Kunneth formula for homology

Statement

Suppose $X$ and $Y$ are topological spaces. We then have, for any $n\geq 0$:

$H_{n}(X\times Y)\cong \sum _{i+j=n}H_{i}(X)\otimes H_{j}(Y)\oplus \sum _{p+q=n-1}{\text{Tor}}(H_{p}(X),H_{q}(Y))$

Results used

The Kunneth formula combines the Kunneth theorem and the Eilenberg-Zilber theorem.

Proof

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