Antipodal vector bundle monomorphisms
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 \(\widetilde N=S^1\times N\) and \(\widetilde\alpha= \xi\otimes\alpha\). Then it is clear that an antipodal monomorphisms \(\alpha\hookrightarrow \beta\) over \(N\) exists if and only if there is any monomorphism \(\widetilde\alpha \hookrightarrow\beta\) over \(\widetilde N\) and also it follows from Theorem A that if \(n+1<2(b-a)\), then this holds if and only if \(\omega(\widetilde \alpha,\beta) =0\) (Theorem C). These three theorems follow rather from the author's prior results. In this paper the author focuses his attention on the following problem: How many different antipodal monomorphisms \(\alpha\hookrightarrow\beta\) exist and what is their percentage among all monomorphisms (Question D)? The key technique to settle this is to study a long exact sequence of normal bordism groups associated with the pair \((P(\widetilde \alpha)\), \(P(\alpha))\) equipped with the Gysin isomorphism. Using the above three obstructions and the homomorphisms which occur in this sequence, the author gives an answer to this question (Theorem D and Corollary). But this solution on the bordism level is too hard to compute, though it is sharp. In order to make this computable, the author devises ``weak stable versions'' of the above three obstructions. The study of these gives two basic results (Theorem E and Proposition F). And also the two examples (Examples G,H) given show that these results are useful consequences.
scientific article; zbMATH DE number 1786373
scientific article