Poincaré duality and Serre fibrations
type: Item
instance of: scholarly article
title: Poincaré duality and Serre fibrations
zbMATH Open document ID: 0915.54011
DOI: 10.1016/S0166-8641(97)00004-7
author: N. Brodskiy
author: Evgeny V. Shchepin
published in: Topology and its Applications
publication date: 1997-11-06T00:00:00Z
review text: Recall that a continuous map is a Serre fibration provided it has the covering homotopy property for maps of polyhedra and it is a Serre \(k\)-fibration if it has the property for maps of polyhedra of dimension at most \(k\). The paper presents an analogue in the setting of maps between manifolds of the detection using Poincaré duality of spheres as those manifolds that are homotopically connected through the middle dimension. The main result is: if \(f: M^m\to N^n\) is a proper Serre \(k\)-fibration of manifolds without boundary such that \(m-n\leq k\), then \(f\) is a Serre fibration if \(m= 2k+1\) or \(m= 2k+2\) and \(f\) has no singular points (i.e., no point-inverse \(f^{-1}(y)\) contains an isolated point). A version for manifolds-with-boundary is also included.
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 57N15
zbMATH DE Number: 1083968
zbMATH Keywords: homotopically regular map
zbMATH Keywords: \(n\)-soft map
zbMATH Keywords: Serre fibration
reviewed by: John Walsh
MaRDI profile type: Publication
cites work: Q3883165
cites work: Q3725165
cites work: Continuous selections. II
cites work: Continuous selections. III
cites work: Q5553895
cites work: Soft maps of manifolds
cites work: Q3787143
description: scientific article; zbMATH DE number 1083968