The Topological Singer Construction

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 $H_*(B; \F_p)$ of finite type, we prove that $(B^{\wedge p})^{tC_p}$ is a topological model for the Singer construction $R_+(H^*(B; \F_p))$ on $H^*(B; \F_p)$. There is a stable map $\epsilon_B \: B\to (B^{\wedge p})^{tC_p}$ inducing the $\Ext_\A$-equivalence $\epsilon\: R_+(H^*(B; \F_p))\to H^*(B; \F_p)$. Hence $\epsilon_B$ and the canonical map $\\Gamma \: (B^{\wedge p})^{C_p}\to (B^{\wedge p})^{hC_p}$ are $p$-adic equivalences.

2010 Mathematics Subject Classification: 55P42, 55P91, 55S10, 55T15

Keywords and Phrases: Singer construction, Tate construction, limit of Adams spectral sequences, $\Ext$-equivalence.

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