A fiberwise analogue of the Borsuk - Ulam theorem for sphere bundles over a 2-cell complex
type: Item
instance of: scholarly article
title: A fiberwise analogue of the Borsuk - Ulam theorem for sphere bundles over a 2-cell complex
zbMATH Open document ID: 1120.55008
author: Ryuichi Tanaka
published in: Topology and its Applications
publication date: 2007-05-10T00:00:00Z
review text: Let \(\alpha\) be a finite dimensional real vector bundle over a CW-complex \(B\), and \(S(\alpha)\) be its sphere bundle with respect to some metric on \(\alpha\). The index of \(\alpha\), denoted by ind\,\(\alpha\), is defined to be the largest integer \(m\) for which there exists a \(\mathbb{Z}/2\)-map from \(S^{m-1}\) to \(S(\alpha)\), where \(S(\alpha)\) and \(S^{m-1}\) are regarded as \(\mathbb{Z}/2\)-spaces by the antipodal map (on each fibre). Let \(M^n(k)=S^{n-1}\cup_ke^n\) be the mapping cone of the degree \(k\) map \(k:S^{n-1}\to S^{n-1}\). In this paper, the following is proved: The equality ind\,\(\alpha=\dim\alpha\) holds for every vector bundle \(\alpha\) over the complex \(M^n(k)\), where \(n\geq 2\), if and only if (1) \(k\) is odd, (2) \(k=0\) and \(n\neq 2,3,4,5,8,9\), or (3) \(k\) is even, \(k>0\) and \(n\neq 2,3,4,8\).
reviewed by: Takuo Miwa
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 55R70
Mathematics Subject Classification ID: 55M35
zbMATH DE Number: 5151668
zbMATH Keywords: vector bundle
zbMATH Keywords: sphere bundle
zbMATH Keywords: \(\mathbb Z/2\)-map
zbMATH Keywords: index
MaRDI profile type: Publication
full work available at URL: j.topol.2007.01.016
OpenAlex ID: W2006378671
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DOI: 10.1016/J.TOPOL.2007.01.016
description: scientific article; zbMATH DE number 5151668