Obstructions for constructing equivariant fibrations
type: Item
instance of: scholarly article
title: Obstructions for constructing equivariant fibrations
zbMATH Open document ID: 1257.57039
author: Aslı Güçlükan İlhan
published in: Algebraic \& Geometric Topology
publication date: 2012-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.3880
reviewed by: Robert D. Little
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 55R91
zbMATH DE Number: 6068599
zbMATH Keywords: equivariant fibration
zbMATH Keywords: Bredon cohomology
zbMATH Keywords: obstruction theory
zbMATH Keywords: group action
MaRDI profile type: Publication
OpenAlex ID: W1997787679
arXiv ID: 1110.3880
cites work: Periodic complexes and group actions
cites work: Equivariant cohomology theories
cites work: Groups which act freely on \({\mathbb{R}}^ m\times S^{n-1}\)
cites work: Partitions of unity in the theory of fibrations
cites work: Constructing free actions of \(p\)-groups on products of spheres
cites work: Transformation groups and algebraic \(K\)-theory
cites work: The topological spherical space form problem. II: Existence of free actions
cites work: Q4255482
cites work: Q4002713
cites work: A classification theorem for fibre spaces
cites work: Higher Whitehead torsion and the geometric assembly map
cites work: Periodic resolutions for finite groups
cites work: A Note on Paired Fibrations
cites work: A strong homotopy equivalence and extensions for Hurewicz fibrations
cites work: Equivariant Classifying Spaces and Fibrations
cites work: Equivariant Fibrations and Transfer
DOI: 10.2140/AGT.2012.12.1313
review text: If \(G\) is a finite group and \({\mathcal H}\) a collection of subgroups of \(G\) closed under conjugation and taking subgroups, then a \(G\)-fibration over a \(G\)-\(CW\)-complex \(B\) is a \(G\)-equivariant fibration over \(B\) with fibers equivariantly homotopy equivalent to a space in a compatible family of \(H\)-spaces, \(H\in{\mathcal H}\). If the isotropy groups are in \({\mathcal H}\) and \(B^H\) is simply connected for every isotropy group \(H\), then there is an obstruction cocycle defined for a \(G\)-fibration over \(B^n\). The cocylce is an element of the Bredon cochain group at level \(n+ 1\) with coefficients in a local system of homotopy groups at level \(n\).NEWLINENEWLINE The main result of this paper is that this Bredon cocycle behaves like a classical obstruction cocycle, that is the cocycle vanishes if and only if the \(G\)-fibration extends to a \(G\)-fibration over \(B^{n+1}\) and the cohomology class of the cocycle vanishes if and only if the restriction of the \(G\)-fibration to \(B^{n-1}\) extends to a \(G\)-fibration over \(B^{n+1}\). In both cases, the total space of the extension is \(G\)-homotopy equivalent to a \(G\)-\(CW\)-complex.
description: scientific article; zbMATH DE number 6068599