Universal Spaces and Splittings of Equivariant Spectra

TL;DR

This paper revisits the splitting of rational G-spectra, explores its behavior under weaker localizations, and introduces a systematic method for computing equivariant cohomology and homotopy groups for dihedral groups.

Contribution

It provides a new systematic approach to compute $RO(G)$-graded rings of equivariant spectra, extending previous partial results and applying to dihedral groups.

Findings

Explicit computation of $oldsymbol{b{ ext{Z}}}$ and $HA_G$ homotopy groups for $D_{2p}$.

Systematic method for $RO(G)$-graded ring computations.

Analysis of splitting behavior under weaker localizations.

Abstract

Let $G$ be a finite group. We re-analyze the splitting of rational $G$-spectra, which is discussed in Barne's thesis and traces back to Greenlees and May. We will study how the splitting behaves under a localization which is weaker than rationalization and discuss its application in computing equivariant cohomology of $G$-spectra. In particular, we will explicitly compute $\pi_\bigstar(H\underline{\mathbb{Z}})$ and $\pi_\bigstar(HA_G)$ for the dihedral group $G=D_{2p}$. The additive structures and partial multiplicative structures are already computed by Kriz, Lu, and Zou. Our new method will provide a systematic way to compute them as $RO(D_{2p})$-graded rings.