The equivariant category of proper \(G\)-spaces. (Q1414911)

Let \(G\) be a topological group and \(X\) a \(G\)-space. There is a far developed equivariant version of Lusternik--Schnirelmann theory for the case of \(G\) being compact, but almost nothing is done for \(G\) non-compact. The authors develop the corresponding theory for an interesting class of \(G\)-spaces without the assumption of compactness, in fact, for \(G\)-spaces with proper actions. Two of the main results (Theorems 4.2 and 4.4) are the following; \(\text{cat}(X/G)=n+1\) if: (a) \(X\) is an acyclic \(n\)-manifold, \(n\geq 3\) and \(G\) is a discrete group acting freely and properly with \(X/G\) compact; (b) \(X\) is a (co)homology \(n\)-sphere and \(G\) is a finite group acting freely on \(X\).