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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Pontryagin classes of vector bundles over $ B{\rm Sp}(n)$


Author: Duane O’Neill
Journal: Proc. Amer. Math. Soc. 40 (1973), 315-318
MSC: Primary 55F40
MathSciNet review: 0317327
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Abstract: Let $ X$ be a finite skeleton of the classifying space of $ \operatorname{BSp} (n),{\gamma _0} \to \operatorname{BSp} (n)$, the classifying bundle for $ \operatorname{Sp} (n)$ vector bundles and $ \gamma \to X$ the restriction of $ {\gamma _0}$ over $ X$. If $ \xi \to X$ is another $ \operatorname{Sp} (n)$ vector bundle, the Pontryagin classes $ {p_q}(\xi )$ must be congruent to $ d_1^q{p_q}(\gamma )$ modulo certain odd primes. Equality obtains if $ \xi $ is the restriction over $ X$ of a $ {\xi _0} \to {\operatorname{BSp}}(n)$. In particular, $ {\operatorname{Sp}}(m)$ vector bundles $ \theta $ over $ {\operatorname{BSp}}(n)$ have $ p(\theta ) = 1$ if $ m < n$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0317327-8
Keywords: Pontryagin classes, symplectic cobordism, Landweber operations
Article copyright: © Copyright 1973 American Mathematical Society