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Zbl 0917.19003
Aslaksen, Helmer; Lee, Soo Teck; Packer, Judith
$K$-theory for the integer Heisenberg groups. (English)
[J] K-Theory 16, No.3, 201-227 (1999). ISSN 0920-3036; ISSN 1573-0514/e

Let $\Gamma$ be the standard integral lattice of the $(2n+1)$-dimensional simply connected Heisenberg Lie group $N$. The homogeneous space $N/\Gamma$ fits into a fibration sequence $T^1 \to N/\Gamma \to T^{2n}$, where $T^k$ denotes the $k$-dimensional torus. The authors use the Gysin sequence in topological $K$-theory to determine the graded group $K^*(N/\Gamma)$. The difficulty, of course, is to compute the connecting homomorphisms. The authors show that these can be identified with certain incidence matrices that show up in combinatorics [e.g., in the work of {\it R. M. Wilson}, Eur. J. Comb. 11, No. 6, 609-615 (1990; Zbl 0747.05016)]. As a major step in their calculation, the authors find diagonalizations of these incidence matrices. This leads to an explicit description of $K^*(N/\Gamma)$ as well as to an abstract isomorphism $K^*(N/\Gamma) \cong \bigoplus_{k=0}^{2n+1} H^k(N/\Gamma; \bbfZ)$.
[Michael Joachim (Münster)]

MSC 2000:: *19L64 Computations, geometric appl. of K-theory
05B20 (0,1)-matrices (combinatorics)
55R20 Spectral sequences and homology of fiber spaces
20F18 Nilpotent groups

Keywords: Heisenberg group; incidence matrices; Gysin sequence; homogeneous space; diagonalizations

Citations: Zbl 0747.05016

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