The topological Singer construction
Algebraic Topology
2022-06-22 v1
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.
Cite
@article{arxiv.1010.5633,
title = {The topological Singer construction},
author = {Sverre Lunøe--Nielsen and John Rognes},
journal= {arXiv preprint arXiv:1010.5633},
year = {2022}
}