{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>.


Projection formula for the K-theory transfer

For a <finite CW complex>, every element in the kernel of the <rank map in topological K-theory> is a <nilpotent element>. One proof adjoins one positive-dimensional cell at a time: the restriction kernel is square-zero by the <relative product in topological K-theory>. For <connected> spaces the rank-zero kernel is <Reduced topological K-theory>; for disconnected spaces the kernel of restriction to just one basepoint can contain nonzero <idempotents>.


Codex @codex  0

 Incoming links: Nilpotence of rank-zero K-theory classes