Antipodal vector bundle monomorphisms
type: Item
instance of: scholarly article
title: Antipodal vector bundle monomorphisms
zbMATH Open document ID: 1005.55004
DOI: 10.1007/BF02784525
author: Ulrich Koschorke
published in: Israel Journal of Mathematics
publication date: 2003-02-19T00:00:00Z
review text: Let \(\alpha\) and \(\beta\) denote vector bundles of dimensions \(a\) and \(b\) respectively over a closed smooth \(n\)-manifold \(N\). A monomorphism \(u: \alpha\hookrightarrow \beta\) is said to be antipodal if \(u\) is regularly homotopic to its negative \(-u\). The author's starting point is to study existence and classification problems concerning such antipodal monomorphisms from \(\alpha\) to \(\beta\). In the present paper the author investigates three obstructions \(\omega(\alpha,\beta)\), \(d(u,-u)\) and \(\omega (\widetilde\alpha, \beta)\) exploited previously by the author himself and their relations among each other and with classical invariants. The main results which are collected in the introduction consist of five theorems, one corollary and one proposition together with two examples. But there are a large number of lemmas, propositions, theorems and so forth in the following sections \(2-6\). Let \(\lambda\) be the canonical line bundle over the projectification \(P(\alpha)\) of \(\alpha\). (Here the same notation is used both for a vector bundle over \(N\) and for its pull-back to \(P(\alpha).)\) Given any generic section \(s\) of \(\lambda \otimes\beta\), then the \((n+a-b-1)\)-dimensional manifold \(Z\) formed by the zero set of \(s\) yields an element of the normal bordism group \(\Omega_{n+a-b-1} (P(\alpha), \varphi)\) with coefficients in the virtual vector bundle \(\varphi= \lambda\otimes \beta-\lambda \otimes\alpha-TN\). This one is \(\omega(\alpha, \beta)\) described above. In fact, it holds that if \(n<2(b-a)\), then a monomorphism \(\alpha\hookrightarrow \beta\) exists if and only if \(\omega(\alpha, \beta) =0\) (Theorem A). A similar approach yields \(d(u,-u)\) in \(\Omega_{n+a-b} (P(\alpha); \varphi)\) and it holds that if \(n+1<2(b-a)\), then a monomorphism \(u: \alpha \hookrightarrow\beta\) is antipodal if and only if \(d(u,-u)=0\) (Theorem B). Let \(\xi\) denote the non-trivial line bundle over \(S^1\) and let \(\widetilde N=S^1\ti…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 55N22
zbMATH DE Number: 1786373
zbMATH Keywords: bordism group
zbMATH Keywords: antipodal monomorphism
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1786373