= Nilpotence of rank-zero K-theory classes
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>.
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