Cohomology classification of self maps of sphere bundles over spheres
type: Item
instance of: scholarly article
title: Cohomology classification of self maps of sphere bundles over spheres
zbMATH Open document ID: 0862.55015
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1997-05-28T00:00:00Z
review text: Let \(E\) denote the \(q\)-sphere bundle over the \(n\)-sphere determined by an element of \(\pi_{n-1} (\text{SO} (q+1))\). Any self-map of \(E\) determines an automorphism of \(\widetilde H^* (E)\), modulo torsion. The purpose of this work is to determine what automorphisms of \(\widetilde H^* (E)\), modulo torsion, arise in this way. The results express the answers in terms of numerical conditions on the multiples of the generators obtained by applying a possible self-map. Special attention is given to the cases of real, complex and quaternionic Stiefel manifolds of 2-frames. A final section looks at the self-maps of the suspension of \(E\).
Mathematics Subject Classification ID: 55S37
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55R45
Mathematics Subject Classification ID: 55Q52
zbMATH DE Number: 927934
zbMATH Keywords: sphere bundle
zbMATH Keywords: self-map
author: Kaoru Morisugi
author: Hideaki Ōshima
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195163184
OpenAlex ID: W2065351405
cites work: On the groups \(J(X)\). IV
cites work: The metastable homotopy of 𝑂(𝑛)
cites work: Some Calculations of Homotopy Groups of Symmetric Spaces
cites work: Q4843498
cites work: Q4884063
cites work: Q5847727
cites work: Note on the Whitehead product
cites work: On the normal bundle of a homotopy sphere embedded in Euclidean space
cites work: ON THE HOMOTOPY GROUPS OF CERTAIN PAIRS AND TRIADS
cites work: On sphere-bundles over spheres
cites work: The Homotopy Theory of Sphere Bundles Over Spheres (I)
cites work: The Homotopy Theory of Sphere Bundles Over Spheres (II)
cites work: Some non-stable homotopy groups of Lie groups
cites work: Some Whitehead products in \(S^ n\)
cites work: Q4009832
cites work: Note on some Whitehead products
cites work: Compositional Methods in Homotopy Groups of Spheres. (AM-49)
cites work: On products in homotopy groups
cites work: Q4192669
cites work: On certain theorems of G. W. Whitehead
DOI: 10.2977/PRIMS/1195163184
description: scientific article; zbMATH DE number 927934