Self-homotopy equivalences of the total spaces of a sphere bundle over a sphere
type: Item
instance of: scholarly article
title: Self-homotopy equivalences of the total spaces of a sphere bundle over a sphere
zbMATH Open document ID: 0562.55004
author: Sasao Seiya
published in: Kodai Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The group \({\mathcal E}(X)\) of homotopy classes of self-homotopy equivalences is calculated here up to two extensions in the case where \(X=S^ m\cup e^ n\cup e^{m+n}\) is an \(S^ m\) bundle over \(S^ n\) \((3<m+1<n<2m- 2):\) the extensions are not given. The group \({\mathcal E}(X)\) was known in the case \(m=3\) and \(n=7\) for rank 2 H-spaces and others: for the Stiefel manifolds \(V_{p,2}=S^{p-2}\cup e^{p-1}\cup e^{2p-3}\) and \(W_{p,2}=S^{2p-3}\cup e^{2p-1}\cup e^{4p-3}\) (most values of p) up to extension: and, up to one extension in special cases for \(m=3\). Rather than making two successive applications of the standard technique using the Barcus-Barratt sequence, the proof uses a modification. The extensions are based on the sequences \(1\to {\mathcal E}_+(X)\to {\mathcal E}(X)\to^{H_*}aut H_*(X)\) and \(0\to H_{\xi}\to {\mathcal E}_+(X)\to G_{\xi}\to \{1\}\) where \({\mathcal E}_+(X)\) is the kernel of \(H_*\), \(G_{\xi}\) is a subgroup of a quotient of \(\pi_ n(S^ m)\), and \(H_{\xi}\) is a quotient of \(\pi_{m+n}(X)\).
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 3895934
zbMATH Keywords: sphere bundles over spheres
zbMATH Keywords: homotopy classes of self-homotopy equivalences
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138036956
OpenAlex ID: W2024763831
cites work: On the Homotopy Classification of the Extensions of a Fixed Map
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cites work: The group of self-homotopy equivalences of principal three sphere bundles over the seven sphere
cites work: On self-homotopy equivalences of \(S^ 3\)-principal bundles over \(S^ n\)
cites work: Self homotopy equivalences of Stiefel manifolds \(W_{n,2}\) and \(V_{n,2}\)
cites work: On the group of self-homotopy equivalences of principal \(S^ 3\)-bundles over spheres
DOI: 10.2996/KMJ/1138036956
description: scientific article; zbMATH DE number 3895934