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# Kunneth theorem

How is homology group of $X\times Y$ related to the homology groups of $X$ and $Y$?

**Kunneth Theorem.** Suppose $X$, $Y$ are cell complexes. Let $H_*(X)$ be the homology with coefficients in a principal ideal domain $R$. Then, for any $k>0$ the following is an exact sequence, over $R$:
$$0\to \bigoplus_{i+j=k} H_i(X)\otimes H_j(Y)\to H_k(X\times Y) \to \bigoplus_{i+j=k-1}\mathrm{Tor}(H_i(X),H_j(Y))\to 0,$$
where $\oplus$ is the direct sum, $\otimes$ is the tensor product over $R$, and $\mathrm{Tor}$ denotes the Tor functor. Furthermore this sequence splits, i.e. the middle term is isomorphic to the direct sum of the left and right terms.

If $R$ is a field, the Tor functor is always trivial and the theorem turns into the following.

**Kunneth formula.** Over a field,
$$H_k(X\times Y)\simeq \bigoplus_{i+j=k} H_i(X)\otimes H_j(Y).$$