Group actions on finite acyclic simplicial complexes

In this paper we develop some homological techniques to obtain fixed points for groups acting on finite Z-acyclic complexes. In particular we show that if a group G acts on a finite 2-dimensional acyclic simplicial complex D, then the fixed point set of G on D is either empty or acyclic. We supply some machinery for determining which of the two cases occurs. The Feit-Thompson Odd Order Theorem is used in obtaining this result.