Equivariant infinite loop space theory, the space level story

TL;DR

This paper advances equivariant infinite loop space theory by constructing genuine G-spectra for finite groups, comparing different infinite loop space machines, and providing detailed proofs and corrections to existing literature.

Contribution

It generalizes equivariant infinite loop space theory, constructs genuine G-spectra for finite G, and compares Segal and operadic machines with detailed proofs and corrections.

Findings

Equivariant infinite loop space theory is reworked and generalized.

Genuine G-spectra are constructed for finite groups.

Segal and operadic machines are proven equivalent with detailed comparison.

Abstract

We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\Omega$-$G$-spectra for any topological group $G$, but our focus is on the construction of genuine $\Omega$-$G$-spectra when $G$ is finite. We also show what is and is not true when $G$ is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$ but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.