= 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 $0\to\bigoplus_{p+q=n}H^p(X;\mathbb Z)\otimes H^q(Y;\mathbb Z)\to H^n(X\times Y;\mathbb Z)\to\bigoplus_{p+q=n+1}\operatorname{Tor}_1(H^p(X;\mathbb Z),H^q(Y;\mathbb Z))\to0$. 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>.
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