Pro-isomorphisms of homology towers (Q1909546)

In this well-written paper the following question is investigated: Given two towers of spaces, when are their homology towers pro-isomorphic? Here homology is always with \(R =\mathbb{F}_p\) coefficients. Recall that a map \(f:\{G_s\}\to \{H_s\}\) of two towers of groups is a pro-isomorphism if it induces an isomorphism \(\text{colim Hom}(H_s,A) \cong \text{colim Hom}(G_s,A)\) for each group \(A\). Let \(X\mapsto R_\infty X\) be the Bousfield-Kan functor for \(R\). Recall that \(R_\infty X\) is the limit of a tower \(\{R_s X\}\) of fibrations. The main result of the paper is Theorem: Let \(f:\{X_s\}\to \{Y_s\}\) be a map of towers of spaces. Assume that \(\{H_* X_s\}\) is pro-constant and \(\{H_* Y_s\}\) is pro-finite type. Assume further that \(\{H_* X_s\}\) is pro-finite type or \(\{H_* Y_s\}\) is pro-constant. Then \(f\) induces a pro-homology isomorphism iff it induces a homotopy equivalence \(\text{holim }R_sX_s\to \text{holim }R_sY_s\). As applications we obtain conditions on a tower \(\{X_s\}\) of spaces ensuring that the map \(H_*X\to \lim H_*X_s\) is an isomorphism, where \(X =\text{holim }X_s\). These results in turn are used to prove a convergence result for the homology spectral sequence of a cosimplicial space \(X^\bullet\) extending a convergence theorem of Bousfield to the case where the total space is not simply connected.