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On sectioning multiples of vector bundles and more general homomorphism bundles. (English) Zbl 0809.55009

For a vector bundle \(\beta\) over a paracompact space \(X\), let \(n \beta\) denote the \(n\)-fold Whitney sum \(\beta \oplus \cdots \oplus \beta\) and \(\text{span} (n \beta)\) the maximal number of linearly independent cross-sections of \(n\beta\). The author proves the following estimates: (1) If \(\text{span} (n \beta) \geq 1\) then \(\text{span} (n \beta) \geq \rho (n)\) (with \(\rho (n) = 2^ c + 8d\) for \(n = (2a + 1) 2^{c + 4d}\) and \(a,c,d \geq 0\) and \(c \leq 3)\); (2) If one of the vector bundles \(\alpha, \beta\) is a \(p\)-Clifford module and \(\text{span} (\operatorname{Hom} (\alpha, \beta)) \geq 1\) then \(\text{span} (\operatorname{Hom} (\alpha, \beta)) \geq p + 1\).

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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References:

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