# The topological Singer construction

### Sverre Lunøe-Nielsen

### John Rognes

Dept. of Mathematics Dept. of Mathematics University of Oslo University of Oslo Norway Norway

## 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 $H_{∗}(B;F_{p})$ of finite type, we prove that $(B_{∧p})_{tC_{p}}$ is a topological model for the Singer construction $R_{+}(H_{∗}(B;F_{p}))$ on $H_{∗}(B;F_{p})$. There is a stable map $ϵ_{B}:B→(B_{∧p})_{tC_{p}}$ inducing the $Ext_{A}$-equivalence $ϵ:R_{+}(H_{∗}(B;F_{p}))→H_{∗}(B;F_{p})$. Hence $ϵ_{B}$ and the canonical map $Γ:(B_{∧p})_{C_{p}}→(B_{∧p})_{hC_{p}}$ are $p$-adic equivalences.

## Cite this article

Sverre Lunøe-Nielsen, John Rognes, The topological Singer construction. Doc. Math. 17 (2012), pp. 861–909

DOI 10.4171/DM/384