Local control of group cohomology
A theorem of P. Webb states: If \(G\) is a finite group acting simplicially on a finite simplicial complex, \(\Delta\), in such a way that any element of \(G\) which stabilizes a cell fixes it pointwise, then the following conditions are equivalent for a fixed prime \(p\): (i) For all subgroups \(H\) of \(G\) with \(p\) dividing \(|H|\) and \(H/O_p (H)\) cyclic, \(\chi (\Delta^H)=1\); (ii) For all finitely generated \(\mathbb{Z} G\)-modules \(M\) and integers \(n\), \[ \widehat {H}^n (G, M)_p= \sum_{\sigma\in \Delta/G} (-1)^{\dim \sigma} \widehat {H}^n (G_\sigma, M)_p. \] [This is an equality of elements of the Grothendieck group of finite \(p\)-groups with respect to direct sum decompositions.] In the present paper, the author proves that under certain conditions (ii) may hold for the special case \(M= \mathbb{Z}\) even when (i) fails to hold. He applies these results to the Mathieu group, \(M_{12}\), and the Hall-Janko-Wales group, \(J_2\), acting on appropriate complexes. The method is to prove first, under appropriate hypotheses, an isomorphism of Tate cohomology and equivariant cohomology with trivial integer coefficients. The spectral sequence for equivariant cohomology is then analysed to obtain the required results.
finite group acting simplicially on finite simplicial complex
finitely generated \(\mathbb{Z} G\)-modules
scientific article; zbMATH DE number 810248