The \(K(n)\)-Euler characteristic of extraspecial \(p\)-groups
Let \(E\) be a finite \(p\)-group which occurs in a central extension \(1 \to N \to E \to V \to 1\) where \(N\) is cyclic of order \(p\) and \(V\) is an elementary abelian group of order \(p^{2m}\). \(E\) is an extraspecial group if \(Z(E)\) is of order \(p\). The cohomology of such groups is known if \(p = 2\) [\textit{D. Quillen}, Math. Ann. 194, 197-212 (1971; Zbl 0225.55015)] but not for odd primes. The paper under review calculates the Morava \(K\)-theory Euler characteristic, \(\chi=\text{rank}(K(n)^{\text{even}}(BE)) - \text{rank} (K(n)^{\text{odd}} (BE))\), of such a group for \(p\) odd, giving a computable formula in terms of the number of \(B\)-isotropic subspaces of an \({\mathbb F}_p\) vector space where \(B\) is a nondegenerate alternating form. The author suggests that this result will lead to information about the mod \(p\) cohomology of \(E\) via the Atiyah-Hirzebruch spectral sequence.
scientific article; zbMATH DE number 1563006
scientific article