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Note on a generalization of the generalized vector field problem. (English) Zbl 0795.55016
For a vector bundle \(\xi\) let \(\text{span} (\xi)\) denote the maximal number of linearly independent sections. Let \(\zeta_{n,k}\) be the line bundle over \(G_{n,k}\) associated with the double covering \(\widetilde G_{n,k} \to G_{n,k}\), where \(G_{n,k}\) \((\widetilde G_{n,k})\) is the Grassmann manifold of (oriented) \(k\)-spaces in \(\mathbb{R}^ n\). Let \(d=k(n-k)=\dim (G_{n,k} )\).
The author proves the following results: Suppose \(3 \leq k \leq {n \over 2}\).
(i) If \(d \equiv 1(4)\) then \(\text{span} ((d+t) \zeta_{n,k})\geq 2+t\) for all \(t \geq 1\).
(ii) If \(n\) is even, \(k\) is odd and \(3 \leq q \leq r+2\), then there exists a map \(f:G_{n,k} \to G_{d+r,q}\) such that \(f^*(\zeta_{d+r,q})\) and \(\zeta_{n,k}\) are equivalent.
The method of proof is classical obstruction theory and characteristic classes.
Reviewer: U.Kaiser (Siegen)
MSC:
55S40 Sectioning fiber spaces and bundles in algebraic topology
57R22 Topology of vector bundles and fiber bundles
57R25 Vector fields, frame fields in differential topology
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