Relative topological K-theory ($K^i(X,A)$)

For a nonempty subcomplex $A\subset X$, relative Topological K-theory is $K^i(X,A)={\tilde{K}}^i(X/A)$, where the collapsed subcomplex is the basepoint. It fits into the K-theory six-term exact sequence.

The relative topological K-theory productis 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.