The ℓ^{2}-cohomology of hyperplane complements

M. W. Davis ^{∗} T. Januszkiewicz^{†} I. J. Leary ^{‡}

December 14, 2006

Abstract

We compute the ℓ^{2}-Betti numbers of the complement of a finite collection of affine hyperplanes in ℂ^{n}. At most one of the ℓ^{2}-Betti numbers is non-zero. AMS classification numbers. Primary: 52B30 Secondary: 32S22, 52C35, 57N65, 58J22. Keywords: hyperplane arrangements, L^{2}-cohomology.

1 Introduction

Suppose X is a finite CW complex with universal cover X̃. For each p ≥ 0, one can associate to X a Hilbert space, H^{p}(X̃), the p-dimensional “reduced ℓ^{2}-cohomology”, cf. [3]. Each H^{p}(X̃) is a unitary π_{1}(X)-module. Using the π_{1}(X)-action, one can attach a nonnegative real number called“von Neumann dimension” to such a Hilbert space. The “dimension” of H^{p}(X̃) is called the p^{th} ℓ^{2}-Betti number of X.

Here we are interested in the case where X is the complement of a finite number of affine hyperplanes in ℂ^{n}. (Technically, in order to be in compliance with the first paragraph, we should replace the complement by a homotopy equivalent finite CW complex.) Let A be the finite collection of hyperplanes, Σ(A) their union and M(A) := ℂ^{n} − Σ(A). The rank of A is the maximum codimension l of any nonempty intersection of hyperplanes in A. It turns out that the ordinary (reduced) homology of Σ(A) vanishes except in dimension l − 1 (cf. Proposition 2.3). Let β(A) denote the rank of \overline{H}_{l−1}(Σ(A)). Our main result, proved as Theorem 5.2, is the following.

∗The first author was partially supported by NSF grant DMS 0405825.

†The second author was partially supported by NSF grant DMS 0405825.

‡The third author was partially supported by NSF grant DMS 0505471.

Theorem A. Suppose A is an affine hyperplane arrangement of rank l. Only the l^{th} ℓ^{2}-Betti number of M(A) can be nonzero and it is equal to β(A).

This is reminiscent of a well-known result about the cohomology of M(A) with coefficients in a generic flat line bundle ( “generic” is defined in Section 4). This result is proved as Theorem 4.3. We state it below.

Theorem B. Suppose that L is a generic flat line bundle over M(A). Then H^{∗}(M(A); L) vanishes except in dimension l and dim H^{l}(M(A); L) = β(A).

Both theorems have similar proofs. In the case of Theorem A the basic fact is that the ℓ^{2}-Betti numbers of S^{1} vanish. (In other words, if the universal cover ℝ of S^{1} is given its usual cell structure, then H^{∗}(ℝ) = 0.) Similarly, for Theorem B, if L is a flat line bundle over S^{1} corresponding to an element λ ∈ ℂ^{∗}, with λ ≠ 1, then H^{∗}(S^{1}; L) = 0. By the Künneth Formula, there are similar vanishing results for any central arrangement. To prove the general results, one considers an open cover of M(A) by “small” open neighborhoods each homeomorphic to the complement of a central arrangement. The E_{1}- page of the resulting Mayer-Vietoris spectral sequence is nonzero only along the bottom row, where it can be identified with the simplicial cochains with constant coefficients on a pair (N(U), N(V)), which is homotopy equivalent to (ℂ^{n}, Σ). It follows that the E_{2}-page can be nonzero only in position (l, 0). (Actually, in the case of Theorem A, technical modifications must be made to the above argument. Instead of reduced ℓ^{2}-cohomology one takes local coefficients in the von Neumann algebra associated to the fundamental group and the vanishing results only hold modulo modules which don’t contribute to the ℓ^{2}-Betti numbers.)

In [2] the first and third authors proved a similar result for the ℓ^{2}- cohomology of the universal cover of the Salvetti complex associated to an arbitrary Artin group (as well as a formula for the cohomology of the Salvetti complex with generic, 1-dimensional local coefficients). This can be interpreted as a computation of the ℓ^{2}-cohomology of universal covers of hyperplane complements associated to infinite reflection groups. Although the main argument in [2] uses an explicit description of the chain complex of the Salvetti complex, an alternative argument, similar to the one outlined above, is given in [2, Section 10].

2 Certain covers and their nerves

A hyperplane arrangement A is a finite collection of affine hyperplanes in ℂ^{n}. A subspace of A is a nonempty intersection of hyperplanes in A. Denote by L(A) the poset of subspaces, partially ordered by inclusion. An arrangement is central if L(A) has a minimum. Given an element G ∈ L(A), its rank, rk(G), is the codimension of G in ℂ^{n}. The minimal elements of L(A) are a family of parallel subspaces and they all have the same rank. The rank of an arrangement A is the rank of a minimal element in L(A). A is essential if rk(A) = n.

The singular set Σ(A) of the arrangement is the union of hyperplanes in A (so that Σ(A) is a subset of ℂ^{n}). The complement of Σ(A) in ℂ^{n} is denoted M(A). When there is no ambiguity we will drop the “A” from our notation and write L, Σ or M instead of L(A), Σ(A) or M(A).

Given G ∈ L(A), define a subarrangement, A_{G} := {H ∈ A | G ⊂ H}. Define a small neighborhood of G to be a tubular neighborhood of G in ℂ^{n} which has empty intersection with any hyperplane not in A_{G}. The next lemma is well-known.

Lemma 2.1. Suppose V_{G} is a small neighborhood of some G ∈ L(A). Let V̂_{G} := V_{G} − Σ(A). Then π_{1}(V̂_{G}) is a retract of π_{1}(M(A)).

Proof. The composition of the two inclusions, V̂_{G} ↪ M(A) ↪ M(A_{G}) is a homotopy equivalence.

Suppose U = {U_{i}}_{i∈I} is a cover of some space X (where I is some index set). Given a subset σ ⊂ I, put U_{σ} := ⋂_{i∈σ} U_{i}. Recall that the nerve of U is the simplicial complex N(U), defined as follows. Its vertex set is I and a finite, nonempty subset σ ⊂ I spans a simplex of N(U) if and only if U_{σ} is nonempty.

We shall need to use the following well-known lemma several times in the sequel, see [5, Cor. 4G.3 and Ex. 4G(4)]

Lemma 2.2. Let U be a cover of a paracompact space X and suppose that either (a) each U_{i} is open or (b) X is a CW complex and each U_{i} is a subcomplex. Further suppose that for each simplex σ of N(U), U_{σ} is contractible. Then X and N(U) are homotopy equivalent.

Next we define an open cover V of a certain neighborhood of Σ. For each H ∈ A, let V_{H} be a small convex neighborhood of H in ℂ^{n}. Let V (Σ) be the union of the V_{H} and let V := {V_{H}}_{H∈A} be the corresponding open cover. Clearly, V (Σ) deformation retracts onto Σ. For each σ ∈ N(V), V_{σ} := ⋂_{H∈σ} V_{H} is a small neighborhood of G(σ) := ⋂_{H∈σ} H. Each V_{σ} is also contractible (being an intersection of convex sets).

Denote by |L(A)| (or simply |L|) the geometric realization of the poset L.

Proposition 2.3. The spaces Σ, N(V) and |L| are homotopy equivalent. Moreover, each is homotopy equivalent to a wedge of (l − 1)-spheres, where l = rk(A). (So, when A is essential, the spheres are (n − 1)-dimensional.)

Proof. The proof is based on Lemma 2.1. V is an open cover of V (Σ) and V_{σ} is contractible for each simplex σ of N(V); so it follows that N(V) ∼ V (Σ) ∼ Σ, where ‘∼’ means ‘is homotopy equivalent’. For each G ∈ L, we have the subposet L_{≤G} := {G′ ∈ L | G′ ≤ G}. Its geometric realization is a cone (and hence is contractible) since it contains the maximum element G. Consider the cover of |L| by the subcomplexes {|L_{≤H}|}_{H∈A}. For any σ ⊂ A, ⋂_{H∈σ} |L_{≤H}| = |L_{≤G(σ)}|. So, this cover has the same nerve as V. Since all nonempty intersections are contractible, N(V) ∼ |L|.

The proof that any one of these spaces is homotopy equivalent to a wedge of spheres follows from the usual “deletion-restriction” argument and induction. If the rank l is 1, then dim |L| = 0. Hence, it is a wedge of 0-spheres. Similarly, if l = 2, then |L| is a connected graph; hence, a wedge of 1-spheres. So, assume by induction that l > 2. Choose a hyperplane H ∈ A, let A′ = A − {H} and let A′′ be the restriction of A to H (i.e., A′′ := {H′ ∩ H | H′ ∈ A′}). Put Σ′ = Σ(A′), Σ′′ = Σ(A′′), l′ = rk(A′) and l′′ = rk(A′′). We can also assume by induction on Card(A) that Σ′ and Σ′′ are homotopy equivalent to wedges of spheres. If l′ < n and H is transverse to the minimal elements of L(A′), then l′′ = l, the arrangement splits as a product, Σ = Σ′′ × ℂ, and we are done by induction. In all other cases l′ = l and l′′ = l − 1. We have Σ = Σ′ ∪ H and Σ′ ∩ H = Σ′′. H is simply connected and since l > 2, Σ′ is simply connected and Σ′′ is connected. By van Kampen’s Theorem, Σ is simply connected. Consider the exact sequence of the pair (Σ, Σ′):

→ H_{∗}(Σ′) → H_{∗}(Σ) → H_{∗}(Σ, Σ′) → .

There is an excision isomorphism, H_{∗}(Σ, Σ′) ≅ H_{∗}(H, Σ′′). Since H is contractible it follows that H_{∗}(H, Σ′′) ≅ \overline{H}_{∗−1}(Σ′′). By induction, \overline{H}_{∗}(Σ′) is concentrated in dimension l − 1 and \overline{H}_{∗}(Σ′′) in dimension l − 2. So, \overline{H}_{∗}(Σ) is also concentrated in dimension l−1. It follows that Σ is homotopy equivalent to a wedge of l − 1 spheres.

Next we want to extend V to an open cover U of the ambient space. There is one technicality before we do this. Since we want to deal with finite covers we first want to replace Σ by its “compact core.” One way to do this is to first choose a closed disk D^{2n} ⊂ ℂ^{n} of large enough radius so that it intersects every element of L(A). Then Σ and Σ∩D^{2n} are homotopy equivalent, so, we simply replace Σ by Σ ∩ D^{2n} (without changing notation). Similarly, replace M(A) by M(A) ∩ D^{2n} = D^{2n} − Σ.

To define U, choose a finite collection of open balls {B_{i}}_{i∈I} such that (a) B_{i} ⊂ ℂ^{n} − Σ and (b) {B_{i} ∩ D^{2n}} covers D^{2n} − V (Σ). Put U_{i} = B_{i} ∩ D^{2n} and

U = V ∪ {U_{i}}_{i∈I}.

The key point is that each element of U is a convex subset of ℂ^{n} and hence, so is any nonempty intersection of such elements. The following is now an immediate consequence of Lemma 2.1 and Proposition 2.3.

Lemma 2.4. N(U) is a contractible simplicial complex and N(V) is a subcomplex. Moreover, H_{∗}(N(U), N(V)) is concentrated in dimension l, where l = rk A.

Definition 2.5. β(A) is the rank of H_{l}(N(U), N(V)).

Equivalently, β(A) is the rank of H_{l}(ℂ^{n}, Σ(A)) (or of \overline{H}_{l−1}(Σ(A)). Also, it is not difficult to see that (−1)^{l}β(A) = χ(D^{2n}, Σ) = 1 − χ(Σ) = χ(M), where χ( ) denotes the Euler characteristic.

Remark 2.6. Suppose A_{ℝ} is an arrangement of real hyperplanes in ℝ^{n} and Σ_{ℝ} ⊂ ℝ^{n} is the singular set. Then ℝ^{n} − Σ_{ℝ} is a union of open convex sets called chambers and β(A_{ℝ}) is the number of bounded chambers. If A is the complexification of A_{ℝ}, then Σ(A) ∼ Σ(A_{ℝ}). Hence, β(A) = β(A_{ℝ}).

By intersecting the elements of V with M (= D^{2n} − Σ) we get an induced cover V̂ of M. An element of V̂ is a deleted small neighborhood V̂_{H} (= V_{H}−H) for some H ∈ A. Similarly, by intersecting U with M we get an induced cover Û of D^{2n}. The key observation is the following.

Observation 2.7. N(Û) = N(U) and N(V̂) = N(V).

3 The Mayer-Vietoris spectral sequence

Let X be a space, π = π_{1}(X) and r : X̃ → X the universal cover. Given a left π-module A, define

C^{∗}(X; A) := Hom_{π}(C_{∗}(X̃), A),

the cochains with local coefficients in A. Taking cohomology gives H^{∗}(X; A).

Let U be an open cover of X and N = N(U) its nerve. Let N^{(p)} denote the set of p-simplices in N. There is an induced cover Ũ := {r^{−1}(U)}_{U∈U} with the same nerve. Suppose that for each simplex σ of N, π_{1}(U_{σ}) → π_{1}(X) is injective. There is a Mayer-Vietoris double complex

C_{p,q} = ⨁_{σ∈N^{(p)}} C_{q}(r^{−1}(U_{σ}))

(cf. [1, §VII.4]) and a corresponding double cochain complex with local coefficients:

C^{p,q}(A) := Hom_{π}(C_{p,q}; A).

The cohomology of the total complex is H^{∗}(X; A). This gives a spectral sequence with E_{1}-page

E_{1}^{p,q} = ⨁_{σ∈N^{(p)}} H^{q}(U_{σ}; A).  (1)

Here H^{q}(U_{σ}; A) means the cohomology of Hom_{π}(C_{∗}(r^{−1}(U_{σ})), A) or equivalently the cohomology of Hom_{π_{1}(U_{σ})}(C_{∗}(Ũ_{σ}); A). The spectral sequence converges to H^{∗}(X; A).

In the next two sections we will apply this spectral sequence to the case where X is M(A) and the open cover is Û of the previous section. Moreover, the π-module A will be such that for any simplex σ in N(V̂), H^{q}(V̂_{σ}; A) = 0 for all q (even for q = 0) while for a simplex σ of N(Û) which is not in N(V̂), H^{q}(U_{σ}; A) = 0 for all q > 0 and is constant (i.e., independent of σ) for q = 0. Thus E_{1}^{p,q} will vanish for q > 0 and E_{1}^{∗,0} can be identified with the cochain complex C^{∗}(N(U), N(V)) with constant coefficients.

4 Generic coefficients

Here we will deal with 1-dimensional local coefficient systems. We begin by considering such local coefficients on S^{1}. Let α be a generator of the infinite cyclic group π_{1}(S^{1}). Suppose k is a field of characteristic 0 and λ ∈ k^{∗}. Let A_{λ} be the k[π_{1}(S^{1})]-module which is a 1-dimensional k-vector space on which α acts by multiplication by λ.

Lemma 4.1. If λ ≠ 1, then H^{∗}(S^{1}; A_{λ}) vanishes identically.

Proof. If S^{1} has its usual CW structure with one 0-cell and one 1-cell, then in the chain complex for its universal cover both C_{0} and C_{1} are identified with the group ring k[π_{1}(S^{1})] and the boundary map with multiplication by 1 − t, where t is the generator of π_{1}(S^{1}). Hence, the coboundary map C^{0}(S^{1}; A_{λ}) → C^{1}(S^{1}; A_{λ}) is multiplication by 1 − λ.

Next, consider M(A). Its fundamental group π is generated by loops a_{H} for H ∈ A, where the loop a_{H} goes once around the hyperplane H in the “positive” direction. Let α_{H} denote the image of a_{H} in H_{1}(M(A)). Then H_{1}(M(A)) is free abelian with basis {α_{H}}_{H∈A}. So, a homomorphism H_{1}(M(A)) → k^{∗} is determined by an A-tuple Λ ∈ (k^{∗})^{A}, where Λ = (λ_{H})_{H∈A} corresponds to the homomorphism sending α_{H} to λ_{H}. Let ψ_{Λ} : π → k^{∗} be the composition of this homomorphism with the abelianization map π → H_{1}(M(A)). The resulting local coefficient system on M(A) is denoted A_{Λ}. The next lemma follows from Lemma 4.1.

Lemma 4.2. Suppose A is a nonempty central arrangement and Λ is such that ∏_{H∈A} λ_{H} ≠ 1. Then H^{q}(M(A)) vanishes for all q.

Proof. Without loss of generality we can suppose the elements of A are linear hyperplanes. The Hopf bundle M(A) → M(A)/S^{1} is trivial (cf. [7, Prop. 5.1, p. 158]); so, M(A) ≅ B × S^{1}, where B = M(A)/S^{1}. Let i : S^{1} → M(A) be inclusion of the fiber. The induced map on H_{1}( ) sends α to ∑ α_{H}. Thus, if we pull back A_{Λ} to S^{1}, we get A_{λ}, where λ = ∏_{H∈A} λ_{H}. The condition on Λ is λ ≠ 1, which by Lemma 4.1 implies that H^{∗}(S^{1}, A_{λ}) vanishes identically. By the Künneth Formula H^{∗}(M(A), A_{Λ}) also vanishes identically.

Returning to the case where A is a general arrangement, for each simplex σ in N(V̂), let A_{σ} be the corresponding central arrangement (which is diffeomorphic to V_{σ}). Given Λ ∈ (k^{∗})^{A}, put

λ_{σ} := ∏_{H∈A_{σ}} λ_{H}.

Call Λ generic if λ_{σ} ≠ 1 for all σ ∈ N(V).

Theorem 4.3. (Compare [8, Thm. 4.6, p. 160]). Let A be an affine arrangement of rank l and Λ a generic A-tuple in k^{∗}. Then H^{∗}(M(A); A_{Λ}) is concentrated in degree l and

dim_{k} H^{l}(M(A); A_{Λ}) = β(A).

Proof. We have an open cover of M̃(A), {r^{−1}(U)}_{U∈Û}. Its nerve is N(Û). By Lemma 4.2 and the last paragraph of Section 3, the E_{1}-page of the Mayer-Vietoris spectral sequence is concentrated along the bottom row where it can be identified with C^{∗}(N(Û), N(V̂); k). By Observation 2.7, (N(Û), N(V̂)) = (N(U), N(V)). So, the E_{2}-page is concentrated on the bottom row and E_{2}^{p0} = H^{p}(N(U), N(V); k). By Lemma 2.4 these groups are nonzero only for p = l and dim_{k} H^{l}(N(U), N(V); k) = β(A).

Remark 4.4. When k = ℂ, a 1-dimensional local coefficient system on X is the same thing as a flat line bundle over X.

5 ℓ^{2}-cohomology

For a discrete group π, ℓ^{2}π denotes the Hilbert space of complex-valued, square integrable functions on π. There are unitary π-actions on ℓ^{2}π by either left or right multiplication; hence, ℂπ acts either from the left or right as an algebra of operators. The associated von Neumann algebra N π is the commutant of ℂπ (acting from, say, the right on ℓ^{2}π).

Given a finite CW complex X with fundamental group π, the space of ℓ^{2}-cochains on its universal cover X̃ is the same as C^{∗}(X; ℓ^{2}π), the cochains with local coefficients in ℓ^{2}π. The image of the coboundary map need not be closed; hence, H^{∗}(X; ℓ^{2}π) need not be a Hilbert space. To remedy this, one defines the reduced ℓ^{2}-cohomology H^{∗}(X̃) to be the quotient of the space of cocycles by the closure of the space of coboundaries. We shall also use the notation H^{∗}(X; ℓ^{2}π) for the same space.

The von Neumann algebra admits a trace. Using this, one can attach a “dimension,” dim_{N π} V, to any closed, π-stable subspace V of a finite direct sum of copies of ℓ^{2}π (it is the trace of orthogonal projection onto V). The nonnegative real number dim_{N π}(H^{p}(X; ℓ^{2}π)) is the p^{th} ℓ^{2}-Betti number of X.

A technical advance of Lück [6, Ch. 6] is the use local coefficients in N π in place of the previous version of ℓ^{2}-cohomology. Given a finitely generated N π-module A, Lück decomposes it into a projective N π-module P A and a “torsion” submodule T A, where

T A := {x ∈ A | f(x) = 0 for all f ∈ Hom_{N π}(A, N π)}
P A := A/T A.

(An equivalent approach is given by Farber in [4].) There is a well-defined dimension function on N π-modules, A → dim_{N π} A, such that dim_{N π} A = dim_{N π} P A. This gives the same answer for ℓ^{2}-Betti numbers, i.e., for each p one has that dim_{N π} H^{p}(X; N π) = dim_{N π} H^{p}(X; ℓ^{2}π). This dimension function satisfies an additivity property for short exact sequences, and vanishes on the class T , the “torsion” N π-modules or modules of the form T A. The dimension function allows one to define ℓ^{2}-Betti numbers for spaces more general than finite complexes. The class T is a Serre class of N π-modules [9], which allows one to compute ℓ^{2}-Betti numbers by working with spectral sequences modulo T .

Lemma 5.1. Suppose A is a nonempty central arrangement. Then, for all q ≥ 0, H^{q}(M(A); N π) lies in T . In other words, all ℓ^{2}-Betti numbers of M(A) are zero.

Proof. The proof is along the same line as that of Lemma 4.2. It is wellknown that the reduced ℓ^{2}-cohomology of ℝ vanishes. Since M(A) = S^{1} × B, the result follows from the Künneth Formula for ℓ^{2}-cohomology in [6, 6.54 (5)].

Theorem 5.2. Suppose A is an affine hyperplane arrangement. Then

H^{∗}(M(A); N π) ≅ H^{∗}(N(U), N(V)) ⊗ N π (mod T )

Hence, for l = rk(A), the ℓ^{2}-Betti numbers of M(A) vanish except in dimension l, where dim_{N π} H^{l}(M̃(A)) = β(A).

Proof. As in the proof of Theorem 4.3, it follows from Lemma 5.1 that the E_{1}-page of the spectral sequence consists of modules in T , except that E_{1}^{∗,0} is identified with C^{∗}(N(U), N(V)) ⊗ N (π). Similarly, the E_{2}-page consists of modules in T , except that E_{2}^{∗,0} is identified with H^{∗}(N(U), N(V)) ⊗ N π. For each subsequent differential, either the source or the target is a module in T , and hence for each i and j one has that E_{∞}^{i,j}/T E_{∞}^{i,j} ≅ E_{2}^{i,j}/T E_{2}^{i,j}. The claim follows since the filtration of H^{∗}(M(A); N π) given by the E_{∞}-page of the spectral sequence is finite.

References

[1] K. S. Brown, Cohomology of Groups, Springer-Verlag, Berlin and New York, 1982.

[2] M. W. Davis and I. J. Leary, The ℓ^{2}-cohomology of Artin groups, J. London Math. Soc. (2) 68 (2003), 493–510.

[3] B. Eckmann, Introduction to ℓ_{2}-methods in topology: reduced ℓ_{2}- homology, harmonic chains, ℓ_{2}-Betti numbers, Israel J. Math. 117 (2000), 183–219

[4] M. Farber, Homological algebra of Novikov-Shubin invariants and Morse inequalities, Geom. Funct. Anal. 6 (1996), 628–665.

[5] A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2001.

[6] W. Lück, L^{2}-invariants and K-theory, Springer-Verlag, Berlin and New York, 2002.

[7] P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin and New York, 1992.

[8] V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones Math. 106 (1991), 139–194.

[9] J.-P. Serre, Groupes d’homotopie et classes de groupes abéliens, Ann. Math. 58 (1953) 258–294.

M. W. Davis mdavis@math.ohio-state.edu
T. Januskiewicz tjan@math.ohio-state.edu
I. J. Leary leary@math.ohio-state.edu

Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus Ohio 43210