  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.
