\(K\)-theory for the integer Heisenberg groups (Q1283356)

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 \textit{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 \(K^*(N/\Gamma) \cong \bigoplus_{k=0}^{2n+1} H^k(N/\Gamma; \mathbb{Z})\).