The topological Singer construction

TL;DR

This paper establishes a topological model for the Singer construction on cohomology using Tate spectra of p-th smash powers, linking algebraic and topological perspectives in stable homotopy theory.

Contribution

It introduces a topological model for the Singer construction via Tate spectra of p-th smash powers of spectra, connecting algebraic and topological methods.

Findings

(B^p)^{tC_p} models R_+(H^*(B)) topologically

The map epsilon_B induces an Ext_A-equivalence

Canonical maps are p-adic equivalences

Abstract

We study the continuous (co-)homology of towers of spectra, with emphasis on a tower with homotopy inverse limit the Tate construction X^{tG} on a G-spectrum X. When G=C_p is cyclic of prime order and X=B^p is the p-th smash power of a bounded below spectrum B with H_*(B) of finite type, we prove that (B^p)^{tC_p} is a topological model for the Singer construction R_+(H^*(B)) on H^*(B). There is a map epsilon_B : B --> (B^p)^{tC_p} inducing the Ext_A-equivalence epsilon : R_+(H^*(B)) --> H^*(B). Hence epsilon_B and the canonical map Gamma : (B^p)^{C_p} --> (B^p)^{hC_p} are p-adic equivalences.