Note

The Smash-Power Segal Conjecture without Finite-Type Hypotheses

Author: Wei Yang · Added: August 24, 2026

Article date: July 28, 2026.

Abstract

Let $G\neq 1$ be a finite $p$ -group and let $B$ be a bounded-below spectrum. We prove that the proper Tate diagonal

\Delta_G(B)\colon B\longrightarrow \bigl(B^{\wedge G}\bigr)^{\tau G}

is the stable $p$ -completion map. Equivalently, the comparison from the categorical fixed points of the Hill-Hopkins-Ravenel norm $N_e^G B$ to its homotopy fixed points is a mod- $p$ equivalence. The published theorem of Bergsaker-Rognes proves this under a degreewise finite mod- $p$ homology hypothesis. The new argument uses their theorem only for finite-type Eilenberg-Mac Lane input.

The finite-type hypothesis is removed by studying the representation-sphere obstruction

Q_G(B)=F_G(S^{\infty\bar\rho_G},N_e^G B)^G.

Finite representation stages, a uniform pro-zero argument, exactness, and a Postnikov-continuity statement specific to norm inputs imply that $Q_G(B)/p\simeq0$ for every bounded-below $B$ . Isotropy separation and induction on $|G|$ then finish the argument. An appendix records why the stronger assertion that proper Tate constructions commute with arbitrary bounded-below Postnikov towers is false.