External product in topological K-theory ($⊠:K^0(X)\times K^0(Y)\to K^0(X\times Y)$)

For compact Hausdorff topological spaces $X,Y$ and complex vector bundles $E\to X$, $F\to Y$, the external product in Topological K-theory iswhere $p_X,p_Y$ are the two projections. Pullback places both bundles over the same product space, so their fibrewise tensor product is defined. The construction respects direct sums and therefore extends bilinearly to differences of bundle classes in the Grothendieck group.

For example, external multiplication by the trivial line bundle on a point just pulls the other class back along $X\times \{\ast \}\to X$. When $X=Y$, pulling the external product back along the diagonal gives the internal ring product on $K^0(X)$. The external product combines information from separate spaces; its existence alone does not assert a Künneth isomorphism for every pair of spaces.