Sections of maps with fibers homeomorphic to a two-dimensional manifold
type: Item
instance of: scholarly article
title: Sections of maps with fibers homeomorphic to a two-dimensional manifold
zbMATH Open document ID: 0998.54010
DOI: 10.1016/S0166-8641(01)00009-8
published in: Topology and its Applications
publication date: 2002-06-27T00:00:00Z
review text: The paper provides a partial answer to the question whether a fibration \(p: E\to B\), having constant (up to homeomorphism) fibres \(p^{-1}(b)\), \(b\in B\), is locally trivial. It was conjectured (by Shchepin) that this holds provided \(p\) is a Serre fibration, the base \(B\) is locally path connected and the (typical) fibre is a compact at most 4-dimensional manifold. The main result of the paper states that if \(p:E\to B\) is a Serre fibration of \(LC^0\)-compact metric spaces, the constant fiber is a compact 2-dimensional manifold and \(B\) is an absolute neighborhood retract, then \(p\) admits local sections, i.e. for each \(b\in B\), there is a section of \(p\) over a neighborhood of \(b\). The proofs make use of the quite involved technique of graph approximations to set-valued maps.
reviewed by: Wojciech Kryszewski
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 55R05
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54C10
zbMATH DE Number: 1760997
zbMATH Keywords: Serre fibration
zbMATH Keywords: low-dimensional manifolds
zbMATH Keywords: graph approximations
author: N. Brodskiy
MaRDI profile type: Publication
cites work: Cell-like maps. The problem of raising dimension
cites work: Regular mappings and the space of homeomorphisms on a 2-manifold
cites work: Some theorems on absolute neighborhood retracts
cites work: Graph-approximation of set-valued maps on noncompact domains
cites work: Q5553895
cites work: Q4219111
cites work: Q4360773
cites work: Q4345803
cites work: Q3787143
full work available at URL: s0166-8641(01)00009-8
OpenAlex ID: W2046088623
Wikidata QID: Q126781567
description: scientific article; zbMATH DE number 1760997