Some remarks on finite 1-acyclic and collapsible complexes (Q1317459)

The author's summary: ``Let \(G\) be a finite group. Assume that \(G\) acts (simplicially) on a finite simplicial complex \(D\). We show that if \(\dim(D)=2\) and \(D\) is collapsible, \(G\) fixes a point of \(| D|\). We show that if \(G\) has no composition factor of Lie-type and Lie-rank 1, or the sporadic \(J_ 1\), \(\dim(D)=3\) and \(D\) is collapsible, \(G\) fixes a point of \(| D|\). In addition we obtain various results on collapsible complexes and a certain tree decomposition of a finite connected simplicial complex \(D\), such that \(H_ 1(D)=0\).