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The cross product takes classes on two factors to a class on their product. Its multiplication rule has the graded tensor product sign: $(a\times b)(c\times d)=(-1{)}^{|b||c|}(ac)\times (bd)$. The Künneth theorem makes these products an additive basis when the integral cohomology of the finite cell complexes is torsion-free.

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.