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 XtG on a G-spectrum X. When G = Cp is cyclic of prime order and X = B^p is the p-th smash power of a bounded below spectrum B with H*(B; Fp) of finite type, we prove that (B^p)tCp is a topological model for the Singer construction R+(H*(B; Fp)) on H*(B; Fp). There is a stable map ∈B: B → (B^p)tCp inducing the ExtA -equivalence ∈: R+(H*(B; Fp)) → H*(B; Fp). Hence ∈B and the canonical map Γ(B^p)Cp → (B^p)hCp are p-adic equivalences.