Relative product in topological K-theory
= Relative product in topological K-theory
The <relative topological K-theory> product
$$
K^i(X,A)\otimes K^j(X,A)\longrightarrow K^{i+j}(X,A)
$$
is induced by the reduced diagonal $X/A\to(X/A)\wedge(X/A)$. If $X/A$ is a positive-dimensional <sphere>, this diagonal is <null-homotopic>, so the kernel of $K^0(X)\to K^0(A)$ is square-zero.