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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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