## 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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