On the group of equivariant self equivalences of free actions
type: Item
instance of: scholarly article
title: On the group of equivariant self equivalences of free actions
zbMATH Open document ID: 0622.55005
author: K. Tsukiyama
author: Hideaki Ōshima
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: Let \(q: P\to B\) be a numerable principal G-bundle, and \(F_ G(P)\) (resp. \(E_ G(P))\) the group of unbased (resp., based) G-equivariant self- homotopy equivalence classes under the free G-action on P. In the case where \(B=SZ\) is a suspended 1-connected CW complex and q is classified by \(k\in [Z,G]\), a commutative diagram of exact sequences (of groups except at \(\theta)\) is obtained (to complicated to state here). This extends earlier results of Tsukiyama in the case where P is simply connected. Note the special case where G is 1-connected. Calculations are made in several cases. A further result given here is that \(F_ G(P)\) and \(E_ G(P)\) are finitely presented in the case where B is a 1-connected, and G a path-connected, finite CW complex, and E(B).k is a finite set. The proof relies on the known result for E(X) in case X is a 1-connected finite CW complex.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 4009319
zbMATH Keywords: group of G-equivariant self-homotopy equivalence classes
zbMATH Keywords: numerable principal G-bundle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195177261
OpenAlex ID: W2084442930
cites work: The generalized Whitehead product
cites work: On maps of H-spaces
cites work: Q5667770
cites work: A note on the Samelson product in the classical groups
cites work: Q5658847
cites work: Partitions of unity in the theory of fibrations
cites work: A Study of Function Spaces by Spectral Sequences
cites work: Q4105006
cites work: Some non-stable homotopy groups of Lie groups
cites work: The evaluation map and E H P sequences
cites work: The group of self-equivalences of certain complexes
cites work: Q5822449
cites work: Q5831641
cites work: Infinitesimal computations in topology
cites work: A topological proof of theorems of Bott and Borel-Hirzeburch for homotopy groups of unitary groups
cites work: On the group of fibre homotopy equivalences
cites work: Equivariant self equivalences of principal fibre bundles
cites work: Applications of minimal simplicial groups
DOI: 10.2977/PRIMS/1195177261
description: scientific article; zbMATH DE number 4009319