K-theory transfer of a finite covering ($p_{!}:K^0(Y)\to K^0(X)$)

For a finite covering map $p:Y\to X$, define the fiber of the transferred vector bundle by $(p_{!}V{)}_x={⨁}_{y\in p^{-1}(x)}{V}_y$. Local trivializations of the covering map make this a vector bundle, and compatibility with direct sums extends it to Topological K-theory.

For a finite covering map $p:Y\to X$, the K-theory transfer of a finite covering obeysOn each fibre, this is the natural linear isomorphism distributing a tensor product over the direct sum of the fibers of $b$. It gives the formula for vector bundles and hence for their virtual differences.

Now suppose $X$ is a finite CW complex and the covering map has constant degree $n\ge 1$. The class $p_{!}(1)$ is the permutation vector bundle of rank $n$, so $\eta =p_{!}(1)-n$ has rank zero. By nilpotence of rank-zero K-theory classes, some power of $\eta$ is zero. After localization of a ring at $n$, the class $p_{!}(1)=n+\eta$ is a unit, with inverse given by a finite geometric sum in $-\eta /n$. If $p^{\ast }a=0$, the projection formula gives $a{p}_{!}(1)=0$, hence $a=0$ after this localization. Thus pullback is injective after inverting $n$. The finite-CW hypothesis supplies the nilpotence step.