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

Smith, Mi-soo Bae; Smith, Larry

doi:10.1090/S0002-9939-1970-0267598-9

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

J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI https://doi.org/10.1016/0040-9383%2866%2990004-8

H. Cartan and J. C. Moore, Périodicité des groupes d’homotopie stables des groupes classiques, d’après Bott, Séminaire H. Cartan, Secrétariat mathématique, Paris, 1961. MR 28 #1092.

P. Hoffman, On the unstable $e$-invariant, Topology 4 (1965), 343–349. MR 198471, DOI https://doi.org/10.1016/0040-9383%2866%2990032-2
John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0096594
Larry Smith, On characteristic numbers of almost complex manifolds with framed boundaries, Topology 10 (1971), 237–256. MR 275454, DOI https://doi.org/10.1016/0040-9383%2871%2990008-5
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
Hirosi Toda, A survey of homotopy theory, Sūgaku 15 (1963/64), 141–155 (Japanese). MR 169241