Stable extendibility of normal bundles over lens spaces
type: Item
instance of: scholarly article
title: Stable extendibility of normal bundles over lens spaces
zbMATH Open document ID: 1204.55012
author: Mitsunori Imaoka
published in: Topology and its Applications
publication date: 2010-09-24T00:00:00Z
full work available at URL: TopolAppl 157 2435.pdf
review text: Let \(L^n(p)=S^{2n+1}/(\mathbb Z/p)\) be the \((2n+1)\)-dimensional standard lens space for an odd prime \(p\). For any \(\mathbb R\)-vector bundle \(\zeta\) over \(L^n(p)\), \(s(\zeta)\) denotes the maximal number \(m\) such that \(\zeta\) is stably extendible to \(L^m(p)\). In the paper under review, the author studies \(s(\nu_n^t(p))\) for the normal bundle \(\nu _n^t(p)\) of an immersion of \(L^n(p)\) in the Euclidean space \(\mathbb R^{2n+1+t}\). First the author shows that if \((p-1)/2\leq n \leq p\) for \(p\geq5\) and \(0\leq n \leq 5\) for \(p=3\), then \(s(\nu_n^t(p))=\infty\) for \(t\geq 2n+1\). To prove the theorem, the author shows that \(\nu_n^t(p)\), under these conditions, is stably equivalent to a bundle which is the Whitney sum of \(2\)-plane bundles and a trivial bundle. For the case of \(p=3\) or \(5\), a more explicit result is given. From the theorem, the author addresses the conjecture that \(s(\nu_n^t(p))=\infty\) if \(n\leq p\). Next the author shows that if \(n\geq 2p\) and \(t\leq 2n+2\), then \(s(\nu_n^t(p))<\infty\) for \(p\geq7\). Finally, the author studies actual values of \(s(\nu_n^t(p))\) for the case of \(s(\nu_n^t(p))<\infty\).
reviewed by: Yutaka Hemmi
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 5791184
zbMATH Keywords: extendible vector bundle
zbMATH Keywords: normal bundle
zbMATH Keywords: lens space
MaRDI profile type: Publication
OpenAlex ID: W1968101013
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DOI: 10.1016/J.TOPOL.2010.07.031
description: scientific article; zbMATH DE number 5791184