The topological Singer construction

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 $X=B^{\wedge p}$ is the $p$-th smash power of a bounded below spectrum $B$ with $H_{\ast }(B;{\mathbb{F}}_p)$ of finite type, we prove that $(B^{\wedge p}{)}^{t{C}_p}$ is a topological model for the Singer construction $R_{+}(H^{\ast }(B;{\mathbb{F}}_p))$ on $H^{\ast }(B;{\mathbb{F}}_p)$. There is a stable map ${ϵ}_B:B\to (B^{\wedge p}{)}^{t{C}_p}$ inducing the ${\mathrm{Ext}}_{\mathcal{A}}$-equivalence $ϵ:R_{+}(H^{\ast }(B;{\mathbb{F}}_p))\to H^{\ast }(B;{\mathbb{F}}_p)$. Hence ${ϵ}_B$ and the canonical map $\Gamma :(B^{\wedge p}{)}^{C_p}\to (B^{\wedge p}{)}^{h{C}_p}$ are $p$-adic equivalences.