= Projection formula for the K-theory transfer
{title2=$p_!(p^*a\cdot b)=a\cdot p_!b$}
The <K-theory transfer of a finite covering> distributes over a pulled-back <tensor product of vector bundles>:
$$
p_!(p^*a\cdot b)=a\cdot p_!b.
$$
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>.
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