The topological Singer construction (Q1946055)

The Tate construction is naturally found in equivariant stable homotopy, however it turns out to have close connections with non-equivariant topology. In particular, when the group considered is cyclic of prime order~\(p\), the Tate construction \((B)^{tC_p}\) on a bounded below spectrum \(B\) has cohomology isomorphic over the Steenrod algebra \(\mathcal{A}\) to the algebraic Singer construction \(R_+(H^*(B))\) on \(H^*(B)=H^*(B;{\mathbb F}_p)\), and there is a map \(B\to(B)^{tC_p}\) inducing an \(\text{Ext}_{\mathcal{A}}\)-equivalence \(R_+(H^*(B))\to H^*(B)\). It follows that there is a \(p\)-adic equivalence \((B)^{C_p}\to(B)^{tC_p}\), a result which can be viewed as a generalisation of the Segal conjecture for \(C_p\). The methods used to prove these results involve towers of spectra and the use of inverse systems of Adams spectral sequences. The technicalities should be of wider applicability.