The Cohomology of the Integer Heisenberg Groups

JOURNAL OF ALGEBRA 184, 230᎐250Ž. 1996 ARTICLE NO. 0258

Soo Teck Lee and Judith A. Packer

Department of Mathematics, National Uni¨ersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore

Communicated by Walter Feit Received November 14, 1995

We give a closed formula for the cohomology groups of the standard integer lattice in the simply-connected Heisenberg Lie group of dimension 2n q 1, n g Zq. We also provide a recursion relation involving n for these cohomology groups. ᮊ 1996 Academic Press, Inc.

0. INTRODUCTION

The Heisenberg Lie group and its discrete subgroups have appeared in many areas of mathematics and mathematical physics, not only serving as a useful source of examples, but also playing a key role in applications to a variety of theoriesŽ listed inwx 7 , for example. . It is our purpose in this paper to study the cohomology of the standard integer lattice ⌫ of the simply-connected Heisenberg Lie group N of dimension 2n q 1 for arbitrary n g Zq. In particular we shall give a closed formula for the cohomology groups of ⌫ with coefficients in the integers. We first became interested in this problem when studying the related problem of computing the ranks of the K-groups of certain C*-algebras associated with ⌫ wx14 , and learned that very recently two papers w 3, 8 x , by A. Dupre´ and R. Howe, respectively, had results from which one could immediately drive formulas for the real cohomology groups H k Ž.⌫, R , kgZq. Inwx 3 , Dupre´ calculates H*Ž.N, R , which by a deep result of G. Mostowwx 15 is isomorphic to H*Ž.⌫, R . Inwx 8 , Howe calculates the Lie algebra cohomology H*Ž.ᒋ, R for the Lie algebra ᒋ of N by considering the representation theory of Sp2 nŽ.C , and by a theorem of K. Nomizuwx 16 , H*Ž.ᒋ,R(H* ŽNr⌫,R ., and since Nr⌫ is a classifying space for ⌫, these groups are isomorphic to H*Ž.⌫, R . Neither of the methods ofwx 3 or wx 8 immediately generalize to the case when the coefficient module is Z, however, and in his paper, Dupre´ noted that the problem of computing H*Ž.⌫,Zwas still an open one. In this paper we shall solve this problem, k giving a closed formula for H Ž.⌫, Z for all k g ZqjÄ40 and for arbitrary 230

0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. COHOMOLOGY OF HEISENBERG GROUPS 231 n g Zq, where 2n q 1 denotes the rank of ⌫. We note here for complete- ness that B. Kostant computed the Lie algebra cohomology of all nilradi- calswx 10a . At this point, we shall mention that because the two-step nilpotent group ⌫ is easily expressed as a central extension of Z2 n by Z, in principle, the Gysin long exact sequence in cohomologywx 9, Theorem 4 has been available for over forty years to allow the computation of the cohomology groups of ⌫. Likewise, in theory, it is possible to apply the results of J. Huebschmannwx 10 on the cohomology rings of two-step nilpotent groups to obtain a ‘‘small’’ free resolution of ⌫, and thus calculate its cohomology ring. However, in practice, one quickly runs into computational difficulties in carrying through either the Gysin sequence or Huebschmann’s algo- rithm, even for groups with a relatively small number of generatorsŽ cf.w 10, paragraph 2, p. 408, and last paragraph, p. 441x. . Indeed, to date many of the calculations of the cohomology of specific nilpotent groups have been done by computerŽ cf. recent work of L. Lambe´ wx 11, 12. , and it has proved very difficult to obtain closed formulas for the cohomology of general families of nilpotent groups. Our way of circumventing this difficulty and obtaining our formulas will be to reduce the problem of computing the cokernels of connecting maps in the Gysin sequence to a problem in combinatorial matrix theory, which will enable us to apply a very elegant result of R. Wilson on diagonal forms of certain incidence matriceswx 18 . From this, we are able to obtain our main result without too much difficulty. In particular, we obtain the following main theorem:

THEOREM 1.8. Let n g Zq and let ⌫ be the integer Heisenberg group of rank 2n q 1. Then the cohomology groups of ⌫ with coefficients in Z ïewed as a triïal module are giën by

2 n 2 n wxkr2 y ¡ ž/ž/ky2jky2jy2 Ž.Zj ,0FkFn, [js0

2n 2n 2n 2n y wŽ.nq1r2x y Ž.n ž/ny2 ž/ž/nq1y2jny1y2j Z [ Ž.Zj , ½5[js1 k HŽ.⌫,Zs~ ksnq1, 2n 2n 2n 2n y wŽ.2nykq2r2x y ž/ž/ky1 kq1 ž/ž/kq2jy2 kq2j Z [ Ž.Zj , ½5[js1 nq2FkF2nq1, ¢0, k G 2n q 2. 232 LEE AND PACKER

Here Z j represents the quotient group ZrjZ so that Z01( Z and Z is the trivial group, and we follow the convention that m0 for j - 0 and ž/j s j)m. In a final section we indicate how our method can be used to calculate n the cohomology groups for a certain subgroup ⌫d of ⌫ of index d where the parameter d represents an arbitrary positive integer. We also derive a recursive formula for the cohomology groups of ⌫.

1. THE GYSIN SEQUENCE FOR THE INTEGER HEISENBERG GROUPS AND COMBINATORIAL MATRIX THEORY

Let ⌫ be the integer lattice in the 2n q 1-dimensional Heisenberg n group N for fixed n g Zq. We will think of ⌫ as ÄŽ.r, s, t : r g Z, s, t g Z 4 with group operation given by

Ž.Ž.Žr,s,tиrЈ,sЈ,tЈsrqrЈq²:t,sЈ,sqsЈ,tqtЈ.Ž..1.1

Here ²:и , и represents the standard inner product on Rnnrestricted to Z , ²:Ј Ýn X ⌫ so that t, s s is1tsii. A matrix representation for is given by the following embedding into SLŽ. n q 2, Z :

1 ttии и tr ¡12 n¦ 1 s1

1 s2 и и Ž.Žr,s,tsr,s1,...,sn,t1,...,tn.ª ` ии `ии 1sn ¢§1

gSLŽ. n q 2,Z .

If one allows the parameters to take on real rather than integer values, the corresponding group is N. It is well known that ⌫ is co-compact in N and that the nilmanifold Nr⌫ is a classifying space for ⌫ wx1. If we denote the center of ⌫ by Z, then Z s wx⌫, ⌫ s ÄŽ.r,0,0: rgZ4 2 n (Z, so that ⌫rZ s Z . Thus ⌫ has the structure of a central extension

2 n 0 ª Z ª ⌫ ª Z ª 0.Ž. 1.2 COHOMOLOGY OF HEISENBERG GROUPS 233

Using the Gysin sequence derived from the Hochschild᎐Serre sequence in group cohomology, one obtains the long exact sequence for H*Ž.⌫, Z w9, Theorem 3x , X

d 6 l 6 Hky22Ž.Znk,Z2 H Ž.Z2nk,Zk H Ž.⌫,Z X X

r6 d6 kHky12Ž.Znk,Z2Hq12 Ž.Ž.Zn,Z,1.3

X and furthermore the differential d2 is given by the cup product with 2 2n ywxcgHŽZ,Z., where wxc is the class of extensionŽ. 1.2wx 9, Theorem 4 . k Ž 2 n . kq2 Ž 2 n . We denote by Lk : H Z , Z ª H Z , Z the map defined by the cup Ž. product with ywxc ; then, since for all k, ker Lk is a subgroup of the finitely generated free abelian group H k ŽZ2 n, Z., the extension defining Hk Ž.⌫,Zsplits and we obtain

H k Ž.⌫, Z coker L ker L .1.4Ž. ( ky2[ ky1 X Using the notation Lk for our map d2 is a deliberate choice, as the map turns out to be related to the Lefshetz decomposition theory for harmonic forms on Kahler¨ manifoldsŽ cf.wx 17, 5. . Since Nr⌫ is a compact orientable manifold of dimension 2n q 1, we k Ž. kŽ. know H ⌫, Z s 0 for k ) 2n q 1, and if we write H ⌫, Z s Fkk[ T , where Fkkis the torsion-free part and T is the torsion subgroup, then

Poincare´ duality shows that Fk s F2 nq1ykkand T s T2nq2ykwx6 . There- k fore, it suffices to compute H Ž.⌫, Z for 0 F k F n q 1. Indeed, the free summands Fk can be immediately obtained from the work of Dupre´ wx 3 , Mostowwx 15 , and the universal coefficient formula, but since our methods produce the rank of Fk with no extra effort, we have chosen to make our proof self-contained. To use the sequenceŽ. 1.3 we first must discuss the ring structure of Ž 2 n . UÄ4 H*Z,Z, which is well known to be isomorphic to H Z e12,...,e n, the exterior algebra on 2n generators over Z. The explicit isomorphism in 1 2 n 2n degree one on H ŽZ , Z.Žs Hom Z , Z.is given by the homomorphism ŽŽ .. ŽŽ .. Ž . Z2 n esiii,tss,esqni,tst,1FiFn, s,tg . Using this corre- spondence, the rank of H jŽZ2n, Z.as a finitely generated free abelian group is exactly2 n , 0 j 2n, and a basis for H jŽ.Z2 n, Z is given by ž/j F F ÄÄ4 4Ä 4 eK :K:1,...,2n and <

Ýn 2ŽZ2nZ. Proof. We first note that wxc s w is1ciixwhere c g Z , is defined by

X X X X X csiŽ.Ž.1,...,sn,t1,...,tn,Ž.s1,...,sn,t1,...,tniists,1FiFn.

To prove the lemma, it is enough to show that wxciinsye ne qi,1FiFn. But this can be easily verified by direct calculation, given the definition of cup product given inwx 1 .

Our problem has been transformed into a problem in exterior algebra k Ä4 involving an analysis of the map from H Z e12,...,e ninto kq2Ä4 Ýn HZ e12,...,e nidefined by the wedge product with s1einn e qi.We kÄ4 now decompose H Z e12,...,e ninto a direct sum of groups in order to make this problem more tractable from the point of view of combinatorial matrix theory. In the subsequent paragraph we adapt some work of A. Weil on exterior algebras over Hermitian vector spaces to our settingw 17, Chap. 1x .

DEFINITION 1.2. Fix n g Zq. We define the pairs of the finite set Ä41, 2, . . . , 2n to be the doubleton sets Ä4i, n q i ,1FiFn. We say a subset C:Ä41, 2, . . . , 2n is pair-free if Ä4i, n q i f C, for all i,1FiFn. By the pigeonhole principle, if C is pair-free, <

Let K be a subset ofÄ4 1, . . . , 2n of order k, and let eK represent the k Ä4 corresponding basis element of H Z e12,...,e n. Then up to multiplication by y1, we can write eK as eC n eJ , where J is the set obtained by taking the union of all pairs inside K, and C s K _ J is pair-free. Thus J can be Ä4 Ä written as a disjoint union J12j J , where J 1: 1,...,n and J2s i q n: 4 igJ11.If <

DCsÄ4jgÄ4Ä41,...,n : j, jqn lCsл .1.5Ž.

Ä4 Note <

ceeeeeиии e e .1.6Ž. JsCnjj11nqnjn 22n jqnjjnn ppn qn COHOMOLOGY OF HEISENBERG GROUPS 235

k Ä 4 kŽ. Let BC s cJ : J : DC , <

n k 2 p rankŽ.V kkŽ.C <

kÄ4 Thus we have decomposed H Z e12,...,e nas a direct sum

wxkr2 kk HZÄ4e12,...,e ns V Ž.C .1.8 Ž . [[p0½5Cpair-free s <

From Eq.Ž. 1.8 , we see that

wŽ.kq2r2x kq2 kq2 HZÄ4e12,...,e ns V Ž.C [[p0½5Cpair-free s <

wxkr2q1 k2 s VqŽ.C [[p1½5Cpair-free s <

k2 [ VqŽ.C Cpair-free[ <

wxkr2 k2 s VqŽ.C[Z,1.9 Ž . [[p0½5Cpair-free s <

k2 k2 n where Z s V qŽ.C has rank 2q , so that Z s 0 for [ Cpair-free ž/k q 2 <

LEMMA 1.3. Let C be a pair-free subset of Ä41,...,2n of order k y 2 p. k Ž. kq2Ž. Then Lk : V C ª V C . 236 LEE AND PACKER

k Proof. We calculate the action of Lk on the basis elements in BC . For Ä4 Jsj1,..., jr :DC,

LcŽ. Le e e иии e e kJs kŽ.Cnjj11n qnjjn n ppn qn

sŽ.eCnejjnenjjnиии n e n e niin Ýe n e n 11q ppqž/q is1 n eee иии e e e e sÝ C njj11nqnjjnn ppn qniin n qn is1 eeeиии e e e e sÝ C njj11nqnjjnn ppn qniin n qn igDC_J

kq2 sÝcJjÄi4gVŽ.C. igDC_J

As an immediate consequence of Lemma 1.3 and the decompositions Ž.1.8 and Ž. 1.9 , it follows that if we let Lk Ž,C .denote the restriction of the k map Lk to V Ž.C , then

wxkr2 kq2 k cokerŽ.Lk s V Ž.C r Lk Ž,C .Ž.V Ž.C [Z, [[p0½5Cpair-free s <

wxkr2 kerŽ.Lk s kerŽ.LkŽ.Ž.,C . 1.11 [[p0½5Cpair-free s <

We thus have reduced our problem further to that of calculating kerŽLk Ž,C ..and coker ŽLk Ž,C ... k k 2 We now consider the map LkŽ.,C:V Ž.C ªV qŽ.C. We first make the crucial observation that upon suitably choosing our bases, the matrix defining this map is a well-known incidence matrix from combinatorics. k Ž. Ä 4 Recall our basis for V C is given by cJ : J : DC , <

kq2 Ž. Ä 4 basis for V C can be indexed by cI : I : DC , <

1, J : I, aI, J s ½ 0, otherwise.

␯ For non-negative integers t k ␯, R. Wilson inwx 18 defined the = F F ž/t ␯ incidence matrix W Ž.␯ Žu .as follows: the rows of W Ž.␯ are ž/k tk s TK tk indexed by the t-subsets T of a ␯-set X, the columns of WtkŽ.␯ are indexed by the k-subsets K of X, and

1, T : K, uT , K s ½ 0, otherwise. Ž. Ž.t Therefore, Lk,C swWnp, pq1 ykq2px, and we can use the results of Wilson to compute the kernel and cokernel of our maps LkŽ.,C. LEMMA 1.4. For 0 F k F n y 1 and 0 F p F wxkr2,

kerŽ.LkŽ.,C s0 and

nykq2 pnykq2p pq1 y ž/ž/pyiq1 pyi cokerŽ.LkŽ.,C ( Ž.Zi . [is0

Here Zi s ZriZ so that Z01( Z and Z is the tri¨ial group. Proof. Since k F n y 1, we have p F Ž.Ž.n y k q 2 p y p q 1 , so that we can apply Theorem 2 ofwx 18 to conclude that LkŽ.,C is equivalent via n k 2 p elementary row and column operations to a y q = n y k q 2 p ž/pq1 ž/p diagonal matrix

b ¡ 1 ¦ b2 и 0 и , и b nykq2 p ž/p ¢§0 238 LEE AND PACKER where the diagonal entries are given by

n y k q 2 pnykq2p bjspq1yi, q1FjF , ž/ž/iy1 i 0FiFp.Ž. 1.12

nykq2p nykq2p Note that 1 occurs as a diagonal entry y times. ž/p ž/py1 From this diagonal matrix one immediately reads off that LkŽ.,C is injective for 0 F k F n y 1Ž this fact has been repeatedly rediscovered; the first recorded proof that we have found is inwx 4. , and also that

n k 2 p y q nykq2 pnykq2p p1 p y ž/q ž/ž/iiy1 cokerŽ.LkŽ.,C sZ Ž.pq1yiZ , [is1 which upon changing the index of summation becomes

nykq2 p nykq2 pq1 y ž/pyiq1 ž/pyi Ž.Zi , [is0 as desired. We now present some conbinatorial formulas which will be of use to us:

LEMMA 1.5. For 0 F k F 2n,

wxkr2 k2p n nykq2p 2n Ý2y s .Ž. 1.13 ž/ky2pž/p ž/k ps0

Proof. We know that

2n rank k Ä4e ,...,e ž/k sH Z 12n

wxkr2 k sÝÝrankŽ.V Ž.C Ž.by Ž 1.8 . ps0Cpair-free <

wxkr2 k2p n nykq2p sÝ2y Ž. byŽ. 1.7 , ž/ky2pž/p ps0

n which gives the resultŽ. note by convention, s0 for k y 2 p)n . ž/ky2p COHOMOLOGY OF HEISENBERG GROUPS 239

LEMMA 1.6. For 0 F k F 2n,

wxkr2 k2p n nykq2p nykq2p Ý2y y ž/ky2pž/pq1 ž/p psy1 2n 2n sy. ž/ž/kq2 k

Proof. We simplify and change the index of summation, keeping in n y k q 2 p mind that s 0: ž/y1

wxkr2 k2p n nykq2p nykq2p Ý2y y ž/ky2pž/pq1 ž/p psy1

wŽ.kq2r2x k22p n nyŽ.kq2q2p sÝ2qy ž/kq2y2pž/p ps0

wxkr2 k2p n nykq2p yÝ2y ž/ky2pž/p ps0 2n 2n sy, by Lemma 1.5. ž/ž/kq2 k

We now are prepared to compute the kernel and cokernel of Lk : k Ä4kÄ4 PROPOSITION 1.7. Let Lk : H Z e12,...,e nªHZe12,...,en be the Ž. ŽÝn . map defined by Lki¨ s ¨ n s1eiin e qn. Then ¡0, 0 F k F n y 1, 2n 2n ker Lk(~ y Ž.1.14 Ž.n ž/ny2 ¢Z , ksn, and

Ž. 2 n 2 n w kq2r2x y ¡ ž/ž/ky2iq2 ky2i Ž.Zi ,0FkFny2, [is0 coker Lk(~ Ž.1.15 Ž. 2n 2n wnq1r2x y ž/ž/ny2iq1 ny1y2i Ž.Zi ,ksny1. ¢[is1 240 LEE AND PACKER

Proof. We use Eqs.Ž.Ž. 1.10 , 1.11 , and Lemma 1.6. For k F n y 1, Lemma 1.4 shows that LkŽ.,C is injective so that by Eq. Ž. 1.11 Lk is injective. For k s n,

wxnr2 ker Lns kerŽ.LnŽ.,C [[p0½5Cpair-free s <

wxnr2 t ( kerž/Wp,pq1Ž.2 p [[ps0½5Cpair-free <

wxnr2 n2p n t2y ker W 2 p ž/ny2p. ( ž/p,pq1Ž. [ps0

Ž.t One easily checks by taking complements in a ␯-set that wWt, k ␯ xs Ž.␯ Ž.t Ž. W␯yk, ␯ytp, so that wW ,pq12 p xs Wpy1, p 2 p , whose kernel has rank 2 p 2 p p y, by eitherwxwx 18, Theorem 2 or 4, Corollary 1 . Therefore ž/ ž/py1

wxnr2 2p ny2p n 2p rankŽ. ker Lns Ý2 y ž/ny2pž/p ž/py1 ps0

wŽ.ny2r2x n22p n sÝ2yy ž/ny2y2p psy1

n y Ž.n y 2 q 2 p nyŽ.ny2q2p = y ž/pq1 ž/p

2n2n sy , ž/n ž/ny2 by Lemma 1.6, completing the proof of Eq.Ž. 1.14 . For k F n y 2, by Eq.Ž. 1.1 we have

wxkr2 cokerŽ.Lk s coker LkŽ.,C [Z, [[p0½5Cpair-free s <

nykq2pnykq2p wxkr2 pq1 y ž/ž/pyiq1 pyi cokerŽ.Lkis Ž.Z [ Z [[ps0Cpair-free ½5 [is0 <

n nykq2pnykq2p wxkr2pq1 y k2pž/ky2p ž/ž/pyiq1 pyi ¡Ž.Z2y ¦ ([[~¥i ps0¢§is0

n ž/kq2 2kq2 [Z0

n nykq2pnykq2p wxk2 p1 y r q ky2p pi1 pi ¡ 2ky2pž/ž/ž/yq y ¦ Ž.Z , ([[~¥i psy1¢§is0 which, upon interchanging the order of summation, becomes

n nykq2 pnykq2p k21 k2 wxrq wxr k2p pi1y pi ¡ ky2pž/y ž/ž/yq y ¦ Ž.Z2 . [[~¥i is0¢§psiy1

We thus want to simplify the sum

wxkr2 k2p n nykq2pnykq2p Ý2y y .Ž. 1.16 ž/ky2pž/ž/pyiq1 pyi psiy1 Upon making the substitutions pЈ s p y i and kЈ s k y 2i, the expres- sionŽ. 1.16 becomes

wxkЈr2 kЈ2pЈ n nykЈq2pЈ nykЈq2pЈ Ý2y y , ž/kЈy2pЈž/ž/pЈq1 pЈ pЈsy1 which, by Lemma 1.6 is exactly 2n 2n 2n 2n ys y . ž/ž/žkЈq2 kЈ ky2iq2 /ž/ky2i Therefore, for 0 F k F n y 2, we see that

Ž. 2 n 2 n w kq2r2x y ž/ž/ky2iq2 ky2i cokerŽ.Lkis wxZ . [is0 242 LEE AND PACKER

For k s n y 1, the summand Z appearing in Eq.Ž. 1.10 is 0, and we get upon calculation

n 2 pq12pq1 wŽ.n12xp1 y yr q ny1y2p pi1 pi ¡2ny1q2pž/ž/ž/yq y ¦ cokerŽ.L Z , ny1 s [[~¥wxi ps0¢§is0 which is the same as

n 2 pq12pq1 wŽ.n12xp1 y yrq ny1y2p pi1 pi 2ny1q2pž/ž/ž/yq y wxZi ; p[[sy1 is0 and by exactly the same reasoning as for the case k - n y 1, this becomes

Ž. 2 n 2 n Ž. 2 n 2 n w nq1r2xwyynq1r2x ž/ž/ny2iq1 ny1y2in ž/ž/y2iq1ny1y2i Ž.ZiisŽ.Z, [[is0is1

2n2n since ys0. This completes the proof of Eq.Ž. 1.15 and ž/ž/nq1 ny1 hence of Proposition 1.7. We can now deduce our main theorem almost immediately.

THEOREM 1.8. Let n g Zq and let ⌫ denote the 2n q 1 integer Heisen- berg group. Then the cohomology group of ⌫ with coefficients in the integers are gi¨en by

2 n 2 n wxkr2 y ¡ ž/ž/ky2jky2jy2 Ž.Zj , [js0 0FkFn, 2n 2n 2n 2n y wŽ.nq1r2x y Ž.n ž/ny2 ž/ž/nq1y2jny1y2j Z [ Ž.Zj , k ½5[js1 HŽ.⌫,Zs~ ksnq1, 2n 2n 2n 2n y wŽ.2nykq2r2x y ž/ž/ky1 kq1 ž/ž/kq2jy2 kq2j Z [ Ž.Zj , ½5[js1 nq2FkF2nq1, ¢0, k G 2n q 2.

kŽ.⌫ Z Ž. Proof. Since H , ( coker Lky2 [ ker Lky1 by Eq. 1.4 , for k F n q 1, we can obtain the result directly from Proposition 1.7. The formulas COHOMOLOGY OF HEISENBERG GROUPS 243 for n q 2 F k F 2n q 1 then follow by Poincare´ duality, as mentioned earlier.

COROLLARY 1.9. Let n g Zq, and let ⌫ denote the rank 2n q 1 integer Heisenberg group. Then for k F n, any integer k-cocycle on ⌫ is cohomolo- 2n gous to a k-cocycle lifted from the quotient group ⌫rwx⌫, ⌫ ( Z .

Proof. For k F n, we have seen in Proposition 1.7 that ker Lky1 s 0, k Ž.⌫ Z Ž . Ž. so that H , ( coker Lky2 and thus by the Gysin sequence 1.3 k Ž 2 n . consists of elements in the image of the inflation map lk : H Z , Z ª Hk Ž.⌫,Z.

Remark 1.10. We have included the trivial group summand Z1 in our formulas, because its exponent will appear in some of our generalized formulas in the next section.

2. APPLICATIONS AND OBSERVATIONS

We indicate now how our methods can also be used to calculate the cohomology groups for certain lattices in the Heisenberg Lie groups. Fix q ÄŽ. dgZ, and let ⌫d denote the subgroup of ⌫ defined by r, s, dt : r g Z, n4 ŽŽ.. s,tgZ ⌫ddwas denoted by Hd,...,d inwx 13 . ⌫ is a subgroup of ⌫ ^`_n times of index d n. The same methods used to compute the cohomology of ⌫ can be used to compute the cohomology of ⌫dd. In particular, ⌫ is again a central extension of Z2 n by Z, and one easily calculates that the class of 2 Ä4 Ýn this extension in H Z e12,...,e niis given by y s1deinn e qis ŽÝn . ydis1 einneqi. Thus if we use the Gysin exact sequence to compute k HŽ.⌫d,Zas in Eq. Ž. 1.3 , we obtain

k ⌫ Z H Ž.dk, s coker ŽdL y2 .[ ker ŽdLky1 ..2.1 Ž.

We can again use the same combinatorial approach, obtaining

wxkr2 cokerŽ.dLk s cokerŽ.dLŽ k, C .[ Z,2.2 Ž. [[p0½5Cpair-free s <

wxkr2 kerŽ.dLk s kerŽ.LkŽ.,C ,2.3Ž. [[p0½5Cpair-free s <

k 2 Z s V q Ž.C .2.4Ž. Cpair-free[ <

It is easy to check that for 0 F k F n y 1 and 0 F p F wxkr2,

nykq2pnykq2p pq1 y ž/ž/pyiq1 pyi cokerŽ.dLŽ. k, C ( Ž.Z di , [is0 so that

Ž. 2 n 2 n w kq2r2x y ¡ ž/ž/ky2iq2 ky2i ÝŽ.Zdi ,0FkFny2, is0 cokerŽ.dLk(~ Ž. 2n 2n wnq1r2x y ž/ž/nq1y2iny1y2i ÝŽ.Zdi ,ksny1 ¢ is1 and we obtain from Eq.Ž. 2.1 the following result, which generalizes Example 7.2 ofwx 10 :

q n THEOREM 2.1. Let n g Z and let ⌫d be the subgroup of index d of the integer Heisenberg group ⌫ of rank 2n q 1 as defined in the preceding paragraph. Then the cohomology groups of ⌫d with coefficients in the integers are gi¨en by

2 n 2 n wxkr2 y ¡ ž/ž/ky2iky2iy2 Ž.Zdi , [is0 0FkFn, 2n 2n Ž.n12 2n 2n Ž.n y w qrx y ž/ny2 ž/ž/nq1y2iny1y2i Z [ Ž.Zdi , k ½5[is1 HŽ.⌫d,Zs~ ksnq1, 2n 2n Ž. 2n 2n y w 2nykq2r2x y ž/ž/ky1 kq1 ž/ž/kq2iy2 kq2i Z [ Ž.Zdi , ½5[is1 nq2FkF2nq1, ¢0, k G 2n q 2. COHOMOLOGY OF HEISENBERG GROUPS 245

Knowing the cohomology groups for ⌫d with coefficients in Z and Poincare´ duality will allow us to easily compute the cohomology groups for ⌫ddwith coefficients in any trivial ⌫ -module M by means of the universal coefficient theoremwx 1, p. 60, Exercise 3 . As an example, we now give formulas for these groups for the case M s T, the circle group. For locally compact groups G, the cohomology groups HG2 Ž.,Tand HG3Ž.,Thave proved important in both the theory of group representations and C*-alge- braswx 13, 2 :

COROLLARY 2.2. The cohomology groups of ⌫d with coefficients in T are gi¨en by

2 n 2 n Ž.k12 2 n 2 n ¡ y w qrx y ž/žkky2 / ž/ž/kq1y2iky1y2i T [ Ž.Zdi , ½5[is1 0FkFn, 2n2n 2n2n y wŽ.ny1r2x Ž.n ž/ny2 ž/ž/n12iny 32i T Ž.Zyy yy , k [ di H⌫,T ½5[is1 Ž.d s~ ksnq1, 2n 2n Ž. 2n 2n y w 2nykq1r2x y ž/ž/ky1 kq1 ž/ž/kq2iy1 kq1q2i T m Ž.Zdi , ½5[is1 nq2FkF2nq1, ¢0, k G 2n q 2.

Proof. The result follows immediately from Theorem 2.1 and the universal coefficient theorem. We note that for k s 2, this result is a special case of Theorem 2.11 ofwx 13 . For our final remarks we return to the case d s 1 and consider the problem of how the groups H*Ž.⌫, Z change as the rank 2n q 1 increases. We thank Jon Berrick for pointing out to us that this matter was of interest, and also for noting that it might be possible to obtain our formulas for the cohomology groups of ⌫ through induction on n. For this point we use the notation ⌫Ž.n to denote the integer Heisenberg group of rank 2n q 1. For n G 2 the groups ⌫Ž.n can be written as a semidirect product

⌫Ž.n ( Ž.⌫ Žn y 1 .= Z i Z,2.5Ž. 246 LEE AND PACKER where the isomorphism is given by

Ž.r, s1,...,sny1,sn,t1,...,tny1,tn

ªŽ.Ž.Ž.r,s1,...,sny11,t ,...,tny1,snn,t , and the action of Z on the direct product ⌫Ž.n y 1 = Z is generated by automorphism ␣: ⌫Ž.n y 1 = Z ª ⌫ Ž.n y 1 = Z defined by

␣ Ž.Ž.r, s1,...,sny11,t ,...,tny1,snnsŽ.rqs ,s1,...,sny11,t ,...,tny1. Ž.2.6

In the usual manner, the decompositionŽ. 2.5 implies that H*ŽŽ..⌫ n , Z fits into a short exact sequence

1 k 1 k 0 ª H Ž.Z, H y Ž.Ž.⌫Ž.n y 1 = Z, Z ª H ⌫Ž.n , Z 0k ªHŽ.Z,HŽ.⌫Ž.ny1=Z,Zª0,Ž. 2.7 but in principle, the groups H*ŽŽ.⌫ n , Z .will be determined only up to an extension by the sequenceŽ. 2.7 . On the other hand, through direct calculation, we have been able to verify the recursion relations

k 2 k 1 2 ¡H y Ž.Ž.⌫Ž.n y 1,Z [ Hy ⌫ Ž.ny1,Z k [HŽ.⌫Ž.ny1,Z, 0FkFny1, n 2 n 1 2 k H y Ž.Ž.⌫Ž.n 1,Z Hy ⌫ Ž.n 1,Z HŽ.⌫Ž.n,Zs~y [ y [TnŽ.⌫Ž.ny1, ksn, nn2 1 HŽ.Ž.⌫Ž.ny1,Z [Hq⌫ Ž.ny1,Z [Qn Ž., ¢ksnq1 Ž.2.8

k Žwe use the convention that H s 0 for k - 0. where as in Section 1, ŽŽ .. nŽŽ . . Tn ⌫ny1 represents the torsion subgroup of H ⌫ n y 1,Z, and QnŽ.is defined by

2Ž.ny12 Ž.ny1 wŽ.n12x y qr ž/ž/nq1y2jny1y2j QnŽ.s Zj .2.9Ž. [js1 COHOMOLOGY OF HEISENBERG GROUPS 247

We briefly comment on the relationshp betweenŽ. 2.7 and Ž. 2.8 . The exact sequenceŽ. 2.7 can be written as

␣ k ⌫ Z ␣ 0 ª coker Ž.ky1* y Id ª H Ž.Ž.n , ª ker Ž.k* y Id ª 0, Ž.2.10

qŽ. kŽŽ . . kŽŽ where for each k g Z , ␣k * y Id: H ⌫ n y 1 = Z, Z ª H ⌫ n y 1..=Z,Zdenotes the map Ž.␣ * y identity, where ␣ is the automorphism of ⌫Ž.n y 1 = Z defined in Eq. Ž. 2.6 . By the Kunneth¨ formula there are isomorphisms

k 11k0( 6 k Hy ŽŽ⌫ny1, .Z .mH ŽZ,Z .mH ŽŽ⌫ ny1, .Z .mH ŽZ,Z .H ŽŽ⌫ny1 .=Z,Z . 55

k1kk␬qi6 HyŽŽ⌫ny1, .Z .[H ŽŽ⌫ ny1, .Z .H Ž⌫ Žny1 .=Z,Z ., Ž.2.11

k k where i: H ŽŽ⌫ n y 1, .Z .ªH ŽŽ⌫ ny1 .=Z,Z .is the inflation map, and

k 1 k ␬ : H y Ž.Ž.⌫Ž.n y 1,Z ªH ⌫ Ž.ny1 =Z,Z is defined on cocycles by

␬ ␥ ␥ ␥ ␥ ␥ и Ž.Žf Ž.12, q .Ž, 22, q .,..., Žkk,q .sf Ž1,..., ky1 .qn, ky1ŽŽ . . Ž . Ž . for f g Z ⌫ n y 1,Z, ␥ii,q g⌫ ny1 =Z, and 1 F i F k. For k kFny1, Corollary 1.9 shows that all elements of H ŽŽ⌫ n y 1, .Z .are inflated from elements of Z k ŽZ2Ž ny1., Z., and thus one easily calculates U that ␣k y Id is the zero map for k F n y 1, which shows that

U U kerŽ.␣kky Id s coker Ž.␣ y Id k1 k (HyŽ.Ž.⌫Ž.ny1,Z [H ⌫ Ž.ny1,Z, kFny1, byŽ. 2.11 . Thus our recursion formulaŽ. 2.8 for k F n y 1 and the fact that Ž U . 2Ž ny1. elements in ker ␣k y Id , being inflated from Z = Z, can easily be lifted to cocycles on ⌫Ž.n , imply that the exact sequenceŽ. 2.10 splits for k F n y 1. A similar analysis, whose details we omit, shows that ␣ ny2 ⌫ Z ny1 ⌫ Z cokerŽ.Ž.ny1* y Id ( H Ž.Ž.Ž.n y 1, [H Ž.ny1, and

ny1 kerŽ.Ž.␣nn* y Id ( H Ž.Ž.⌫Ž.n y 1,Z [T ⌫ Ž.ny1, so thatŽ. 2.10 also splits for k s n. 248 LEE AND PACKER

For k s n q 1, the situation is somewhat more complicated, but we briefly indicate the reasoning which led to the recursive formulaŽ. 2.8 in n 1 this case. By Corollary 1.9, every element of H y ŽŽ⌫ n y 1, .Z .is inflated 2Žn1. from an Ž.n y 1 cocycle on Z y; thus one can easily calculate that Ž. Ž. Žny1ŽŽ . .. nŽŽ . . ␣n *s␣* acts on ␬ H ⌫ n y 1,Z :H ⌫ ny1 =Z,Z as the U Ž ny1ŽŽ . .. identity, so that ␣n y Id ' 0on␬ H ⌫ny1,Z . By Theorem 1.8, nŽŽ⌫ . . ŽŽ⌫ .. ŽŽ⌫ .. H ny1,Z (Tnnny1 [ker L y1, where Tnn y 1 s ny2ŽZ2Žny1.Z.nŽZ2Žny1.Z. coker Lny2 for Lny2 : H , ª H , , and Lny1: ny1Ž 2Ž ny1. . nq1Ž 2Ž ny1. .ŽŽ.. HZ,ZªHZ,Z. As before, elements of Tn ⌫ n y 1 nŽ 2Ž ny1. .Ž. are inflated from elements of H Z , Z so that ␣n * y Id is identi- ŽŽ .. cally zero restricted to Tn ⌫ n y 1 . On the other hand, given ␻ g ␶ ker Lny1, the proof ofwx 9, Theorem 3 provides a splitting : ker Lny1 ª n HŽŽ⌫ny1, .Z .defined on cocycles by

␶␻Ž.ŽŽ.r11,¨ .Ž,r 22,¨ .,..., Žrnn,¨ .sr12␻ Ž¨ ,...,¨n .,

ny1Ž2Žny1.. 2Žny1. for ␻ g Z Z , Z , riig Z, ¨ g Z ,1FiFn, and one calculates that

␣ ␶␻ ␬ ␻ ␬ ny1 ⌫ Z Ž.Žnn .* y Id Ž. Ž.s Ž.ly1 Ž.g Ž.H Ž.Ž.ny1, n ;HŽ.⌫Ž.ny1=Z,Z,

ny1ŽZ2Žny1.Z.ny1ŽŽ⌫ .Z . where lny1: H , ª H n y 1, is the inflation map defined in Eq.Ž. 1.3 . So using the isomorphism

n ny1 H Ž.Ž.⌫Ž.n y 1 = Z, Z ( H ⌫Ž.n y 1,Z [TnŽ.⌫ Ž.ny1

[ker Lny1 , we have

␣ ␻ ␻ Ž.Ž.nn* y Id Ž. Žf , t, .s Žl y1 Ž.,0,0 . .

Ž. Ž. Since im Lny3 l ker Lny1 s 0, we see that

ny1 kerŽ.Ž.␣nn* y Id s H Ž.Ž.⌫Ž.n y 1,Z [T ⌫ Ž.ny1

and

␣ ny1 ⌫ Z cokerŽ.Ž.nn* y Id ( H Ž.Ž.Ž.n y 1, rly1ker Lny1 n [HŽ.⌫Ž.ny1,Z. COHOMOLOGY OF HEISENBERG GROUPS 249

ny1ŽŽ⌫ .Z . ny1ŽZ2Žny1.Z.Ž . By Theorem 1.8, H n y 1, (H , rim Lny3 . Hence ny1ŽŽ⌫ .Z . the map lny1: ker Lny1 ª H n y 1, is one-to-one, and maps n 1 onto a free subgroup of H y ŽŽ⌫ n y 1, .Z ., of rank

2Ž.n y 12 Ž.ny12 Ž.ny12 Ž.ny1 ys y ž/ž/ž/ž/ny1 nq1 ny1 ny3 n 1 srankŽ.H y Ž.⌫Ž.n y 1.

ny1ŽŽ⌫ .Z . Ž . Thus H n y 1, rlny1 ker Lny1 is a torsion group, and the example n s 3 shows that this group is non-trivial in general. Write ny1ŽŽ⌫ .Z . Ž . ŽŽ⌫ .. H ny1, rlny1 ker Lny1 s Eny1 n y 1 . One similarly calculates that

␣ n ⌫ Z ⌫ kerŽ.Ž.nq1* y Id ( H Ž.Ž.Ž.Ž.n y 1, [Tnq1ny1 ⌫ [Gnq1Ž.Ž.ny1, where

⌫ ␶ ␶ Gnq1Ž.Ž.n y 1 ( Ž.ker Ž.Ž.ŽLnnl im L y2: ker Ln . n2 :HqŽ.⌫Ž.ny1,Z.

ŽŽ⌫ .. Since Gnq1 n y 1 is a free group whose rank is equal to the rank of n 1 Hq ŽŽ⌫ny1, .Z ., we see that

␣ n ⌫ Z nq1 ⌫ Z kerŽ.Ž.nq1* y Id ( H Ž.Ž.Ž.Ž.n y 1, [H ny1, and thus the sequenceŽ. 2.10 for k s n q 1 becomes

⌫ n ⌫ Z nq1 ⌫ Z 0 ª Eny1Ž.Ž.Ž.Ž.n y 1 [ H Ž.n y 1, ªH Ž.ny1, n n 1 ªHŽ.Ž.⌫Ž.ny1,Z [Hq ⌫ Ž.ny1,Z ª0.Ž. 2.12

Because of our recursion relation for H nq1ŽŽ.⌫ n , Z ., and because the n n 1 torsion subgroup of H ŽŽ⌫ n y 1, .Z .[H qŽŽ⌫ ny1, .Z .again corre- sponds to cocycles inflated from Z2Ž ny1. = Z, we conjecture that the Ž. ŽŽ..Ž.⌫ Ž. sequence 2.12 splits and Eny1 n y 1 ( Qn, where Qn is as inŽ. 2.9 .

ACKNOWLEDGMENT

We would like to thank Professor Jon Berrick for useful conversations. 250 LEE AND PACKER

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