^{p} (X_{k} : G) ⟶^{i_{j,k}^{∗}} \overline{H}_{(2)}^{p} (X_{j} : G)) ) ,

where i_{j,k} : X_{j} → X_{k} is the inclusion for j ≤ k. There is an identification

\overline{H}_{(2)}^{p} (X_{j} : G) = H_{(2)}^{p} (X_{j}; N (G)).

Notice that for a G-map f : Y → Z of G-CW -complexes of finite type H_{p}^{(2)} (Y; N (G)) can be identified with H_{(2)}^{p} (Y; N (G)) and analogously for Z and that under these identifications H_{(2)}^{p} (f) = (H_{p}^{(2)} (f))^{∗}. We conclude from Additivity

dim_{N (G)} (im (H_{(2)}^{p} (f)) )

= dim_{N (G)} (im (H_{p}^{(2)} (f)) ) .

This implies

dim_{N (G)} (im (\overline{H}_{(2)}^{p} (X_{k} : G) ⟶^{i_{j,k}^{∗}} \overline{H}_{(2)}^{p} (X_{j} : G)) )
= dim_{N (G)} (im (H_{p}^{G}(X_{j}; N (G))

⟶^{(i_{j,k})_{∗}} H_{p}^{G}(X_{k}; N (G))) ) .

Hence we conclude from Theorem 2.13 that the definitions in Cheeger-Gromov(1986) and in Definition 2.15 for a countable free simplicial complex X with free simplicial G-action agree.

3. Survey on Further Results and Conjectures

Theorem 3.1 (Approximation Theorem) Let X be a free G-CW -complex of finite type. Suppose that G is residually finite, i.e. there is a nested sequence

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ …

of normal subgroups of finite index. Then G_{n}\X is a CW -complex of finite type and for any such sequence (G_{n})_{n≥1}

b_{p}^{(2)} (X; N (G)) = lim_{n→∞} \frac{b_{p}(G_{n}\X)}{[G : G_{n}]} .

Remark 3.2 We have already seen in the first lecture that there are no relations between b_{p}^{(2)} (X; N (G)) and b_{p}(G\X) for a finite G-CW -complex X except for the Euler-Poincaré formula

χ(G\X) = _{p≥0}^{∑}(−1)^{p} · b_{p}^{(2)} (X; N (G))
= _{p≥0}^{∑}(−1)^{p} · b_{p}(G\X).

One decisive difference between the ordinary Betti numbers and L^{2}-Betti numbers is that the ordinary ones are not multiplicative under finite coverings, whereas the L^{2}- Betti numbers are, i.e. for a d-sheeted covering p : X → Y we get

b_{p}^{(2)} (X̃; N (π_{1}(X)) = d · b_{p}^{(2)} (Ỹ; N (π_{1}(Y)).

With the expression lim_{n→∞} \frac{b_{p}(G\X)}{[G:G_{n}]} we try to force the Betti numbers to be multiplicative by a limit process.

Theorem 3.1 says that L^{2}-Betti numbers are asymptotic Betti numbers. It was conjectured by Gromov.

Example 3.3 Consider S^{1} and the nested sequence

π_{1}(S^{1}) = ℤ ⊃ 2^{1} · ℤ ⊃ 2^{2} · ℤ ⊃ 2^{3} · ℤ ⊃ ….

Then

b_{p}^{(2)} (S̃^{1}; N (ℤ)) = lim_{n→∞} \frac{b_{p}((2^{n} · ℤ)\S̃^{1})}{[π_{1}(S^{1}) : 2^{n} · ℤ]}
= lim_{n→∞} \frac{b_{p}(S^{1})}{2^{n}}
= 0.

Theorem 3.4 (Schick-Lück) Let (X, A) be a pair of finite free G-CW - complexes. Suppose that G\(X, A) is a Poincaré pair of dimension 4m. Suppose that G is residually finite, i.e. there is a nested sequence

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ …

of normal subgroups of finite index. Then G_{i}\(X, A) is a finite Poincaré pair of dimension 4m and for any such sequence (G_{n})_{n≥1}

sign ^{(2)} (X, A; N (G)) = lim_{n→∞} \frac{sign (G\(X, A))}{[G : G_{i}]} .

Remark 3.5 In the case that Y = ∅ and G\X = M for a closed orientable manifold M, Theorem 3.4 follows from the index theorem of Atiyah which says

sign ^{(2)} (X, A; N (G)) = \frac{sign (G_{n}\(X, A))}{[G : G_{n}]} .

In particular the signature is multiplicative under finite coverings.

If G\(X, Y) = (M, ∂M) for a compact manifold with non-empty boundary or if Y = ∅ and X is a Poincaré complex (which is not necessarily a closed orientable manifold), then the equation above is not true and the signature is not multiplicative under finite coverings.

Given a group G, let FIN (G) be the set of finite subgroups of G. Denote by

\frac{1}{|FIN (G)|} ℤ ⊂ ℚ

the additive subgroup of ℝ generated by the set of rational numbers \{\frac{1}{|H|} | H ∈ FIN (G)}.

Conjecture 3.6 (Strong Atiyah Conjecture) A group G satisfies the strong Atiyah Conjecture if for any matrix A ∈ M(m, n, ℚG) the von Neumann dimension of the kernel of the G-equivariant bounded operator r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}, x ↦ xA satisfies

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

∈ \frac{1}{|FIN (G)|} ℤ.

Remark 3.7 If G is torsionfree, then

\frac{1}{|FIN (G)|} ℤ = ℤ.

Lemma 3.8 Let G be a group. Then the following statements are equivalent:

1. For any cocompact free proper G-manifold M without boundary we have

b_{p}^{(2)} (M; N (G)) ∈ \frac{1}{|FIN (G)|} ℤ;

2. For any cocompact free proper G-CW - complex X we have

b_{p}^{(2)} (X; N (G)) ∈ \frac{1}{|FIN (G)|} ℤ;

3. The Atiyah Conjecture 3.6 is true for G.

Remark 3.9 Atiyah asked originally the following question. Let G → \overline{M} → M be a Gcovering of a closed Riemannian manifold M. Is then

b_{p}^{(2)} (M) = lim_{t→∞} ∫_{F} tr_{ℂ}(e^{−tΔ_{p}} (x, x)) dvol

a rational number?

Lemma 3.10 A group G satisfies the strong Atiyah Conjecture if and only if for any finitely presented ℚG-module M

dim_{N (G)}(N (G) ⊗_{ℚG} M) ∈ \frac{1}{|FIN (G)|} ℤ.

Proof: Given a matrix A ∈ M(m, n, ℚG), let

r_{A}: ℚG^{m} → ℚG^{n}

resp.

r_{A}^{N (G)} : N (G)^{m} → N (G)^{n}

be the associated ℚG- resp. N (G)-map given by right multiplication with A. Since the tensor product N (G) ⊗_{ℚG} − is right exact, coker (r_{A}^{N (G)} ) is N (G)-isomorphic to N (G)⊗_{ℚG}coker (r_{A}). We conclude from Additivity

dim_{N (G)} (ker (r_{A}^{N (G)} ))

= m−n+dim_{N (G)} (N (G) ⊗_{ℚG} coker (r_{A})) .

As N (G) is semihereditary, ker (r_{A}^{N (G)} ) is finitely generated projective. We have

dim_{N (G)} (ker (r_{A}^{N (G)} ))
= dim_{N (G)} (ker (r_{A}^{(2)} )) .

Therefore

dim_{N (G)} (ker (r_{A}^{(2)} )) ∈ \frac{1}{|FIN (G)|} ℤ

if and only if

dim_{N (G)} (N (G) ⊗_{ℚG} coker (r_{A})) ∈ \frac{1}{|FIN (G)|} ℤ.

Conjecture 3.11 (Kaplanski Conjecture) The Kaplanski Conjecture for a torsionfree group G and a field F says that the group ring F G has no non-trivial zero-divisors.

Lemma 3.12 The Kaplanski Conjecture holds for G and the field ℚ if the strong Atiyah Conjecture 3.6 holds for G.

Proof: Let x ∈ ℚG be a zero-divisor. Let r_{x}^{(2)} : l^{2}(G) → l^{2}(G) be given by right multiplication with x. We get

0 < dim_{N (G)} (ker (r_{x}^{(2)} )) ≤ 1

Since by assumption dim_{N (G)} (ker (r_{x}^{(2)} )) ∈

ℤ, we conclude dim_{N (G)} (ker (r_{x}^{(2)} )) = 1.

Since ker (r_{x}^{(2)} ) is closed in l^{2}(G), we conclude ker (r_{x}^{(2)} ) = l^{2}(G) and hence x = 0.

Definition 3.13 Let C be the smallest class of groups which contains all free groups and is closed under directed union and extensions with elementary amenable quotients.

Theorem 3.14 (Linnell) Let G be a group such that there is an upper bound on the orders of finite subgroups and G belongs to C. Then the strong Atiyah Conjecture 3.6 holds for G.

Definition 3.15 Let G be the smallest class of groups which contains the trivial group and is closed under the following operations:

1. Amenable quotient Let H ⊂ G be a normal subgroup. Suppose that H ∈ G and the quotient G/H is amenable. Then G ∈ G;

2. Colimits If G = colim _{i∈I} G_{i} is the colimit of the directed system {G_{i} | i ∈ I} of groups indexed by the directed set I and each G_{i} belongs to G, then G belongs to G; 3. Inverse limits

If G = lim_{i∈I} G_{i} is the limit of the inverse system {G_{i} | i ∈ I} of groups indexed by the directed set I and each G_{i} belongs to G, then G belongs to G;

4. Subgroups If H is isomorphic to a subgroup of the group G with G ∈ G, then H ∈ G;

5. Quotients with finite kernel Let 1 → K → G → Q → 1 be an exact sequence of groups. If K is finite and G belongs to G, then Q belongs to G.

Theorem 3.16 (Schick) Let {G_{i} | i ∈ I} be a directed system of groups such that each G_{i} belongs to the class G and satisfies the strong Atiyah Conjecture 3.6. Then both its colimit and its inverse limit satisfy the strong Atiyah Conjecture 3.6.

The lamplighter group L is defined by the semidirect product

L := ⊕_{n∈ℤ}ℤ/2 ⋊ ℤ

with respect to the shift automorphism of ⊕_{n∈ℤ}ℤ/2, which sends (x_{n})_{n∈ℤ} to (x_{n−1})_{n∈ℤ}. Let e_{0} ∈ ⊕_{n∈ℤ}ℤ/2 be the element whose entries are all zero except the entry at 0. Denote by t ∈ ℤ the standard generator of ℤ.Then {e_{0}t, t} is a set of generators for L. The associate Markov operator M: l^{2}(G) → l^{2}(G) is given by right multiplication with \frac{1}{4} · (e_{0}t + t + (e_{0}t)^{−1} + t^{−1}). It is related to the Laplace operator Δ_{0} : l^{2}(G) → l^{2}(G) of the Cayley graph of G by Δ_{0} = 4 · id −4 · M.

Theorem 3.17 (Grigorchuk-Zuk) The von Neumann dimension of the kernel of the Markov operator M of the lamplighter group L associated to the set of generators {e_{0}t, t} is 1/3. In particular L does not satisfy the strong Atiyah Conjecture 3.6.

Remark 3.18 No counterexample to the strong Atiyah Conjecture 3.6 is known if one replaces \frac{1}{|FIN (G)|} ℤ by ℚ or if one assumes that there is a bound on the orders of finite subgroups of G.

Conjecture 3.19 (Singer Conjecture) If M is an aspherical closed manifold, then

b_{p}^{(2)} (M̃) = 0 if 2p ≠ dim (M).

If M is a closed connected Riemannian manifold with negative sectional curvature, then

b_{p}^{(2)} (M̃) \{ \begin{matrix}= 0 & if 2p ≠ dim (M); \\ > 0 & if 2p = dim (M). \end{matrix}

Because of the Euler-Poincaré formula

χ(M) = _{p≥0}^{∑}(−1)^{p} · b_{p}^{(2)} (M̃)

the Singer Conjecture 3.19 implies the following conjecture in the cases where M is aspherical or has negative sectional curvature.

Conjecture 3.20 (Hopf Conjecture) If M is an aspherical closed manifold of even dimension, then

(−1)^{dim (M)/2} · χ(M) ≥ 0.

If M is a closed Riemannian manifold of even dimension with sectional curvature sec (M), then for ϵ = (−1)^{dim (M)/2}

ϵ · χ(M) > 0  if sec (M) < 0;
ϵ · χ(M) ≥ 0  if sec (M) ≤ 0;
χ(M) = 0  if sec (M) = 0;
χ(M) ≥ 0  if sec (M) ≥ 0;
χ(M) > 0  if sec (M) > 0.

Theorem 3.21 (Jost-Xin) Let M be a closed connected Riemannian manifold of dimension dim (M) ≥ 3. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec (M) ≤ 0 and the Ricci curvature is bounded from above by −b^{2}. If the non-negative integer p satisfies 2p ≠ dim (M) and 2pa ≤ b, then

b_{p}^{(2)} (M̃) = 0.

Theorem 3.22 (Ballmann-Brüning) Let M be a closed connected Riemannian manifold. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec (M) ≤ −b^{2}. If the non-negative integer p satisfies

2p < dim (M) − 1;
p · a < (dim (M) − 1 − p) · b,

then

b_{p}^{(2)} (M̃) = 0.

Remark 3.23 Direct computations show that the Singer Conjecture 3.19 holds for a closed Riemannian manifold M if dim (M) ≤ 3 (assuming Thurston’s Geometrization) or if M is a locally symmetric space or if M carries an S^{1}-action.

Definition 3.24 A Kähler hyperbolic manifold is a closed connected Kähler manifold M whose fundamental form ω is d̃(bounded), i.e. its lift ω̃ ∈ Ω^{p}(M̃) to the universal covering can be written as d(η) holds for some bounded (p − 1)-form η ∈ Ω^{p−1}(M̃).

Example 3.25 The closed manifold M is Kähler hyperbolic if it satisfies one of the following conditions:

1. M is a closed Kähler manifold which is homotopy equivalent to a Riemannian manifold with negative sectional curvature;

2. M is a closed Kähler manifold such that π_{1}(M) is word-hyperbolic and π_{2}(M) is trivial;

3. M̃ is a symmetric Hermitian space of non-compact type;

4. M is a complex submanifold of a Kähler hyperbolic manifold;

5. M is a product of two Kähler hyperbolic manifolds.

Theorem 3.26 (Gromov) Let M be a closed Kähler hyperbolic manifold of complex dimension m and real dimension n = 2m. Then

b_{p}^{(2)} (M̃) = 0 if p ≠ m;
b_{m}^{(2)} (M̃) > 0;
(−1)^{m} · χ(M) > 0;

Theorem 3.27 Let M be a closed Kähler hyperbolic manifold of complex dimension m and real dimension n = 2m.

1. The canonical line bundle L = Λ^{m}T^{∗}M is quasiample, i.e. its Kodaira dimension is m;

2. M satisfies all of the following four assertions (which are equivalent for closed Kähler manifolds):

(a) M is Moishezon, i.e. the transcendental degree of the field M(X) of meromorphic functions is equal to m;

(b) M is Hodge, i.e. the Kähler form represents a class in H^{2}(M; ℂ) which lies in the image of H^{2}(M; ℤ) → H^{2}(M; ℂ);

(c) M can be holomorphically embedded into ℂℙ^{N} for some N;

(d) M is a projective algebraic variety;

3. The fundamental group is an infinite non-amenable group of deficiency ≤ 1. It cannot be a non-trivial free product.

Let X be a topological space and let C_{∗}^{sing} (X; ℝ) be its singular chain complex with real coefficients. Let S_{p}(X) be the set of all singular p-simplices. Then C_{p}(X; ℝ) is the real vector space with S_{p}(X) as basis. Define the L^{1}-norm of an element x ∈ C_{p}(X), which is given by the (finite) sum ∑_{σ∈S_{p}(X)} λ_{σ}· σ, by

||x||_{1} := _{σ}^{∑} |λ_{σ}|.

Define the L^{1}-seminorm of an element y in the p-th singular homology H_{p}^{sing} (X; ℝ) := H_{p}(C_{∗}^{sing} (X; ℝ)) by

||y||_{1} := inf {||x||_{1} | x ∈ C_{p}^{sing} (X; ℝ),
∂_{p}(x) = 0, y = [x]}.

Notice that ||y||_{1} defines only a semi-norm on H_{p}^{sing} (X; ℝ), it is possible that ||y||_{1} = 0 but y ≠ 0. The next definition is due to Gromov and Thurston.

Definition 3.28 Let M be a closed connected orientable manifold of dimension n. Define its simplicial volume to be the non-negative real number

||M|| := ||j([M])||_{1} ∈ [0, ∞)

for any choice of fundamental class [M] ∈ H_{n}^{sing} (M, ℤ) and j : H_{n}^{sing} (M; ℤ) → H_{n}^{sing} (M; ℝ) the change of coefficients map associated to the inclusion ℤ → ℝ.

Theorem 3.29 (Simplical volume of hyperbolic manifolds) Let M be a closed hyperbolic orientable manifold of dimension n. Let v_{n} be the volume of the regular ideal simplex in \overline{ℍ^{n}}. Then

||M|| = \frac{vol(M)}{v_{n}} .

Example 3.30 We have ||S^{2}|| = ||T^{2}|| =

0. Let F_{g} be the closed connected orientable surface of genus g ≥ 1. Then

||F_{g}|| = 2 · |χ(F_{g})| = 4g − 4.

Definition 3.31 Let M be a smooth manifold. Define its minimal volume minvol (M) to be the infimum over all volumes vol(M, g), where g runs though all complete Riemannian metrics on M, for which the sectional curvature satisfies | sec (M, g)| ≤ 1.

Example 3.32 Obviously any closed flat Riemannian manifold has vanshing minimal volume. Hence we get

minvol (T^{n}) = ||T^{n}|| = 0.

Let F_{g} be the closed orientable surface of genus g, then

minvol (F_{g}) = 2π · |χ(F_{g})| = 2π · |2 − 2g| = π · ||F_{g}||

by the following argument. The Gauss-Bonnet formula implies for any Riemannian metric on F_{g} whose sectional curvature satisfies | sec | ≤ 1

vol(F_{g}) ≥ ∫_{F_{g}} | sec |dvol ≥ |∫_{F_{g}} sec dvol|

= |2π · χ(F_{g})|.

If g ≠ 1 and we take the Riemannian metric whose sectional curvature is constant 1 or −1, then the Gauss-Bonnet Theorem shows

|2π · χ(F_{g})| = |∫_{F_{g}} sec dvol| = vol(F_{g}).

Now the claim follows.

Notice that ||S^{2}|| = 0 and minvol (S^{2}) ≠ 0.

We have

minvol (ℝ^{2}) = 2π(1 + √2);
minvol (ℝ^{n}) = 0 for n ≥ 3.

Theorem 3.33 (Gromov-Thurston) Let M be a closed connected orientable Riemannian manifold of dimension n. Then

||M|| ≤ (n − 1)^{n} · n! · minvol (M).

Conjecture 3.34 (Simplical volume and L^{2}-invariants) Let M be an aspherical closed oriented manifold of dimension ≥ 1. Suppose that its simplicial volume ||M|| vanishes. Then

b_{p}^{(2)} (M̃) = 0 for p ≥ 0.

Example 3.35 Let M be an aspherical closed orientable manifold. Then Conjecture 3.34 is true in following cases:

1. Suppose that M carries an S^{1}-action. Then minvol (M) = 0, ||M|| = 0 and b_{p}^{(2)} (M̃) = 0 for all p ≥ 0;

2. Let H ⊂ π_{1}(M) be a normal infinite subgroup. If H is amenable, then ||M|| = 0. If H is elementary amenable subgroup, then b_{p}^{(2)} (M̃) = 0 for all p ≥ 0;

3. If there is a selfmap f : M → M of degree deg (f) different from −1, 0, and 1, then ||M|| = 0.

If any normal subgroup of finite index of π_{1}(M) is Hopfian and there is a selfmap f : M → M of degree deg (f) different from −1, 0, and 1, then b_{p}^{(2)} (M̃) = 0 for all p ≥ 0;

4. Supose that M has dimension 3 (and satisfies Thurston’s Geometrization Conjecture). Then ||M|| = 0 implies that b_{p}^{(2)} (M̃) = 0 for all p ≥ 0;

5. If the minimal volume minvol (M) of M is zero, then ||M|| = 0 and b_{p}^{(2)} (M̃) = 0 for all p ≥ 0.