 Integral cohomological Künneth theorem for finite cell complexes

For finite cell complexes, the cohomological cross product gives a split, but not naturally split, short exact sequence $0\to {⨁}_{p+q=n}{H}^p(X;\mathbb{Z})\otimes H^q(Y;\mathbb{Z})\to H^n(X\times Y;\mathbb{Z})\to {⨁}_{p+q=n+1}{\mathrm{Tor}}_1(H^p(X;\mathbb{Z}),H^q(Y;\mathbb{Z}))\to 0$. The Tor degree is $n+1$ for cohomology, unlike the $n-1$ shift in the homological version. If one factor's cohomology is free, the Tor term vanishes and the resulting ring is the graded tensor product.