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Proceedings of the American Mathematical Society

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

 

 

On the cohomology Chern classes of the $ K$-theory Chern classes


Authors: Mi-soo Bae Smith and Larry Smith
Journal: Proc. Amer. Math. Soc. 26 (1970), 209-214
MSC: Primary 57.32
DOI: https://doi.org/10.1090/S0002-9939-1970-0267598-9
MathSciNet review: 0267598
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Abstract: Let $ \xi $ be a vector bundle over a finite complex and $ {\gamma ^i}\xi $ its $ i$th-$ K$ theory Chern class. We first show that

$\displaystyle {c_n}{\gamma ^i}\xi = (i - 1)!S(n,i){c_n}\xi + {\text{decomposables}},$

where $ S(n,i)$ is a Stirling number of the second kind. We apply this result to show that certain multiples of the $ e$-invariant of a map $ {S^{2m - 1}} \to {S^{2n}}$ must always be integral.

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DOI: https://doi.org/10.1090/S0002-9939-1970-0267598-9
Keywords: $ K$-theory, characteristic classes, Chern classes, $ e$-invariant, Stirling numbers
Article copyright: © Copyright 1970 American Mathematical Society