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For the K-theory transfer of a finite covering, let $V\to Y$ be a vector bundle and defineOn an evenly covered neighborhood the $n$ sheets identify this with a direct sum of $n$ local vector bundles. On overlaps the identifications permute the sheets and apply the transition maps of $V$, so these local descriptions glue to a vector bundle on $X$. This operation respects direct sums, hence extends to a homomorphism of Grothendieck groupsIt obeys the projection formula for the K-theory transfer:For actual vector bundles, this follows fiberwise by distributing the tensor product of vector bundles over the direct sum; it then extends to their Grothendieck groups.

Let $P=p_{!}(1_Y)$ be the permutation vector bundle of the covering. It has rank $n$ on every connected component, but need not be a trivial vector bundle; in particular one cannot replace $p_{!}{p}^{\ast }$ by multiplication by $n$ integrally. Instead writeThe componentwise nilpotence result from the preceding solution gives ${\eta }^N=0$ for some $N$. In the localization of a ring $K^0(X)\otimes \mathbb{Z}[1/n]$, the finite geometric seriesis an inverse. The projection formula for the K-theory transfer givesThus $p^{\ast }(a)=0$ implies $a=0$ after inverting $n$. More explicitly, $(n+\eta {)}^{-1}{p}_{!}$ is a left inverse, proving

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.