 Projection formula for the K-theory transfer

The K-theory transfer of a finite covering distributes over a pulled-back tensor product of vector bundles:For an $n$-sheeted covering map, $p_{!}(1)$ is the permutation vector bundle of rank $n$. Its difference from $n$ is a nilpotent element by nilpotence of rank-zero K-theory classes, so it is a unit after inverting $n$. Thus pullback on Topological K-theory becomes injective after this localization of a ring.