Homotopy monomorphisms and \(H\)-splitting in loop space fibrations
type: Item
instance of: scholarly article
title: Homotopy monomorphisms and \(H\)-splitting in loop space fibrations
zbMATH Open document ID: 1214.55008
author: Luofei Liu
published in: Topology and its Applications
publication date: 2009-03-06T00:00:00Z
review text: Let \(X,\) \(Y\) and \(Z\) be \(H\)-spaces and \(f:Y\to X\) and \(g:Z\to X\) be \(H\)-maps. The composite \(Y\times Z\to X\times X\to X\) is called an \(H\)-splitting if it is a homotopy equivalence and an \(H\)-map. The author considers here the special case of \(\Omega B\times\Omega F\to\Omega E,\) where \(p:E\to B\) is a fibration with fibre \(i:F\to E\) and \(\Gamma\;:\Omega B\to\Omega E\) is a, possibly naturally ocurring, homotopy right inverse of \(\Omega p.\) With these conditions \(\Omega B\times\Omega F\to\Omega E\) is known to be a homotopy equivalence, but is not in general an \(H\)-map. The author gives examples from the literature where it is, or is not, an \(H\)-map. For example, in the case of the extended Hopf fibration \(S^1\to S^3\to \mathbb CP^1\to K({\mathbb Z} ,2),\) the relevant homotopy equivalence \(\Omega S^2\to \Omega K({\mathbb Z},2)\times \Omega S^3\) is not an \(H\)-map, using the loop structure of \(\Omega S^2\), since \(\Omega S^2\) is not homotopy abelian and the other two spaces are. The author considers the case where \(F\) has an \(H\)-structure such that the holonomy action of the fibration \(F\times \Omega B\to F\) is homotopic to \(F\times \Omega B\to F\times F\to F,\) where \( \Omega B\to F\) is the connecting map; this condition is satisfied for the various definitions of principal fibration for example. His main result is that, if also the homotopy right inverse \(\Gamma\;:\Omega B\to\Omega E\) is an \(H\)-map, then \(\Omega B\times\Omega F\to\Omega E\) is an \(H\)-splitting. He also shows, without the assumption that \(\Gamma \) is an \(H\)-map, that \(F\to E\) is a homotopy monomorphism if, and only if, it is a weak homotopy monomorphism. Numerous special cases and examples are given.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55R05
Mathematics Subject Classification ID: 55P45
zbMATH DE Number: 5526281
zbMATH Keywords: \(H\)-space
zbMATH Keywords: \(H\)-splitting
zbMATH Keywords: Fibration
zbMATH Keywords: Homotopy monomorphism
reviewed by: John W. Rutter
MaRDI profile type: Publication
full work available at URL: j.topol.2008.10.017
OpenAlex ID: W2015826337
cites work: Q4896560
cites work: Operators and cooperators in homotopy theory
cites work: On the mapping theorem for Lusternik-Schnirelmann category
cites work: On monomorphisms in homotopy theory
cites work: Q5526617
cites work: Induced Fibrations and Cofibrations
cites work: Monomorphisms and relative Whitehead products
cites work: Nilpotency and H-spaces
cites work: Q3025866
cites work: Principal and induced fibrations
cites work: Holonomy-nilpotent fibrations and rational Lusternik-Schnirelmann category
cites work: On the topology of fibrations with section and free loop spaces
cites work: Principal Fibrations
cites work: Q5613822
cites work: H-FIBRATIONS
cites work: Odd primary torsion in \(\pi_k(S^3)\)
cites work: Q5522742
cites work: Q3292133
cites work: Q5530411
DOI: 10.1016/J.TOPOL.2008.10.017
description: scientific article; zbMATH DE number 5526281