The Künneth Formula

The Künneth formula computes the homology and cohomology of a product space in terms of the factors, providing a multiplicative decomposition analogous to the tensor product of graded algebras.

The Künneth Theorem

Theorem7.7Künneth Formula for Homology

Let $X$ and $Y$ be topological spaces. If $H_*(X; \mathbb{Z})$ is finitely generated and torsion-free (or more generally, if $H_n(X)$ is a flat $\mathbb{Z}$ -module for all $n$ ), then $H_n(X \times Y; \mathbb{Z}) \cong \bigoplus_{p+q=n} H_p(X; \mathbb{Z}) \otimes H_q(Y; \mathbb{Z})$ In general, there is a split short exact sequence $0 \to \bigoplus_{p+q=n} H_p(X) \otimes H_q(Y) \to H_n(X \times Y) \to \bigoplus_{p+q=n-1} \operatorname{Tor}_1(H_p(X), H_q(Y)) \to 0$

The proof relies on the Eilenberg-Zilber theorem, which provides a natural chain homotopy equivalence $C_*(X \times Y) \simeq C_*(X) \otimes C_*(Y)$ , reducing the topological problem to an algebraic one about tensor products of chain complexes.

The Cohomological Künneth Formula

Theorem7.8Künneth Formula for Cohomology

If $H^*(X; R)$ and $H^*(Y; R)$ are finitely generated free $R$ -modules (where $R$ is a principal ideal domain), then there is a ring isomorphism $H^*(X \times Y; R) \cong H^*(X; R) \otimes_R H^*(Y; R)$ where the right side has the tensor product of graded-commutative algebras structure: $(\alpha \otimes \beta) \cdot (\gamma \otimes \delta) = (-1)^{|\beta||\gamma|}(\alpha \smile \gamma) \otimes (\beta \smile \delta)$

ExampleCohomology ring of the torus

The torus $T^2 = S^1 \times S^1$ has $H^*(T^2; \mathbb{Z}) \cong H^*(S^1; \mathbb{Z}) \otimes H^*(S^1; \mathbb{Z}) \cong \mathbb{Z}[\alpha] / (\alpha^2) \otimes \mathbb{Z}[\beta] / (\beta^2)$ where $|\alpha| = |\beta| = 1$ . Thus $H^*(T^2) \cong \Lambda_\mathbb{Z}[\alpha, \beta]$ , the exterior algebra on two generators of degree $1$ . The generator of $H^2(T^2) \cong \mathbb{Z}$ is $\alpha \smile \beta$ .

Applications

ExampleHigher-dimensional tori

For the $n$ -torus $T^n = (S^1)^n$ , iterated application gives $H^*(T^n; \mathbb{Z}) \cong \Lambda_\mathbb{Z}[\alpha_1, \ldots, \alpha_n]$ In particular, $\dim H^k(T^n; \mathbb{Q}) = \binom{n}{k}$ , and the total Betti number is $\sum_k \binom{n}{k} = 2^n$ .

RemarkCross product vs. cup product

The Künneth isomorphism is realized by the cross product $\times : H^p(X) \otimes H^q(Y) \to H^{p+q}(X \times Y)$ , defined by $\alpha \times \beta = \pi_X^*(\alpha) \smile \pi_Y^*(\beta)$ where $\pi_X, \pi_Y$ are projections. The Künneth theorem states that this cross product is an isomorphism under the given hypotheses.