Equivariant covering type and the number of vertices in equivariant triangulations

TL;DR

This paper introduces the equivariant covering type, a new invariant for spaces with group actions, and explores its properties, relations, and applications to minimal G-triangulations and specific G-spaces.

Contribution

It defines the equivariant covering type, proves its invariance under G-homotopy, and relates it to other G-invariants, providing computational methods and applications.

Findings

Equivariant covering type is a G-homotopy invariant.

Computed G-covering type for regular G-graphs and surfaces.

Provided estimates for minimal G-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.