The Euler class of a subset complex

Abstract

The subset complex Δ(G) of a finite group G is defined as the simplicial complex whose simplices are non-empty subsets of G. The oriented chain complex of Δ(G) gives a G-module extension of by , where is a copy of integers on which G acts via the sign representation of the regular representation. The extension class ζG ∈ ExtGG-1 (, ) of this extension is called the Ext class or the Euler class of the subset complex Δ (G). This class was first introduced by Reiner and Webb [The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), 291-327] who also raised the following question: What are the finite groups for which ζG is non-zero?In this paper, we answer this question completely. We show that ζG is non-zero if and only if G is an elementary abelian p-group or G is isomorphic to /9, /4 × /4 or (/2)n × /4 for some integer n ≥ 0. We obtain this result by first showing that ζG is zero when G is a non-abelian group, then by calculating ζG for specific abelian groups. The key ingredient in the proof is an observation by Mandell which says that the Ext class of the subset complex Δ (G) is equal to the (twisted) Euler class of the augmentation module of the regular representation of G.We also give some applications of our results to group cohomology, to filtrations of modules and to the existence of Borsuk-Ulam type theorems. © 2008. Published by Oxford University Press. All rights reserved.