Equivariant covering type and the number of vertices in equivariant triangulations

Abstract

We introduce the notion of the \emph{equivariant covering type} of a space $X$ on which a finite group $G$ acts, and study its properties. The equivariant covering type measures the size of $G$-equivariant good covers of $X$ and is thus an extension of the \emph{covering type} of a space, introduced by Karoubi and Weibel. We show that the equivariant covering type is a $G$-homotopy invariant and describe its relation with other $G$-invariants, like the equivariant LS-category, $G$-genus and the multiplicative structures of equivariant cohomology theories. We also compute the $G$-covering type of regular $G$-graphs, give estimates for orientation-preserving actions on surfaces and for the projectivizations of complex representations of $G$ and cohomology spheres. As an application, we derive estimates of sizes of minimal $G$-triangulations for various $G$-spaces.

Revised version with a new title