Title: The Euler Class of a Subset Complex

Authors: Asl\i \ G\"{u}\c{c}l\"{u}kan and Erg\"{u}n Yal\c{c}\i n

Address: Asli G\" u\c cl\" ukan,
Department of Mathematics,
Bilkent University,
Bilkent, Ankara, Turkey. 

Current Address (until May 2008): University of Rochester,
Department of Mathematics, Rochester, NY, USA


Address: Erg\" un Yal\c c\i n, 
Department of Mathematics,
Bilkent University,
Bilkent, Ankara, Turkey. 

Current Address (until August 2008): 
McMaster University,
Department of Mathematics and Statistics,
Hamilton, ON, Canada.

Status of the paper: Submitted.

Abstract:
The subset complex $\Delta(G)$ of a finite group $G$ is defined as
the simplicial complex whose simplices are nonempty subsets of
$G$. The oriented chain complex of $\Delta(G)$ gives a
$\mathbb{Z}G$-module extension of $\mathbb{Z}$ by
$\tilde{\mathbb{Z}}$ where $\tilde{\mathbb{Z}}$ is a copy of
integers on which $G$ acts via the sign representation of the
regular representation. The extension class $\zeta_{G} \in
\mathrm{Ext}_{\mathbb{Z}G}^{|G|-1}(\mathbb{Z}, \tilde{\mathbb{Z}})$
of this extension is called the Ext class or the Euler class of the
subset complex $\Delta (G)$. This class was first introduced by Reiner and Webb
\cite{Reiner-Webb} who also raised the following question: What are
the finite groups for which $\zeta_{G}$ is nonzero?

In this paper, we answer this question completely. We show that
$\zeta_G$ is nonzero if and only if $G$ is an
elementary abelian $p$-group or $G$ is isomorphic to $\ZZ /9$,
$\mathbb{Z}/4 \times \mathbb{Z}/4$, or $(\mathbb{Z}/2)^n \times
\mathbb{Z}/4$ for some integer $n\geq 0$.
We obtain this result by first showing that $\zeta _G$ is zero
when $G$ is a nonabelian group, then by calculating $\zeta _G$
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 $\Delta (G)$ 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.