The
ā
2
-cohomology of hyperplane complements
M. W. Davis
āT. Januszkiewicz
āI. J. Leary
ā”March 12, 2007
Abstract
We compute the ā2-Betti numbers of the complement of a finite
collection of affine hyperplanes in Cn. At most one of the ā2-Betti
numbers is non-zero.
AMS classification numbers. Primary: 52B30 Secondary: 32S22, 52C35, 57N65, 58J22.
Keywords: hyperplane arrangements, L2-cohomology.
1
Introduction
Suppose X is a finite CW complex with universal cover Xe. For each pā„0, one can associate to X a Hilbert space, Hp(Xe), the p-dimensional āreduced
ā2-cohomology,ā cf. [3]. Each Hp(Xe) is a unitary Ļ
1(X)-module. Using
the Ļ1(X)-action, one can attach a nonnegative real number called āvon
Neumann dimensionā to such a Hilbert space. The ādimensionā of Hp(Xe) is
called the pth ā2-Betti number of X.
Here we are interested in the case where X is the complement of a finite number of affine hyperplanes in Cn. (Technically, in order to be in
com-pliance with the first paragraph, we should replace the complement by a homotopy equivalent finite CW complex. However, to keep from pointlessly complicating the notation, we shall ignore this technicality.) Let A be the finite collection of hyperplanes, Ī£(A) their union and M(A) :=CnāĪ£(A).
āThe first author was partially supported by NSF grant DMS 0405825.
ā The second author was partially supported by NSF grant DMS 0405825.
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 Ī£(A) vanishes except in dimension l ā1 (cf. Proposition 2.1). Let Ī²(A) denote the rank of Hlā1(Ī£(A)). Our main result, proved as Theorem 6.2, is
the following.
Theorem A. SupposeA is an affine hyperplane arrangement of rankl. Only the lth ā2-Betti number of M(A) can be nonzero and it is equal to Ī²(A).
This is reminiscent of a well-known result about the cohomology ofM(A) with coefficients in a generic flat line bundle ( āgenericā is defined in Sec-tion 5). This result is proved as Theorem 5.3. We state it below.
Theorem B. Suppose that Lis a generic flat line bundle over M(A). Then
Hā
(M(A);L)vanishes except in dimensionlanddimCHl(M(A);L) =Ī²(A).
Both theorems have similar proofs. In the case of Theorem A the basic fact is that theā2-Betti numbers ofS1vanish. (In other words, if the universal
cover R of S1 is given its usual cell structure, then Hā(R) = 0.) Similarly, for Theorem B, if Lis a flat line bundle overS1 corresponding to an element
Ī»āCā, withĪ»6= 1, thenHā(S1;L) = 0. By the KĀØunneth Formula, there are
similar vanishing results for any central arrangement. To prove the general results, one considers an open cover ofM(A) by āsmallā open neighborhoods each homeomorphic to the complement of a central arrangement. The E1
-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 (Cn,Ī£). 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 ā2-cohomology one takes local
coefficients in the von Neumann algebra associated to the fundamental group and the vanishing results only hold modulo modules which donāt contribute to the ā2-Betti numbers.)
In [2] the first and third authors proved a similar result for the ā2
-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 Sal-vetti complex with generic, 1-dimensional local coefficients). This can be interpreted as a computation of theā2-cohomology of universal covers of
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].
Our thanks to the referee for finding some mistakes in the first version of this paper.
2
Hyperplane arrangements
Ahyperplane arrangement Ais a finite collection of affine hyperplanes inCn.
A subspace of A is a nonempty intersection of hyperplanes in A. Denote by
L(A) the poset of subspaces, partially ordered by inclusion, and let L(A) :=
L(A)āŖ {Cn}. An arrangement is central if L(A) has a minimum element.
GivenGāL(A), itsrank, rk(G), is the codimension ofGinCn. The minimal elements ofL(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 Ī£(A) of the arrangement is the union of hyperplanes in A (so that Ī£(A) is a subset of Cn). The complement of Ī£(A) in Cn is
denoted M(A). When there is no ambiguity we will drop the āAā from our notation and write L, Ī£ or M instead ofL(A), Ī£(A) or M(A).
Proposition 2.1. Ī£ is homotopy equivalent to a wedge of (lā1)-spheres, where l = rk(A). (So, ifA is essential, the spheres are(nā1)-dimensional.) Proof. The proof follows from the usual ādeletion-restrictionā argument and induction. If the rank l is 1, then Ī£ is the disjoint union of a finite family of parallel hyperplanes. Hence, Ī£ is homotopy equivalent to a finite set of points, i.e., to a wedge of 0-spheres. Similarly, when l = 2, it is easy to see that Ī£ is homotopy equivalent to a connected graph; hence, a wedge of 1-spheres. So, assume by induction that l > 2. Choose a hyperplane
H ā A, let Aā²
= A ā {H} and let Aā²ā²
be the restriction of A to H (i.e.,
Aā²ā²
:={Hā²
ā©H | Hā²
ā Aā²
}). Put Ī£ā²
= Ī£(Aā²
), Ī£ā²ā²
= Ī£(Aā²ā²
), lā²
= rk(Aā²
) and
lā²ā² = rk(Aā²ā²
). We can also assume by induction on Card(A) that Ī£ā² and Ī£ā²ā² are homotopy equivalent to wedges of spheres. If lā²
< n and H is transverse to the minimal elements of L(Aā²
), then lā²ā²
= l, the arrangement splits as a product, Ī£ = Ī£ā²ā²ĆC, and we are done by induction. In all other cases
lā²
= l and lā²ā²
=l ā1. We have Ī£ = Ī£ā²
āŖH and Ī£ā²
ā©H = Ī£ā²ā²
. H is simply connected and since l >2, Ī£ā²
is simply connected and Ī£ā²ā²
van Kampenās Theorem, Ī£ is simply connected. Consider the exact sequence of the pair (Ī£,Ī£ā²):
āHā(Ī£ā²)āHā(Ī£)āHā(Ī£,Ī£ā²)ā.
There is an excision isomorphism, Hā(Ī£,Ī£ā²) =ā¼ Hā(H,Ī£ā²ā²). Since H is
con-tractible it follows that Hā(H,Ī£ā²ā²) ā¼= Hāā1(Ī£ā²ā²). By induction, Hā(Ī£ā²) is
concentrated in dimension lā1 and Hā(Ī£ā²ā²) in dimensionlā2. So,Hā(Ī£) is
also concentrated in dimension lā1. It follows that Ī£ is homotopy equivalent to a wedge of lā1 spheres.
3
Certain covers and their nerves
Suppose U = {Ui}iāI is a cover of some space X (where I is some index
set). Given a subset Ļ ā I, put UĻ := TiāĻUi. 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 Ļ ā I spans a simplex ofN(U) if and only if UĻ is
nonempty.
We shall need to use the following well-known lemma several times in the sequel, see [4, Cor. 4G.3 and Ex. 4G(4)]
Lemma 3.1. Let U be a cover of a paracompact space X and suppose that
either (a) each Ui is open or (b)X is a CW complex and eachUi is a
subcom-plex. Further suppose that for each simplex Ļ of N(U), UĻ is contractible.
Then X and N(U) are homotopy equivalent.
Suppose A is a hyperplane arrangement in Cn. An open convex subset
U inCn issmall (with respect to A) if the following two conditions hold: (i) {GāL(A)|Gā©U 6=ā }has a unique minimum element Min(U). (ii) A hyperplane H ā A has nonempty intersection with U if and only if
Min(U)āH.
The intersection of two small convex open sets is also small; hence, the same is true for any finite intersection of such sets.
Now let U ={Ui}iāI be an open cover ofCn by small convex sets. (Such
covers clearly exist.) Put
Lemma 3.2. N(U) is a contractible simplicial complex and N(Using) is a
subcomplex homotopy equivalent to Ī£. Moreover,Hā(N(U), N(Using))is
con-centrated in dimension l, where l= rkA.
Proof. Using is an open cover of a neighborhood of Ī£ which deformation
retracts onto Ī£. For each simplex Ļ of N(U), UĻ is contractible (in fact,
it is a small convex open set). By Lemma 3.1, N(U) is homotopy equivalent to Cn and N(Using) is homotopy equivalent to Ī£. The last sentence of the lemma follows from Proposition 2.1.
Remark 3.3. Lemma 3.1 can also be used to show that the geometric real-ization of L is homotop[y equivalent to Ī£.
Definition 3.4. Ī²(A) is the rank ofHl(N(U), N(Using)).
Equivalently, Ī²(A) is the rank ofHl(Cn,Ī£(A)) (or of Hlā1(Ī£(A)). Also,
it is not difficult to see that (ā1)lĪ²(A) = Ļ(Cn,Ī£) = 1āĻ(Ī£) = Ļ(M),
where Ļ( ) denotes the Euler characteristic.
Remark 3.5. Suppose AR is an arrangement of real hyperplanes in Rn and Ī£R ā Rn is the singular set. Then RnāĪ£R is a union of open convex sets called chambers and Ī²(AR) is the number of bounded chambers. If A is the complexification of AR, then Ī£(A)ā¼ Ī£(AR). Hence, Ī²(A) = Ī²(AR).
For any small convex open set U, put
b
U :=U āĪ£(A) =U ā©M(A).
Since U is convex, (U, U ā© Ī£(A)) is homeomorphic to (Cn,Ī£(A
G)), where
G= Min(U) and AG is the central subarrangement defined by
AG:={Hā A | GāH}.
(G might be Cn, in which case A
G = ā .) Hence, Ub is homeomorphic to
M(AG), the complement of a central subarrangement.
The next lemma is well-known.
Lemma 3.6. Suppose U is a small open convex set. Then Ļ1(Ub)is a retract
of Ļ1(M(A)).
Proof. The composition of the two inclusions, U Öb ā M(A) Öā M(AG) is a
By intersecting the elements ofU with M (=CnāĪ£) we get an induced
cover UbofM. An element ofUbis a deleted small convex open set Ub for some
U ā U. Similarly, by intersecting Using withM we get an induced coverUbsing
of a deleted neighborhood of Ī£. The key observation is the following.
Observation 3.7. N(Ub) =N(U) and N(Ubsing) =N(Using).
4
The Mayer-Vietoris spectral sequence
Let X be a space, Ļ = Ļ1(X) and r : Xe ā X the universal cover. Given a
left Ļ-moduleA, define
Cā(X;A) := HomĻ(Cā(Xe), A),
the cochains withlocal coefficients in A. Taking cohomology givesHā(X;A). LetU 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 Ue := {rā1(U)}
UāU
with the same nerve. Suppose that for each simplex Ļ of N, UĻ is connected
and thatĻ1(UĻ)āĻ1(X) is injective. (This means thatrā1(UĻ) is a disjoint
union of copies of the universal cover UeĻ.) There is a Mayer-Vietoris double
complex
Cp,q =
M
ĻāN(p)
Cq(r
ā1(U
Ļ))
(cf. [1, Ā§VII.4]) and a corresponding double cochain complex with local coefficients:
Cp,q(A) := Hom
Ļ(Cp,q;A).
The cohomology of the total complex is Hā
(X;A). Now suppose that for each simplex Ļ of N, UĻ is connected and that Ļ1(UĻ)āĻ1(X) is injective.
(This means that rā1(U
Ļ) is a disjoint union of copies of the universal cover
e
UĻ.) We get a spectral sequence with E1-page
E1p,q =
M
ĻāN(p)
Hq(UĻ;A). (1)
Here Hq(U
Ļ;A) means the cohomology of HomĻ(Cā(rā1(UĻ)), A) or
equiva-lently, of HomĻ1(UĻ)(Cā(UeĻ);A). TheE2-page has the formE
p,q=Hp(N;Hq),
whereHqmeans the functorĻāHq(U
Ļ;A). The spectral sequence converges
to Hā
In the next two sections we will apply this spectral sequence to the case where X is M(A) and the open cover is Ub from the previous section. By Lemma 3.6, Ļ1(UbĻ) ā Ļ1(M(A) is injective so we get a spectral sequence
with E1-page given by (1). Moreover, the Ļ-module A will be such that for
any simplexĻinN(Ubsing),Hq(UbĻ;A) = 0 for allq(even forq= 0) while for a
simplex Ļ of N(Ub) which is not inN(Ubsing), Hq(UĻ;A) = 0 for allq >0 and
is constant (i.e., independent ofĻ) for q= 0. Thus E1p,q will vanish forq >0 andE1ā,0 can be identified with the cochain complexCā(N(U), N(Using)) with
constant coefficients.
5
Generic coefficients
Here we will deal with 1-dimensional local coefficient systems. We begin by considering such local coefficients on S1. LetĪ± be a generator of the infinite
cyclic group Ļ1(S1). Suppose k is a field of characteristic 0 and Ī»ākā. Let
AĪ» be thek[Ļ1(S1)]-module which is a 1-dimensionalk-vector space on which
Ī± acts by multiplication by Ī».
Lemma 5.1. If Ī»6= 1, then Hā
(S1;A
Ī») vanishes identically.
Proof. If S1 has its usual CW structure with one 0-cell and one 1-cell, then
in the chain complex for its universal cover both C0 and C1 are identified
with the group ring k[Ļ1(S1)] and the boundary map with multiplication
by 1āt, where t is the generator of Ļ1(S1). Hence, the coboundary map
C0(S1;A
Ī»)āC1(S1;AĪ») is multiplication by 1āĪ».
Next, consider M(A). Its fundamental group Ļ is generated by loops
aH for H ā A, where the loop aH goes once around the hyperplane H
in the āpositiveā direction. Let Ī±H denote the image of aH in H1(M(A)).
Then H1(M(A)) is free abelian with basis {Ī±H}HāA. So, a homomorphism
H1(M(A))ākāis determined by anA-tuple Ī ā(kā)A, where Ī = (Ī»H)HāA
corresponds to the homomorphism sending Ī±H to Ī»H. Let ĻĪ : Ļ ā kā be
the composition of this homomorphism with the abelianization map Ļ ā
H1(M(A)). The resulting local coefficient system on M(A) is denoted AĪ.
The next lemma follows from Lemma 5.1.
Lemma 5.2. Suppose A is a nonempty central arrangement and Ī is such
Proof. Without loss of generality we can suppose the elements ofAare linear hyperplanes. The Hopf bundle M(A) ā M(A)/S1 is trivial (cf. [6, Prop.
5.1, p. 158]); so,M(A)ā¼=BĆS1, whereB =M(A)/S1. Let i:S1 āM(A)
be inclusion of the fiber. The induced map onH1( ) sendsĪ±toPĪ±H. Thus,
if we pull back AĪ toS1, we get AĪ», where Ī»=
Q
HāAĪ»H. The condition on
Ī isĪ»6= 1, which by Lemma 5.1 implies thatHā(S1;A
Ī») vanishes identically.
By the KĀØunneth Formula Hā
(M(A);AĪ) also vanishes identically.
Returning to the case whereAis a general arrangement, for each simplex
Ļ inN(Ub), let AĻ :=AMin(UĻ) be the corresponding central arrangement (so that UbĻ ā¼=M(AĻ)). Given Īā(kā)A, put
Ī»Ļ :=
Y
HāAĻ
Ī»H.
Call Ī generic if Ī»Ļ 6= 1 for all ĻāN(Using).
Theorem 5.3. (Compare [7, Thm. 4.6, p. 160]). Let A be an affine
ar-rangement of rank l and Ī a generic A-tuple in kā
. Then Hā
(M(A);AĪ) is
concentrated in degree l and
dimkHl(M(A);AĪ) =Ī²(A).
Proof. We have an open cover of Mf(A),{rā1(Ub)}
UāU. By Observation 3.7,
its nerve is N(U). By Lemma 5.2 and the last paragraph of Section 4, the
E1-page of the Mayer-Vietoris spectral sequence is concentrated along the
bottom row where it can be identified withCā
(N(U), N(Using);k). So, theE2
-page is concentrated on the bottom row and E2p,0 = Hp(N(U), N(U
sing);k).
By Lemma 3.2 these groups are nonzero only for p=l and
dimkE2l,0 = dimkHl(N(U), N(Using);k) = Ī²(A).
6
ā
2-cohomology
For a discrete group Ļ, ā2Ļ denotes the Hilbert space of complex-valued, square integrable functions on Ļ. There are unitary Ļ-actions on ā2Ļ by
either left or right multiplication; hence, CĻ acts either from the left or right as an algebra of operators. The associated von Neumann algebra NĻ is the commutant of CĻ (acting from, say, the right on ā2Ļ).
Given a finite CW complex X with fundamental group Ļ, the space of
ā2-cochains on its universal cover Xe is the same asCā
(X;ā2Ļ), the cochains
with local coefficients in ā2Ļ. The image of the coboundary map need not be
closed; hence, Hā
(X;ā2Ļ) need not be a Hilbert space. To remedy this, one
defines the reduced ā2-cohomology Hā
(Xe) to be the quotient of the space of cocycles by the closure of the space of coboundaries. We shall also use the notation Hā
(X;ā2Ļ) for the same space.
The von Neumann algebra admits a trace. Using this, one can attach a ādimension,ā dimNĻV, to any closed, Ļ-stable subspace V of a finite direct
sum of copies of ā2Ļ (it is the trace of orthogonal projection onto V). The
nonnegative real number dimNĻ(Hp(X;ā2Ļ)) is thepthā2-Betti number ofX.
A technical advance of LĀØuck [5, Ch. 6] is the use local coefficients in NĻ
in place of the previous version of ā2-cohomology. He shows there is a
well-defined dimension function on NĻ-modules, Aā dimNĻA, which gives the
same gives the same answer for ā2-Betti numbers, i.e., for each p one has
that dimNĻHp(X;NĻ) = dimNĻHp(X;ā2Ļ). Let T be the class of NĻ
-modules of dimension 0. The dimension function is additive with respect to short exact sequences. This allows one to define ā2-Betti numbers for
spaces more general than finite complexes. The class T is a Serre class of
NĻ-modules [8], which allows one to compute ā2-Betti numbers by working
with spectral sequences modulo T.
Lemma 6.1. Suppose A is a nonempty central arrangement. Then, for all
q ā„ 0, Hq(M(A);NĻ) lies in T. In other words, all ā2-Betti numbers of
M(A) are zero.
Proof. The proof is along the same line as that of Lemma 5.2. It is well-known that the reduced ā2-cohomology of R vanishes. Since M(A) = S1Ć
B, the result follows from the KĀØunneth Formula for ā2-cohomology in [5,
Theorem 6.2. SupposeA is an affine hyperplane arrangement. Then
Hā(M(A);NĻ)ā¼=Hā(N(U), N(Using))ā NĻ (modT)
Hence, for l = rk(A), theā2-Betti numbers ofM(A)vanish except in
dimen-sion l, where dimNĻHl(Mf(A)) =Ī²(A).
Proof. For each Ļ āN(Using), let ĻĻ :=Ļ1(UĻ). By Lemma 6.1,
dimNĻĻH
ā
(M(AĻ);NĻĻ) = 0.
Since the NĻ-moduleHā(M(AĻ),NĻ) is induced from Hā(M(AĻ),NĻ),
dimNĻH
ā
(M(AĻ);NĻ) = dimNĻĻH
ā
(M(AĻ);NĻĻ) = 0.
As in the proof of Theorem 5.3, it follows that the E1-page of the
spec-tral sequence consists of modules in T, except that E1ā,0 is identified with
Cā
(N(U), N(Using))ā N(Ļ). Similarly, the E2-page consists of modules in
T, except that E2ā,0 is identified with Hā
(N(U), N(Using))ā NĻ. 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 Ei,j
ā ā¼=E
i,j
2 (modT). The claim follows
since the filtration of Hā
(M(A);NĻ) given by the Eā-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 ā2-cohomology of Artin groups, J.
London Math. Soc. (2) 68 (2003), 493ā510.
[3] B. Eckmann, Introduction to ā2-methods in topology: reduced ā2
-homology, harmonic chains, ā2-Betti numbers, Israel J. Math. 117
(2000), 183ā219
[4] A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2001.
[5] W. LĀØuck,L2-invariants andK-theory, Springer-Verlag, Berlin and New
[6] P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin and New York, 1992.
[7] V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones Math. 106 (1991), 139ā194.
[8] J.-P. Serre, Groupes dāhomotopie et classes de groupes abĀ“eliens, Ann. Math. 58 (1953) 258ā294.
M. W. Davis mdavis@math.ohio-state.edu
T. Januszkiewicz tjan@math.ohio-state.edu
I. J. Learyleary@math.ohio-state.edu
Department of Mathematics, The Ohio State University, 231 W. 18th Ave.,