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

Abstract

Let ξ \xi be a vector bundle over a finite complex and γ i ξ {\gamma ^i}\xi its i i th- K K theory Chern class. We first show that \[ c n γ i ξ = ( i − 1 ) ! S ( n , i ) c n ξ + decomposables , {c_n}{\gamma ^i}\xi = (i - 1)!S(n,i){c_n}\xi + {\text {decomposables}}, \] where S ( n , i ) S(n,i) is a Stirling number of the second kind. We apply this result to show that certain multiples of the e e -invariant of a map S 2 m − 1 → S 2 n {S^{2m - 1}} \to {S^{2n}} must always be integral.