The Segal conjecture for smash powers

TL;DR

This paper proves that for the G-fold smash power of a spectrum, the comparison map between G-fixed points and G-homotopy fixed points becomes an equivalence after specific completions, extending the Segal conjecture.

Contribution

It establishes new equivalences between fixed points and homotopy fixed points for smash powers under p-completion and I(G)-completion, generalizing the Segal conjecture.

Findings

Equivalence after p-completion for finite p-groups.

Equivalence after I(G)-completion for any finite group.

Conditions on spectra for the equivalences to hold.

Abstract

We prove that the comparison map from $G$-fixed points to $G$-homotopy fixed points, for the $G$-fold smash power of a bounded below spectrum $B$, becomes an equivalence after $p$-completion if $G$ is a finite $p$-group and $H_*(B; F_p)$ is of finite type. We also prove that the map becomes an equivalence after $I(G)$-completion if $G$ is any finite group and $\pi_*(B)$ is of finite type, where $I(G)$ is the augmentation ideal in the Burnside ring.