Künneth theorem

For chain complexes of free modules over a principal ideal domain $R$, the Künneth theorem gives a natural split short exact sequencealthough the splitting is not natural. Over a field the Tor term vanishes, and the cohomological cross product gives $H^{\ast }(X\times Y;k)\cong H^{\ast }(X;k){\otimes }_k{H}^{\ast }(Y;k)$ as graded rings under standard finiteness hypotheses.

For classes on $X$ and $Y$, apply the induced maps on cohomology of the coordinate projections and take their cup product on $X\times Y$. The result has degree $|a|+|b|$. On a tensor product of graded rings, the sign rule $(a\otimes b)(c\otimes d)=(-1{)}^{|b||c|}ac\otimes bd$ makes this a ring homomorphism.

If $k$ is a field and $H^{\ast }(F;k)$ has finite total dimension, the cohomology cross product gives the displayed graded-ring isomorphism for arbitrary spaces. Over a field, every chain complex splits into its homology with zero differential and contractible summands. The Eilenberg–Zilber theorem thus replaces the fiber chain complex by a bounded finite-dimensional complex; dualizing its tensor product with $C_{\ast }(X;k)$ gives a finite direct sum of shifted cochain complexes of $X$. Its cohomology is the displayed tensor product. Finiteness prevents the infinite-product obstruction that can occur for two infinite discrete spaces. The standard product formula and its finiteness hypothesis are discussed in pi.math.cornell.edu/~hatcher/AT/ATch3.pdf, Theorem 3.15.

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.

For $d\ge 1$ and $n>1$, no factor-permutation-invariant map $f:(S^d{)}^n\to S^d$ restricts to the identity on the diagonal. The Künneth theorem makes the degree-$d$ cohomology a direct sum of $n$ copies of $\mathbb{Z}$. Symmetry forces $f^{\ast }\alpha =a{\sum }_i{u}_i$ for one integer $a$. Pulling back to the diagonal multiplies this coefficient by $n$, so the identity restriction would give $na=1$. The contradiction uses integral coefficients and applies in both even and odd sphere dimensions.

The singular-chain product induces this natural pairing on homology. Composing it with multiplication of a loop space gives the Pontryagin product on loop-space homology. This is different from the vector cross product.

The singular chain complex of a product is naturally chain-homotopy equivalent to the tensor product of the factors' chain complexes. The shuffle map and the front-face/back-face map implement the equivalence. It is the chain-level input to the Künneth theorem.

For spaces whose integral cohomology has only free and $\mathbb{Z}/2$ summands, let $F$ encode free ranks and $T$ count order-two summands by degree. An additively split integral Künneth theorem gives $F_{A\times B}=F_A{F}_B$ and the displayed rule for $T$. The $t^{-1}$ term is the cohomological degree shift of the Tor functor contribution. The formula requires this torsion type and the usual finite-type hypotheses.