Nonstable and stable monomorphisms of vector bundles
type: Item
instance of: scholarly article
title: Nonstable and stable monomorphisms of vector bundles
zbMATH Open document ID: 0870.55011
DOI: 10.1016/S0166-8641(96)00091-0
author: Ulrich Koschorke
published in: Topology and its Applications
publication date: 1997-09-01T00:00:00Z
review text: Let \(\alpha\) and \(\beta\) be real vector bundles of dimensions \(a\) and \(n\), respectively, \(0<a<n\), over a closed smooth connected \(n\)-dimensional manifold \(M\). The author discusses the questions of existence of a monomorphism from \(\alpha\) to \(\beta\) and of classification of such monomorphisms, both in the nonstable and in the stable case. The stable existence question is that of existence of a monomorphism from \(\alpha \oplus \mathbb{R}^N\) to \(\beta \oplus\mathbb{R}^N\). If \(u \colon \alpha \to \beta\) is a homomorphism, let \(S\) be the set of points where \(u\) fails to be injective. \(S\) is, generically, a smooth closed submanifold of \(M\) with an additional structure inherited from the homomorphism. In such a situation an obstruction \(\omega (\alpha , \beta)\) to the existence of a monomorphism \(\alpha \to \beta\) can be defined; it is an element of a normal bordism group. For the stable existence question, an obstruction element \(\omega^{st} (\alpha , \beta)\) is defined. If \(n>2a\) then a monomorphism \(\alpha \to \beta\) (or \(\alpha \oplus \mathbb{R}^N \to \beta \oplus \mathbb{R}^N\)) exists if and only if \(\omega (\alpha , \beta)\) (respectively, \(\omega^{st} (\alpha , \beta)\)) vanishes. In the classification question the author constructs a difference invariant which establishes a bijection between the set of homotopy classes of such (nonstable or stable, respectively) monomorphisms provided that \(n>2a +1\). Methods of how to compute such invariants are also given. These invariants and results are applied to the questions of existence and of classification of \(a\)-dimensional plane fields on a manifold and to immersion questions. The paper is carefully written and very well organized.
Mathematics Subject Classification ID: 55S35
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 58A30
zbMATH DE Number: 988054
zbMATH Keywords: vector bundle homomorphisms
zbMATH Keywords: plane field
zbMATH Keywords: singularity
zbMATH Keywords: obstruction
zbMATH Keywords: enumeration
zbMATH Keywords: normal bordism
zbMATH Keywords: stabilization
reviewed by: Jan Jaworowski
MaRDI profile type: Publication
cites work: Vector fields with finite singularities
cites work: Q5331105
cites work: On 2-distributions in 8-dimensional vector bundles over 8-complexes
cites work: Q4042369
cites work: Vector fields and other vector bundle morphisms - a singularity approach
cites work: Q3879964
cites work: Q3812943
cites work: Q3810554
cites work: Two-plane sub-bundles of nonorientable real vector-bundles
cites work: Characteristic Classes. (AM-76)
cites work: Q3933645
cites work: Bordism and Geometric Dimension.
cites work: Vector fields on manifolds
cites work: Q5792774
cites work: Q5793196
description: scientific article; zbMATH DE number 988054