JOURNAL OF ALGEBRA 184, 230]250Ž. 1996 ARTICLE NO. 0258 The Cohomology of the Integer Heisenberg Groups 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. Q 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 G of the simply-connected Heisenberg Lie group N of dimension 2n q 1 for arbi- trary n g Zq. In particular we shall give a closed formula for the cohomol- ogy groups of G 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 G 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.G, R , kgZq. Inwx 3 , Dupre calculates H*Ž.N, R , which by a deep result of G. Mostowwx 15 is isomorphic to H*Ž.G, R . Inwx 8 , Howe calculates the Lie algebra cohomology H*Ž.n, R for the Lie algebra n of N by considering the representation theory of Sp2 nŽ.C , and by a theorem of K. Nomizuwx 16 , H*Ž.n,R(H* ŽNrG,R ., and since NrG is a classifying space for G, these groups are isomorphic to H*Ž.G, 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.G,Zwas still an open one. In this paper we shall solve this problem, k giving a closed formula for H Ž.G, Z for all k g ZqjÄ40 and for arbitrary 230 0021-8693r96 $18.00 Copyright Q 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 G. 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 G 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 G. 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 G, 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 G be the integer Heisenberg group of rank 2n q 1. Then the cohomology groups of G with coefficients in Z ¨iewed as a tri¨ial module are gi¨en 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.G,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 Gd of G of index d where the parameter d represents an arbitrary positive integer. We also derive a recursive formula for the cohomology groups of G. 1. THE GYSIN SEQUENCE FOR THE INTEGER HEISENBERG GROUPS AND COMBINATORIAL MATRIX THEORY Let G be the integer lattice in the 2n q 1-dimensional Heisenberg n group N for fixed n g Zq. We will think of G as ÄŽ.r, s, t : r g Z, s, t g Z 4 with group operation given by Ž.Ž.Žr,s,t?r9,s9,t9srqr9q²:t,s9,sqs9,tqt9.Ž..1.1 Here ²:? , ? represents the standard inner product on Rnnrestricted to Z , ²:9 Ýn X G so that t, s s is1tsii. A matrix representation for is given by the following embedding into SLŽ. n q 2, Z : 1 tt?? ? tr ¡12 n¦ 1 s1 1 s2 ? ? Ž.Žr,s,tsr,s1,...,sn,t1,...,tn.ª ` ?? `?? 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 G is co-compact in N and that the nilmanifold NrG is a classifying space for G wx1. If we denote the center of G by Z, then Z s wxG, G s ÄŽ.r,0,0: rgZ4 2 n (Z, so that GrZ s Z . Thus G has the structure of a central extension 2 n 0 ª Z ª G 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.G, Z w9, Theorem 3x , X d 6 l 6 Hky22Ž.Znk,Z2 H Ž.Z2nk,Zk H Ž.G,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.G,Zsplits and we obtain H k Ž.G, 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 KahlerÈ manifoldsŽ cf.wx 17, 5. Since NrG is a compact orientable manifold of dimension 2n q 1, we k Ž. kŽ. know H G, Z s 0 for k ) 2n q 1, and if we write H G, 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.G, 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 <<K s j , where if K s k12- k - ??? - k j: Ä41,...,2n, e represents the product e ??? e jÄ4e ,...,e Kk1n n kjg H Z12n( HjŽZ2n,Z.Ž.