Nonstable and stable monomorphisms of vector bundles
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.
scientific article; zbMATH DE number 988054
scientific article