Equivariant covering type and the number of vertices in equivariant triangulations

We introduce the notion of the 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 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.