A model of Adams-Hilton type for fiber squares
Let p: \(E\to B\) be a filtration, let f: \(X\to B\) be any map, and let q: \(A\to X\) be the pull back of p via f. While Eilenberg and Moore have given a functorial but ''very large'' chain complex whose homology equals that of A, this paper provides an analogous ''small'' chain complex based upon cellular decompositions for X, B, and E. When \(E\simeq X\simeq (pt),\) the new construct reduces to the Adams-Hilton model for \(\Omega\) X. By ''small'' we mean that the new chain complex has finite type whenever E, B, and X have homology of finite type. Over a field we obtain another proof of the existence of the Eilenberg-Moore spectral sequence through an explicit perturbation theory for its \(E_ 1\) term. We also demonstrate that this spectral sequence is a homotopy invariant. As applications, the Eilenberg-Moore spectral sequence is shown to degenerate when f and p are ''p-homogeneous'' maps and we compute the characteristic zero cohomology of the free loop space on any formal space.
Adams-Hilton model of chain complex of loop space induced fibrations
scientific article; zbMATH DE number 3870226