“L^{2}-Betti numbers of mapping tori and groups” by Wolfgang Lück

Abstract: We prove the following two conjectures of Gromov. Firstly, all L^{2}-Betti numbers of a manifold fibered over S^{1} are trivial. Secondly, the first L^{2}-Betti number of a finitely presented group Γ vanishes provided that Γ is an extension {1} ⟶ Δ ⟶ Γ ⟶ π ⟶ {1} of finitely presented groups such that Δ is infinite and π contains ℤ as a subgroup. We conclude for such a group Γ that its deficiency is less than or equal to one and that any closed 4-manifold with Γ as fundamental group satisfies χ(M) ≥ |σ(M)|.

0. Introduction

In his preprint [12, page 152 and page 156] Gromov states the following two conjectures:

Let a compact aspherical manifold M be fibered over the circle S^{1}. Then all L^{2}-Betti numbers b_{p}(M) are trivial.

Let {1} ⟶ Δ ⟶ Γ ⟶ π ⟶ {1} be an extension of infinite groups which are fundamental groups of finite aspherical CW -complexes. Then the first L^{2}-Betti number b_{1}(Γ) is trivial.

We will give affirmative answers to these conjectures. The first conjecture follows from Theorem 2.1 which states that all L^{2}-Betti numbers b_{p}(T_{f}) of a mapping torus T_{f} of an endomorphism f of a finite CW -complex F vanish. We prove in Theorem 4.1 for an extension {1} ⟶ Δ ⟶ Γ ⟶ π ⟶ {1} of finitely presented groups that the first L^{2}-Betti number b_{1}(Γ) vanishes provided that Δ is infinite and π contains ℤ as a subgroup. This implies the second conjecture above. Let Γ be an infinite finitely presented group with trivial first L^{2}-Betti number b_{1}(Γ). As applications we show in Theorem 5.1 that a closed 4-manifold with Γ as fundamental group satisfies χ(M) ≥ |σ(M)| for χ(M) the Euler characteristic and σ(M) the signature. This generalizes a result of Johnson and Kotschick [16]. We prove in Theorem 6.1 that the deficiency of Γ satisfies def(Γ) ≤ 1.

L^{2}-Betti numbers were introduced by Atiyah [1]. In Section 1 we recall their definitions and basic properties from the topological point of view. They also have an analytic meaning, namely, the p-th L^{2}-Betti number of a closed Riemannian manifold measures the size of the space of harmonic L^{2}-integrable smooth p-forms of the universal covering [7]. For general information and applications of L^{2}-Betti numbers, and in particular of conditions that determine when they vanish, the reader may refer for example to [1], [4], [5], [6],[7],[8], [11],[12], [18], [19], [21] and [22].

The author would like to thank John Lott for fruitful discussions and for pointing out Gromov’s preprint [12] to him.

The paper is organized as follows:

1. Preliminaries concerning L^{2}-Betti numbers
2. The vanishing of the L^{2}-Betti numbers of a mapping torus
3. The first L^{2}-Betti number of a total space of a fibration
4. Groups with vanishing first L^{2}-Betti number
5. 4-manifolds satisfying χ(M) ≥ |σ(M)|
6. Deficiency of groups

1. Preliminaries concerning L^{2}-Betti numbers

In this section we give the basic definitions and properties of L^{2}-Betti numbers.

Let Γ be a countable group and l^{2}(Γ) be the Hilbert space of square integrable formal sums ∑_{γ∈Γ} λ_{γ}γ with coefficients λ_{γ} ∈ ℂ. The von Neumann algebra N (Γ) is the algebra B(l^{2}(Γ), l^{2}(Γ))^{Γ} of bounded operators from l^{2}(Γ) ⟶ l^{2}(Γ) which commute with the left Γ-action on l^{2}(Γ). The von Neumann trace tr(f) of an element f ∈ N (Γ) is the complex number ⟨f(e), e⟩ where e ∈ Γ is the unit element. This extends to square matrices over N (Γ) by taking the sum of the traces of the diagonal entries. A Hilbert N (Γ)-module is a Hilbert space M together with a left Γ-action by unitary operators such that there exists an isometric Γ-equivariant embedding into H ⊗ l^{2}(Γ) for a separable Hilbert space H (which is not part of the structure). We call M finitely generated if H can be chosen to be ℂ^{n} for some positive integer n. The von Neumann dimension dim(M) of a finitely generated Hilbert N (Γ)-module M is the non-negative real number tr(pr) for any projection pr in M(n, n, N (Γ)) = B(⊕_{i=1}^{n}l^{2}(Γ), ⊕_{i=1}^{n}l^{2}(Γ))^{Γ} whose image is isometrically Γ-isomorphic to M. A weakly exact sequence 0 ⟶ M ⟶^{i} N ⟶^{p} P ⟶ 0 of Hilbert N (Γ)-modules is a sequence of bounded operators such that i is injective, the closure of the image of i is the kernel of p and the closure of the image of p is P. Given such a sequence of finitely generated Hilbert N (Γ)-modules, the relation dim(M) − dim(N) + dim(P) = 0 holds. We have dim(M) = 0 precisely if M = {0}. A (finitely generated) Hilbert N (Γ)-chain complex C is a chain complex of (finitely generated) Hilbert N (Γ)-modules with bounded Γ-equivariant operators as differential. Its L^{2}-homology is defined to be H_{p}(C) = ker(c_{p})/\overline{im(c_{p+1})}. Notice that one divides by the closure of the image and not just by the image so that the L^{2}-homology is not ordinary homology. Now one can define the p-th L^{2}-Betti number as b_{p}(C) = dim(H_{p}(C)) provided that H_{p}(C) is finitely generated.

Let X be a CW -complex with finite d-skeleton and fundamental group π. Consider a group homomorphism ϕ : π ⟶ Γ. Let X̃ be the universal covering. For dimensions less than or equal to d, define the finitely generated Hilbert Γ-chain complex C(X; ϕ^{∗}l^{2}(Γ)) by l^{2}(Γ) ⊗_{ℤπ} C(X̃). Here the right π-action on l^{2}(Γ) is induced by ϕ : π ⟶ Γ and the Hilbert N (Γ)-structure comes from the identification of l^{2}(Γ) ⊗_{ℤπ} C_{p}(X̃) with ⊕_{i=1}^{n}l^{2}(Γ) given by a cellular ℤπ-basis. The cellular basis is not unique and a different choice of cellular basis will give a different identification. Since the two different identifications differ only by a unitary Γ-equivariant operator, the Hilbert N (Γ)-module structure is independent of the choice of cellular basis. The differentials in dimension less than or equal to d are bounded Γ-equivariant operators. These considerations prompt the following

Definition 1.1 Let X be a CW -complex with finite d-skeleton and ϕ : π_{1}(X) ⟶ Γ be a homomorphism. Define for p < d the p-th L^{2}-Betti number of X with coefficients in ϕ^{∗}l^{2}(Γ) by

b_{p}(X; ϕ^{∗}l^{2}(Γ)) = b_{p}(C(X; ϕ^{∗}l^{2}(Γ))).

In case Γ = π_{1}(X) and ϕ = id, we abbreviate this to read

b_{p}(X) = b_{p}(X; id^{∗}l^{2}(π_{1}(X))).

If Bπ has finite d-skeleton we define for p < d

b_{p}(π) = b_{p}(Bπ).

The next lemma shows in particular that the definition of b_{p}(π) for p < d is independent of the choice of Bπ. Notice that a group π is finitely presented if and only if Bπ has finite 2-skeleton. Most of the claims of the next lemma are already in the literature provided X and Y are finite and Γ = π and ϕ = id. We require this more general setup for Theorem 2.1 which is needed in its present form to prove Theorem 3.1 and Theorem 4.1.

Lemma 1.2 Let X and Y be CW -complexes having finite d-skeletons. Let ϕ : π_{1}(Y) ⟶ Γ be a group homomorphism.

1. Suppose f : X ⟶ Y is s-connected for s ≥ 2. Then for p < min{s, d}

b_{p}(X; (ϕ ∘ f_{∗})^{∗}l^{2}(Γ)) = b_{p}(Y; ϕ^{∗}l^{2}(Γ)).

If s < d, then

b_{s}(X; (ϕ ∘ f_{∗})^{∗}l^{2}(Γ)) ≥ b_{s}(Y; ϕ^{∗}l^{2}(Γ)).

2. If Y has finite 2-skeleton, then for p = 0, 1

b_{p}(π_{1}(Y)) = b_{p}(Y).

3. If i : Γ ⟶ Γ′ is injective, then for p < d

b_{p}(Y; (i ∘ ϕ)^{∗}l^{2}(Γ′)) = b_{p}(Y; ϕ^{∗}l^{2}(Γ)).

4. Let p : \overline{Y} ⟶ Y be a n-sheeted finite covering. Denote by Γ_{n} ⊂ Γ the image of ϕ ∘ p_{∗} and by ϕ_{n} : π_{1}(\overline{Y}) ⟶ Γ_{n} the induced map. If Γ_{n} has index n in Γ, then for p < d

b_{p}(\overline{Y}; ϕ_{n}^{∗}l^{2}(Γ_{n})) = n · b_{p}(Y; ϕ^{∗}l^{2}(Γ)).

In particular for p < d

b_{p}(\overline{Y}) = n · b_{p}(Y).

5. Assume d ≥ 1. If the image of ϕ : π_{1}(Y) ⟶ Γ is finite of cardinality n, then

b_{0}(Y; ϕ^{∗}l^{2}(Γ)) = \frac{1}{n}.

Otherwise

b_{0}(Y; ϕ^{∗}l^{2}(Γ)) = 0.

and in particular

b_{0}(π) = \frac{1}{|π|}.

6. If Y is a finite CW -complex, then

χ(Y) = ∑_{p≥0}(−1)^{p} · b_{p}(Y; ϕ^{∗}l^{2}(Γ)).

\underline{\underline{Proof}} : 1.) In the sequel we write π = π_{1}(Y). Let f̃ : X̃ ⟶ Ỹ be a lift of f to the universal coverings. The induced ℤπ-chain map ℤπ ⊗_{ℤπ_{1}(X)} C(X̃) ⟶ C(Ỹ) is s-connected. Hence it suffices to show the following chain complex analogue (which we will use later): Let f : C ⟶ D be a s-connected ℤπ-chain map of free ℤπ-chain complexes such that the d-dimensional chain complexes obtained by truncating C|_{d} and D|_{d} are finitely generated. Then we have b_{p}(l^{2}(Γ)⊗_{ℤπ}C) = b_{p}(l^{2}(Γ)⊗_{ℤπ}D) if p < min{s, d} and b_{s}(C) ≥ b_{s}(D) if s < d.

The strategy of the proof is precisely the same as in [19, Lemma 2.4, Theorem 2.5 and Lemma 4.3] which we describe briefly. One extends f to a ℤπ-chain homotopy equivalence f′ : C′ ⟶ D such that C|_{s} = C′|_{s} and C′_{s+1} is finitely generated free if s < d. Obviously b_{p}(l^{2}(Γ) ⊗_{ℤπ} C) = b_{p}(l^{2}(Γ) ⊗_{ℤπ} C′) for p < min{s, d} and b_{s}(l^{2}(Γ) ⊗_{ℤπ} C) ≥ b_{s}(l^{2}(Γ) ⊗_{ℤπ} C′) for s < d. Hence it suffices to show b_{p}(l^{2}(Γ) ⊗_{ℤπ} C) = b_{p}(l^{2}(Γ) ⊗_{ℤπ} D) for p < d provided f : C ⟶ D is a homotopy equivalence. We may assume that f is an inclusion with a free contractible quotient D/C, otherwise substitute D by the mapping cylinder. The exact sequence 0 ⟶ C ⟶ D ⟶ D/C ⟶ 0 splits yielding an isomorphism between D and C ⊕ D/C. This reduces the claim to the assertion that b_{p}(l^{2}(Γ) ⊗_{ℤπ} C) = 0 for p < d provided that C is contractible. This follows from the fact that C is a direct sum ⊕_{p≥1}E(p) of free ℤπ-chain complexes E(p) such that E(p) is concentrated in dimensions p and p + 1 and the non-trivial differential is a ℤπ-isomorphism.

2.) follows from 1.) applied to the classifying map Y ⟶ Bπ.

3.) follows from the elementary proof of [19, Lemma 4.6].

4.) In the sequel res denotes restriction for the subgroup π_{1}(\overline{Y}) ⊂ π_{1}(Y) respectively Γ_{n} ⊂ Γ. Define a bounded Γ_{n}-equivariant operator for p ≤ d

I_{p} : l^{2}(Γ_{n}) ⊗_{ℤπ_{1}(\overline{Y})} res(C_{p}(Ỹ)) ⟶ res (l^{2}(Γ) ⊗_{ℤπ_{1}(Y)} C_{p}(Ỹ)) u ⊗ v ↦ u ⊗ v.

This map is a well-defined Γ_{n}-equivariant isometry since C_{p}(Ỹ) is finitely generated free and π_{1}(\overline{Y}) ⊂ π_{1}(Y) and Γ_{n} ⊂ Γ have the same finite index, namely n. As the collection I_{p} is compatible with the differentials, C(\overline{Y}; ϕ_{n}^{∗}l^{2}(Γ_{n})) and res (C(Y; ϕ^{∗}l^{2}(Γ))) have the same L^{2}-Betti numbers over N (Γ_{n}) for p < d. Given a finitely generated Hilbert N (Γ)-module M, we have dim_{N (Γ_{n})}(res(M)) = n · dim_{N (Γ)}(M) since tr_{N (Γ_{n})}(res(k)) = n · tr_{N (Γ)}(k) holds for any bounded Γ-equivariant endomorphism k of ⊕_{i=1}^{l}l^{2}(Γ). This establishes assertion 4.).

5.) We can assume by assertion 3.) that ϕ is surjective. Choose a set of generators s_{1}, s_{2}, … s_{g} for π. Then ϕ(s_{1}), ϕ(s_{2}), … ϕ(s_{g}) is a set of generators for Γ. Moreover, C(Y; ϕ^{∗}l^{2}(Γ)) is given in dimension 1 and 0 by

⊕_{i=1}^{g}l^{2}(Γ) \underrightarrow{⊕_{i=1}^{g}r(ϕ(s_{i}) − 1)} l^{2}(Γ)

where r(ϕ(s_{i}) − 1) is right multiplication with ϕ(s_{i}) − 1. Hence we can assume π = Γ and ϕ = id. It remains to show b_{0}(π) = 0 if π is infinite and b_{0}(π) = 1/|π| if π is finite. This follows from the observation that l^{2}(π)^{π} is zero for infinite π and ℂ with the trivial π-action for finite π.

6.) follows from the additivity of the von Neumann dimension under weakly exact sequences.

Finally we mention the following combinatorial way of computing b_{1}(π) for a finitely presented group π proved in [21]. Let ⟨s_{1}, …, s_{g}|R_{1}, …, R_{r}⟩ be any finite presentation of π. Let A be the (r, g−1)-matrix over ℤπ given by the the Fox derivatives A_{i,j} = \frac{∂R_{i}}{∂s_{j}} for 1 ≤ i ≤ r and 1 ≤ j ≤ g −1.(The index j does not take the value g.) Define for u = ∑_{w∈π} λ_{w} · w ∈ ℝπ its ℝΓ-trace tr_{ℝπ}(u) = λ_{e} ∈ ℝ if e is the unit element in π. This extends to a square (n, n)-matrix B with entries in ℝπ by putting tr_{ℝπ}(B) = ∑_{k=1}^{n} tr_{ℝπ}(b_{k,k}). Let K be any real number satisfying K ≥ ∥A∥ where ∥A∥ is the operator norm of the bounded operator ⊕_{i=1}^{r}l^{2}(π) ⟶ ⊕_{j=1}^{g−1}l^{2}(π) induced by A. A possible choice is the product of \sqrt{g − 1} and the maximum of the word length of the relations R_{i} in terms of the s_{j}. Denote by A^{∗} the matrix obtained from A by transposing and applying to each entry the involution on ℝπ sending ∑_{w∈π} λ_{w} · w to ∑_{w∈π} λ_{w} · w^{−1}. Denote by (I_{g−1} − K^{−2} · A^{∗}A)^{n} the n-fold product of the square (g − 1, g − 1)-matrix (I_{g−1} − K^{−2} · A^{∗}A) for I_{g−1} the unit matrix. Then the sequence of non-negative real numbers tr_{ℝπ} (1 − K^{−2} · A^{∗}A)^{n} is monotone decreasing and converges for n → ∞ to b_{1}(π). In this context we mention Conjectures 9.1 and 9.2 in [19] which imply for torsion-free π that b_{1}(π) is an integer.

2. The vanishing of the L^{2}-Betti numbers of a mapping torus

Given a self map f : F ⟶ F, its mapping cylinder M_{f} is obtained by gluing the bottom of the cylinder F × [0, 1] to F by the identification (x, 0) = f(x). Its mapping torus T_{f} is obtained from the mapping cylinder by identifying the top and the bottom by the identity. If f is a homotopy equivalence T_{f} is homotopy equivalent to the total space of a fibration over S^{1} with fiber F. Conversely, the total space of such a fibration is homotopy equivalent to the mapping torus of the self homotopy equivalence of F given by the fiber transport with a generator of π_{1}(S^{1}). The homotopy type of T_{f} depends only on the homotopy class of f.

Let L denote the colimit of the following system of groups indexed by the integers

… ⟶^{π_{1}(f)} π_{1}(F) ⟶^{π_{1}(f)} π_{1}(F) ⟶^{π_{1}(f)} …

Denote by i : π_{1}(F) ⟶ L the map at the group indexed by zero. The map i is bijective if and only if π_{1}(f) is an isomorphism. Let ℤ operate on L by shifting the sequence. Then π_{1}(T_{f}) is the semidirect product of L and ℤ with respect to the operation above. Consider any factorization of the canonical epimorphism π_{1}(T_{f}) ⟶ ℤ into a composition of epimorphisms π_{1}(T_{f}) ⟶^{ϕ} Γ ⟶^{ψ} ℤ. Denote by \overline{F} the covering of F associated to the homomorphism ϕ ∘ i : π_{1}(F) ⟶ ker(ψ) and by \overline{T_{f}} be the covering of T_{f} associated to the epimorphism ϕ. Let \overline{f} : \overline{F} ⟶ \overline{F} be a lift of f. Then \overline{T_{f}} is the mapping telescope of \overline{F} infinite to both sides, i.e., the identification space

\overline{T_{f}} = ∐_{n∈ℤ} \overline{F} × [n, n + 1]/ ∼

where the identification ∼ is given by (x, n + 1) ∼ (\overline{f}(x), n). The group of deck transformations Γ is a semidirect product of ker(ψ) and ℤ and acts in the obvious way. One easily checks that the cellular ℤΓ-chain complex of \overline{T_{f}} is the mapping cone of the following ℤΓ-chain map

ℤΓ ⊗_{ℤ[ker(ψ)]} C(\overline{F}) ⟶ ℤΓ ⊗_{ℤ[ker(ψ)]} C(\overline{F}) γ ⊗ u ↦ γ ⊗ u − γt ⊗ C(\overline{f})(u)

where t is a lift of the generator of ℤ to Γ.

The next theorem implies a conjecture by Gromov in [12, page 152]. The case of a manifold fibered over S^{1} has already been dealt with in [19, Theorem 4.10].

Theorem 2.1 Let f : F ⟶ F be a self map of a connected CW -complex F with finite d-skeleton for d ≥ 2. Let π_{1}(T_{f}) ⟶^{ϕ} Γ ⟶^{ψ} ℤ be a factorization of the canonical map π_{1}(T_{f}) ⟶ ℤ into epimorphisms. Then the mapping torus T_{f} has a CW -structure with finite d-skeleton and for p < d

b_{p}(T_{f}; ϕ^{∗}l^{2}(Γ)) = 0.

\underline{\underline{Proof}} : Let T_{f}^{n} be obtained from the n-fold mapping telescope of f by identifying the bottom and top by the identity. In this notation T_{f}^{1} is just T_{f}. There is an obvious n-fold covering p : T_{f}^{n} ⟶ T_{f}. Let Γ_{n} be the image of ϕ ∘ p_{∗} and denote by ϕ_{n} : π_{1}(T_{f}^{n}) ⟶ Γ_{n} the induced map. Then Γ_{n} has index n in Γ. Lemma 1.2.4 implies for all integers p < d

b_{p}(T_{f}; ϕ^{∗}l^{2}(Γ)) = \frac{b_{p}(T_{f}^{n}; ϕ_{n}^{∗}l^{2}(Γ_{n}))}{n}.

There is a homotopy equivalence g : T_{f^{n}} ⟶ T_{f}^{n}. Hence we get

b_{p}(T_{f^{n}}; (ϕ_{n} ∘ g_{∗})^{∗}l^{2}(Γ_{n})) = b_{p}(T_{f}^{n}; ϕ_{n}^{∗}l^{2}(Γ_{n})).

The mapping torus T_{f^{n}} has a CW -structure such that its d-skeleton is finite and the number of p-cells is c_{p−1} + c_{p} where c_{p} is the number of p-cells in F. Thus

b_{p}(T_{f^{n}}; (ϕ_{n} ∘ g_{∗})^{∗}l^{2}(Γ_{n})) ≤ c_{p−1} + c_{p}.

We conclude

0 ≤ b_{p}(T_{f}; ϕ^{∗}l^{2}(Γ)) ≤ \frac{c_{p−1} + c_{p}}{n} .

Taking the limit n → ∞ proves the claim.

3. The first L^{2}-Betti number of a total space of a fibration

In this section we prove

Theorem 3.1 Let F ⟶ E ⟶^{p} B be a fibration of connected CW -complexes such that F and B have finite 2-skeletons. Then E has finite 2-skeleton up to homotopy. If the image of π_{1}(F) ⟶ π_{1}(E) is infinite and π_{1}(B) contains ℤ as subgroup, then the first L^{2}-Betti number of E

b_{1}(E) = 0.

The proof of Theorem 3.1 requires some preparations. Let F ⟶ E ⟶^{p} B be a fibration of connected CW -complexes. Denote π = π_{1}(B), Γ = π_{1}(E) and let Δ be the image of π_{1}(F) ⟶ π_{1}(E). We obtain the group extension {1} ⟶ Δ ⟶ Γ ⟶^{p_{∗}} π ⟶ {1}. The pointed fiber transport is a homomorphism of monoids into the monoid of pointed homotopy classes of pointed self maps of the fiber

σ : Γ ⟶ [F, F]^{+}.

We recall its definition. Let w : I ⟶ E be a loop at the base point e ∈ E. Put b = p(e) and F = p^{−1}(b) . Choose a solution h of the following lifting problem

\begin{matrix}F_{b} × \{0\} ∪ \{e\} × I & \underrightarrow{i ∪ w} & E \\ _{j ↓} & [figure: h] & _{↓ p} \\ F × I & \underrightarrow{p ∘ w ∘ pr_{I}} & B \end{matrix}

where i and j are the obvious inclusions. Then σ assigns to the class w ∈ Γ the pointed homotopy class of the map (F, e) ⟶ (F, e) sending x to h(x, 1).

Let \overline{F} ⟶ F be the connected covering of F with Δ as group of deck transformations. Let c(γ) : Δ ⟶ Δ send δ to γδγ^{−1} for γ ∈ Γ. A representative of σ(γ) lifts to a c(γ)- equivariant (pointed) self map of \overline{F}. Its c(γ)-equivariant (free) homotopy class denoted by \overline{σ}(γ) depends only on γ. Given w ∈ π, denote by \overline{w} ∈ Γ some element satisfying p_{∗}(\overline{w}) = w. Define a ℤΓ-chain map U(w) : ℤΓ ⊗_{Δ} C(\overline{F}) ⟶ ℤΓ ⊗_{Δ} C(\overline{F}) by sending γ ⊗ x to γ\overline{w} ⊗ C(\overline{σ}(\overline{w}^{(−1)}))(x). The ℤΓ-chain homotopy class of U(w) is independent of the choice of \overline{w} and of the representative of \overline{σ}(\overline{w}^{(−1)}) and hence depends only on w. This follows from the fact that a representative for \overline{σ}(δ) for δ ∈ Δ is given by left multiplication with δ on \overline{F}. For u = ∑_{γ∈π} λ_{γ}γ ∈ ℤπ define U(u) by ∑_{γ∈π} λ_{γ}U(γ). Then the ℤΓ-chain homotopy classes of U(v) ∘ U(u) and U(uv) agree and U(1) = id. Hence we have constructed an algebra homomorphism

U : ℤπ ⟶ [ℤΓ ⊗_{ℤΔ} C(\overline{F}), ℤΓ ⊗_{ℤΔ} C(\overline{F})]_{ℤΓ}^{op}

into the opposite (reverse multiplication) of the algebra of ℤΓ-chain homotopy classes of ℤΓ-chain self maps of ℤΓ⊗_{ℤΔ}C(\overline{F}). This is called chain homotopy presentation of the given fibration. For more information and (elementary) proofs of some of the claims above we refer to [20, section 1 and 6].

As ℤΓ is free over ℤΔ the natural ℤΓ-map S_{1} : ℤΓ⊗_{ℤΔ}H_{∗}(C(\overline{F})) ⟶ H_{∗}(ℤΓ⊗_{ℤΔ}C(\overline{F})) is bijective. The ℤΓ-isomorphism S_{2} : ℤΓ ⊗_{Δ} H_{∗}(C(\overline{F})) ⟶ ℤπ ⊗_{ℤ} H_{∗}(C(\overline{F})) sends γ ⊗ x to p_{∗}(γ) ⊗ H_{∗}(C(\overline{σ}(γ)))(x) where γ ∈ Γ acts on ℤπ by left multiplication with p_{∗}(γ) ∈ π, on H_{∗}(C(\overline{F})) by H_{∗}(\overline{σ}(γ)) and diagonally on ℤπ ⊗_{ℤ} H_{∗}(C(\overline{F})). Define the ℤΓ-isomorphism

S : H_{∗}(ℤΓ ⊗_{ℤΔ} C(\overline{F})) ⟶ ℤπ ⊗_{ℤ} H_{∗}(C(\overline{F}))

by S = S_{2} ∘ S_{1}^{−1}.

Lemma 3.2 Given u ∈ ℤπ, let r(u) : ℤπ ⟶ ℤπ be right multiplication with u. The following diagram commutes

H_{∗}(ℤΓ ⊗_{ℤΔ} C(\overline{F})) \underrightarrow{H_{∗}(U(u))} H_{∗}(ℤΓ ⊗_{ℤΔ} C(\overline{F}))
S ↓  _{↓ S}

ℤπ ⊗_{ℤ} H_{∗}(C(\overline{F}))  \underrightarrow{r(u) ⊗ id} ℤπ ⊗_{ℤ} H_{∗}(C(\overline{F}))

and has isomorphisms as vertical arrows.

Proof of Theorem 3.1: Choose a set of generators s_{1}, …, s_{g} of π such that the cyclic subgroup ⟨s_{1}⟩ generated by s_{1} is infinite. The following sequence is exact

⊕_{i=1}^{g}ℤπ \underrightarrow{⊕_{i=1}^{g}r(1 − s_{i})} ℤπ ⟶^{ϵ} ℤ ⟶ \{0\}

where ℤ here and in the sequel carries always the trivial Δ-,Γ-, respectively, π-action and the augmentation ϵ sends ∑_{w∈π} λ_{w}w to ∑_{w∈π} λ_{w}. As H_{0}(C(\overline{F})) is ℤΓ-isomorphic to ℤ, Lemma 3.2 proves the exactness of the sequence of ℤΓ-modules

⊕_{i=1}^{g}H_{0}(ℤΓ ⊗_{ℤΔ} C(\overline{F})) \underrightarrow{⊕_{i=1}^{g}H_{0}(U(1 − s_{i}))} H_{0}(ℤΓ ⊗_{ℤΔ} C(\overline{F})) ⟶ ℤ ⟶ \{0\}.

Let cone(f) denote the mapping cone of a chain map f. We derive from the long homology sequence that H_{0}(cone(⊕_{i=1}^{g}U(1 − s_{i}))) is ℤΓ-isomorphic to ℤ. Let Ẽ be the universal covering of E. For a chain complex C let C|_{2} be the 2-dimensional chain complex obtained by truncating C. Notice that C(Ẽ)|_{2} is a 2-dimensional ℤΓ-chain complex such that H_{1}(C(Ẽ)|_{2}) is trivial and H_{0}(C(Ẽ)|_{2}) is ℤ and that cone(⊕_{i=1}^{g}U(1−s_{i})) is a free ℤΓ-chain complex. Hence there is a ℤΓ-chain map f : cone(⊕_{i=1}^{g}U(1 − s_{i}))|_{2} ⟶ C(Ẽ)|_{2} inducing the identity on H_{0}. Since f is 1-connected, we conclude from the proof of Lemma 1.2.1

b_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(⊕_{i=1}^{g}U(1 − s_{i}))) ≥ b_{1}(l^{2}(Γ) ⊗_{ℤΓ} C(Ẽ)) = b_{1}(E).

There is an obvious exact sequence of ℤΓ-chain complexes

0 ⟶ cone(U(1 − s_{1})) ⟶ cone(⊕_{i=1}^{g}U(1 − s_{i})) ⟶ ⊕_{i=2}^{g}Σ (ℤΓ ⊗_{ℤΔ} C(\overline{F})) ⟶ 0

where Σ denotes the suspension. It induces the weakly exact sequence in L^{2}-homology [4, Theorem 2.1 on page 10]

H_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(U(1 − s_{1}))) ⟶ H_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(⊕_{i=1}^{g}U(1 − s_{i})))
⟶ H_{1} (l^{2}(Γ) ⊗_{ℤΓ} ⊕_{i=2}^{g}Σ (ℤΓ ⊗_{ℤΔ} C(\overline{F}))).

Clearly H_{1} (l^{2}(Γ) ⊗_{ℤΓ} Σ (ℤΓ ⊗_{ℤΔ} C(\overline{F}))) agrees with the L^{2}-homology H_{0}(F, (j ∘ l)^{∗}l^{2}(Γ)) where j : Δ ⟶ Γ is the inclusion and l : π_{1}(F) ⟶ Δ is the obvious surjection. This L^{2}- homology group vanishes by Lemma 1.2.5 as Δ is infinite. This implies

b_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(U(1 − s_{1}))) ≥ b_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(⊕_{i=1}^{g}U(1 − s_{i})))

and hence

b_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(U(1 − s_{1}))) ≥ b_{1}(E).

Therefore it remains to show b_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(U(1 − s_{1}))) = 0. Let Γ′ be the subgroup of Γ generated by Δ and \overline{s_{1}} ∈ Γ. Define U′(1 − s_{1}) : ℤΓ′ ⊗_{ℤΔ} C(\overline{F}) ⟶ ℤΓ′ ⊗_{ℤΔ} C(\overline{F}) by mapping γ′ ⊗ x to γ′ ⊗ x − γ′\overline{s_{1}} ⊗ C(\overline{σ}(\overline{s_{1}}^{(−1)}))(x). Then U(1−s_{1}) is obtained from U′(1−s_{1}) by induction with the inclusion of groups i : Γ′ ⟶ Γ. This implies

b_{1}(l^{2}(Γ) ⊗_{ℤΓ′} cone(U′(1 − s_{1}))) = b_{1}(l^{2}(Γ) ⊗_{ℤΓ} cone(U(1 − s_{1}))).

Let T_{σ(\overline{s_{1}})} be the mapping torus of σ(\overline{s_{1}}) : F ⟶ F. (Notice that the free homotopy class of σ(\overline{s_{1}}) and hence the homotopy type of T_{σ(\overline{s_{1}})} depend only on s_{1} ∈ π.) The obvious map

π_{1}(T_{σ(\overline{s_{1}})}) ⟶ ℤ factorizes into the composition of epimorphisms π_{1}(T_{σ(\overline{s_{1}})}) ⟶^{ϕ′} Γ′ ⟶ ℤ. We get from Section 2 that cone(U′(1−s_{1})) is the ℤΓ′-chain complex of the covering of T_{σ(\overline{s_{1}})} associated to ϕ′. So we get

b_{1}(l^{2}(Γ) ⊗_{ℤΓ′} cone(U′(1 − s_{1}))) = b_{1}(T_{σ(\overline{s_{1}})}; (i ∘ ϕ′)^{∗}l^{2}(Γ)).

This finishes the proof of Theorem 3.1 since

b_{1}(T_{σ(\overline{s_{1}})}; (i ∘ ϕ′)^{∗}l^{2}(Γ)) = b_{1}(T_{σ(\overline{s_{1}})}; (ϕ′)^{∗}l^{2}(Γ′)) = 0

follows from Lemma 1.2.3 and Theorem 2.1.

4. Groups with vanishing first L^{2}-Betti number

In this section we give criteria for the triviality of the first L^{2}-Betti number of a finitely presented group π. The next theorem answers a query by Gromov [12, page 154].

Theorem 4.1 Let {1} ⟶ Δ ⟶ Γ ⟶ π ⟶ {1} be an extension of finitely presented groups. Suppose that Δ is infinite and π contains ℤ as a subgroup. Then the first L^{2}-Betti number of Γ

b_{1}(Γ) = 0.

\underline{\underline{Proof}} : Apply Theorem 3.1 to the fibration of classifying spaces BΔ ⟶ BΓ ⟶ Bπ.

We mention that there are counterexamples to the so called Burnside problem, namely, infinite finitely generated groups which do not contain ℤ as a subgroup. At of this writing the author does not know of an example of a infinite finitely presented group which does not contain ℤ as a subgroup.

We call a prime 3-manifold exceptional if it is closed and no finite cover of it is homotopy equivalent to a Haken, Seifert or hyperbolic 3-manifold. No exceptional prime 3-manifolds are known, and standard conjectures (Thurston geometrization conjecture, Waldhausen conjecture) imply that there are none. The next theorem follows from [19, Corollary 7.7].

Theorem 4.2 Let π be the fundamental group of a prime 3-manifold M. Suppose either that ∂M is a union of tori or that M is closed and not exceptional. Then

b_{1}(π) = 0.

Next we investigate closed 4-manifolds with a geometric structure in the sense of Thurston and compute their L^{2}-Betti numbers. Note that b_{p}(M) = b_{4−p}(M) holds for such a manifold by Poincaré duality [19, Proposition 4.2]. The descriptions of the geometries and the computations of the Euler characteristics and signatures can be found in [24, Table 1 on page 122 and Theorem 6.1 on page 135] with a correction in [17].

Theorem 4.3 Let M be a closed orientable 4-manifold with a geometry in the sense of Thurston and fundamental group π. Then the following are the values of the L^{2}-Betti numbers b_{p}, the Euler characteristics χ and the signatures σ of M

\begin{matrix}geometry & b_{0} = b_{4} & b_{1} = b_{3} & b_{2} & χ & σ \\ S^{4} & \frac{1}{|π|} & 0 & 0 & \frac{2}{|π|} & 0 \\ S^{2} × S^{2} & \frac{1}{|π|} & 0 & \frac{2}{|π|} & \frac{4}{|π|} & 0 \\ P^{2}(ℂ) & 1 & 0 & 1 & 3 & 1 \\ S^{2} × ℝ^{2} & 0 & 0 & 0 & 0 & 0 \\ S^{2} × ℍ^{2} & 0 & −\frac{χ(M)}{2} & 0 & < 0 & 0 \\ ℝ^{4} & 0 & 0 & 0 & 0 & 0 \\ ℝ^{2} × ℍ^{2} & 0 & 0 & 0 & 0 & 0 \\ ℍ^{2} × ℍ^{2} & 0 & 0 & χ(M) & > 0 & 0 \\ S^{3} × ℝ^{1} & 0 & 0 & 0 & 0 & 0 \\ ℍ^{3} × ℝ & 0 & 0 & 0 & 0 & 0 \\ S̃L_{2} × ℝ & 0 & 0 & 0 & 0 & 0 \\ Nil^{3} × ℝ & 0 & 0 & 0 & 0 & 0 \\ Nil^{4} & 0 & 0 & 0 & 0 & 0 \\ Sol_{m,n}^{4} & 0 & 0 & 0 & 0 & 0 \\ Sol_{0}^{4} & 0 & 0 & 0 & 0 & 0 \\ Sol_{1}^{4} & 0 & 0 & 0 & 0 & 0 \\ ℍ^{2}(ℂ) & 0 & 0 & χ(M) & χ = 3σ & > 0 \\ ℍ^{4} & 0 & 0 & χ(M) & > 0 & 0 \end{matrix}

In particular b_{1}(M) = b_{1}(π) = 0 in all cases except S^{2} × ℍ^{2}.

\underline{\underline{Proof}} : We first give the proofs of the statements of the L^{2}-Betti numbers. Recall from Lemma 1.2.6 that χ(M) = 2b_{0}(M)−b_{1}(M)+b_{2}(M) holds. The fundamental group π is finite if and only if the underlying manifold of the geometry is compact. If M̃ is the universal covering and π is finite, then b_{p}(M) = \frac{b_{p}(M̃)}{|π|} from Lemma 1.2.4. Now the claim follows for S^{4}, S^{2} × S^{2} and P^{2}(ℂ). If M and N have the same geometry X , then b_{p}(M) = 0 is equivalent to b_{p}(N) = 0 since b_{p}(M) = 0 is equivalent to the fact that X has no harmonic smooth L^{2}-integrable p-forms [7]. Hence it suffices to check b_{p}(M) = 0 for one example of a manifold with a given geometry. Suppose the geometry is a product of two lower dimensional geometries. Then such an example is given by a product of two surfaces or of S^{1} with a 3-manifold. The L^{2}-Betti numbers for a closed orientable surfaces F_{g} of genus g ≥ 1 are b_{0}(F_{g}) = b_{2}(F_{g}) = 0 and b_{1}(F_{g}) = 2g − 2 and the L^{2}-Betti numbers of S^{1} are zero [19, Example 4.11]. Using the Künneth formula for L^{2}-cohomology [22, Theorem 3.16] one obtains the claim for all these cases. Suppose the geometry is Nil^{4}, Sol_{0}^{4}, Sol_{1}^{4} or Sol_{m,n}^{4}. Then an example for M can be constructed which fibers over S^{1}. For such a space all L^{2}- Betti numbers are trivial by Theorem 2.1. One can also argue that the fundamental group of a manifold carrying such a geometry contains a normal infinite amenable subgroup and refer to [6, Theorem 0.2]. In the case X = ℍ^{4}, we get b_{1}(M) = b_{0}(M) = 0 and b_{2}(M) = χ(M) > 0 from [8]. If M carries a ℍ^{2}(ℂ)-geometry, then M is Kähler hyperbolic in the sense of [11] and we derive b_{p}(M) = 0 for p = 0, 1 from [11].

The statements about the Euler characteristic and the signature follow in the cases where all L^{2}-Betti numbers are trivial from the L^{2}-signature theorem [1] and in the cases where the geometry is compact from the multiplicativity of Euler characteristic and signature under finite coverings. The remaining cases are S^{2} × ℍ^{2}, ℍ^{2} × ℍ^{2}, ℍ^{4} and ℍ^{2}(ℂ). Because of the Hirzebruch proportionality principle [15] it suffices to check the examples S^{2} × F_{g} and F_{g} × F_{g} for some g ≥ 2. In the cases S^{2} × ℍ^{2} and ℍ^{2} × ℍ^{2} and one gets χ(M) = t · χ(S^{4}) and σ(M) = t·σ(S^{4}) if M carries a ℍ^{4}-structure and χ(M) = s·χ(P^{2}(ℂ)) and σ(M) = s·σ(P^{2}(ℂ)) if M carries a ℍ^{2}(ℂ)-structure for some non-zero constants t and s. From this the claim follows.

5. 4-manifolds satisfying χ(M) ≥ |σ(M)|

Theorem 5.1 Let M be a closed oriented 4-manifold with fundamental group π. If b_{1}(π) = 0, then the following inequality for the Euler characteristic and the signature holds

χ(M) ≥ |σ(M)|.

\underline{\underline{Proof}} : According to the L^{2}-signature theorem (see [1]), the signature σ(M) is the difference of the von Neumann dimensions of two complementary subspaces of the second L^{2}-cohomology H^{2}(M; l^{2}(π)) of M. This implies

|σ(M)| ≤ dim_{N (π)}(H^{2}(M; l^{2}(π))) = dim_{N (π)}(H_{2}(M; l^{2}(π))) = b_{2}(M).

Since b_{4}(M) = b_{0}(M) and b_{3}(M) = b_{1}(M) by Poincaré duality [19, Proposition 4.2] and b_{1}(M) = 0 by assumption holds, we conclude χ(M) = b_{2}(M) + 2 · b_{0}(M) from Lemma 1.2.6 and the claim follows.

This Theorem 5.1 together with Theorems 4.1, 4.2 and 4.3 generalizes respectively reproves results of [16] (see also [25]). In [16] it is shown among other things that the inequality in Theorem 5.1 holds if π is an extension of a finitely generated group Δ and the integers ℤ. (We have to assume that Δ is finitely presented). If a connected closed oriented 4-manifold has a geometric structure in the sense of Thurston, then it is pointed out in [16] that inspecting the 18 different cases using the results in [24] and the correction in [17] shows that the inequality in Theorem 5.1 holds for M provided the geometry is not S^{2} × ℍ^{2}. A simple proof can be found in [16] that the inequality above holds if π satisfies 3-dimensional oriented Poincaré duality. If π is infinite amenable, b_{1}(π) vanishes by [6, Theorem 0.2] and Theorem 5.1 applies (see also [9]). For further information on lower bounds on the Euler characteristic of a closed 4-manifold we refer to [13].

6. Deficiency of groups

The deficiency of a finitely presented group π is the maximum over all differences g − r where g respectively r is the number of generators respectively relations of a presentation of π.

Theorem 6.1 Let π be a finitely presented group.

1. If Bπ has finite 3-skeleton, then

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

2. def(π) ≤ 1 − b_{0}(π) + b_{1}(π).

3. If b_{1}(π) = 0, then

def(π) ≤ 1.

\underline{\underline{Proof}} : Given a presentation with r relations and g generators, let X be the corresponding connected 2-dimensional CW -complex with fundamental group isomorphic to π which has precisely one cell of dimension 0, g cells of dimension 1 and r cells of dimension 2. Since the classifying map X ⟶ Bπ is 2-connected, we conclude from Lemma 1.2 that

1 − g + r = χ(X) = b_{0}(X) − b_{1}(X) + b_{2}(X) ≥ b_{0}(π) − b_{1}(π) + b_{2}(π)

from which assertion 1.) follows. Assertion 2.) is proven similiarly and does imply assertion 3.).

From Theorem 4.1 and Theorem 6.1.3. we obtain

Corollary 6.2 Let {1} ⟶ Δ ⟶ Γ ⟶ π ⟶ {1} be an extension of finitely presented groups. Suppose that Δ is infinite and π contains ℤ as a subgroup. Then

def(Γ) ≤ 1.

If π is finite, we rediscover from Theorem 6.1.1 the well-known fact that def(π) ≤ 0. The inequality in Theorem 6.1.1 is obviously sharp and def(π) = 1 − χ(Bπ) if Bπ is a finite 2-dimensional CW -complex. If π is a torsion-free one-relator group, the 2-dimensional CW -complex associated with the presentation is aspherical and hence Bπ is 2-dimensional [23, chapter III §§9 -11]. We conjecture for a torsion-free group having a presentation with g ≥ 2 generators and one non-trivial relation that b_{2}(π) = 0 and b_{1}(π) = def(π) − 1 = g − 2 holds (compare [12, page 156]). This would follow from [19, Conjecture 9.2] saying that the L^{2}-Betti numbers of a finite CW -complex with torsion-free fundamental groups are integers. Namely, the kernel of the second differential of the L^{2}-chain complex of Bπ is a proper submodule of l^{2}(π) so that its dimension b_{2}(π) is less than one.

The L^{2}-homological test for the deficiency described in Theorem 6.1 is useful in the situation considered in Corollary 6.2 where the corresponding tests using homology with ℤ/pcoefficients appears insufficient. However, in other situations homology with ℤ/p-coefficients seems to be more useful than L^{2}-homology as illustrated by the following result which is a direct consequence of [10, Theorem 2.5]

Theorem 6.3 Let M be a compact 3-manifold with fundamental group π and prime decomposition

M = M_{1}♯M_{2}♯ … ♯M_{r}.

Let s(M) be the number of prime factors M_{i} with non-empty boundary and t(M) be the number of prime factors which are S^{2}-bundles over S^{1}. Denote by χ(M) the Euler characteristic. Then

def(π_{1}(M)) = dim_{Z/2}(H_{1}(π; Z/2)) − dim_{Z/2}(H_{2}(π; Z/2)) = s(M) + t(M) − χ(M).

Let M be a compact irreducible 3-manifold with infinite fundamental group. Any such manifold is aspherical by the Sphere Theorem [14, page 40]. It follows from Theorem 6.3 that def(π_{1}(M)) is 1 − χ(M) if ∂M is not empty and is zero if ∂M is empty. Thus 1 − b_{0}(M) + b_{1}(M) − b_{2}(M) = 1 − χ(M) since b_{3}(M) = 0 holds by [19, Lemma 4.5]. Hence we rediscover Theorem 6.3 in the case ∂M ≠ ∅ from our L^{2}-homological test in Theorem 6.1. However, if M is closed, the inequality in Theorem 6.1.1 is not sharp.

Finally we mention the in a certain sense complementary result [3, Theorem 2]. If π is a finitely presented group with def(π) ≥ 2, then π can be written as an amalgamated product π = A ∗_{C} B where A, B and C are finitely generated, C is proper subgroup of both A and B and has index greater than two in A or B. In particular π contains a free subgroup of rank 2 and is not amenable. This implies that an amenable finitely presented group has deficiency less or equal to one (see also [2] and [9]). This also follows from Theorem 6.1.3 and the fact that b_{1}(π) = 0 for a finitely presented amenable group [6, Theorem 0.2].

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Current address Wolfgang Lück Fachbereich Mathematik Johannes Gutenberg-Universität Staudingerweg 9 55128 Mainz Bundesrepublik Deutschland email: lueck@topologie.mathematik.uni-mainz.de

Version of November 8, 1996