Notes on equivariant homology with constant coefficients

TL;DR

This paper presents a method for calculating equivariant homology with constant coefficients for finite groups, including explicit computations of geometric fixed points and examples involving inverting representations in specific group contexts.

Contribution

It introduces a new calculation method for equivariant homology with constant coefficients and applies it to complex examples involving geometric fixed points and representation inversion.

Findings

Explicit calculation of geometric fixed points for equivariant spectra.

Method for inverting standard representations in equivariant homology.

Application to split extraspecial groups at prime 2.

Abstract

In this paper, for a finite group, we discuss a method for calculating equivariant homology with constant coefficients. We apply it to completely calculate the geometric fixed points of the equivariant spectrum representing equivariant (co)homology with constant coefficients. We also treat a more complicated example of inverting the standard representation in the equivariant homology of split extraspecial groups at the prime 2.