A model of Adams-Hilton type for fiber squares
type: Item
instance of: scholarly article
title: A model of Adams-Hilton type for fiber squares
zbMATH Open document ID: 0546.55025
author: David J. Anick
published in: Illinois Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55U15
Mathematics Subject Classification ID: 55T20
Mathematics Subject Classification ID: 55R05
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 3870226
zbMATH Keywords: Adams-Hilton model of chain complex of loop space induced fibrations
zbMATH Keywords: Eilenberg-Moore spectral sequence
zbMATH Keywords: characteristic zero cohomology
zbMATH Keywords: free loop space on any formal space
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3870226