Some aspects of equivariant LS-category (Q891255)

The Lusternik-Schnirelmann category (or LS-category, for short) of a space \(X,\) \(\text{cat}(X),\) is the least nonnegative integer \(n\) such that \(X\) admits a cover constituted by \(n\) open subsets that are contractible in \(X.\) The motivation of this numerical homotopy invariant is that, for any compact differentiable manifold \(M\), \(\text{cat}(M)\) provides a lower bound of the number of critical points of any smooth function \(f:M\rightarrow \mathbb{R}.\) An equivariant version of \(\text{cat}\) was introduced by \textit{W. Marzantowicz} [Topology 28, No. 4, 403--412 (1989; Zbl 0679.55001)] for \(G\) a compact Lie group acting on a space \(X.\) The authors of the paper under review deal with the computation of both LS-category and equivariant LS-category for the class of locally standard torus manifolds (as well as a special case of such a class, called quasitoric manifolds). In order to do this they first study some general properties and bounds of the equivariant LS-category, \(\text{cat}_G(X),\) in terms of the set of fixed points \(X^G\), where \(G\) is any compact Hausdorff topological group acting on a Hausdorff space \(X.\) They also discuss the equivariant LS-category of a product by considering the equivariant version of categorical sequences and giving some counterexamples. Some examples of computation of equivariant LS category are also given.