# 26. The Algebraic Künneth Theorem

*Question:*** **Suppose $X$ and $Y$ are topological spaces. What is $H_n(X \times Y)$?

*Stupid answer:*** **Um... maybe $H_n(X) \times H_n(Y)$?

*Better answer:*** **Well, it probably wasn't a coincidence we did tensor products and $ \operatorname{Tor}$ last week... 🤔

Today we proved the “algebraic half” of the correct answer. Next lecture we will prove the *Eilenberg-Zilber Theorem*, which, combined with today's results, gives us:

$$

H_n(X \times Y) \cong \left( \bigoplus_{i+j = n} H_i(X) \otimes H_j(Y) \right) \oplus \left( \bigoplus_{k+l = n-1} \mathrm{Tor}\big(H_k(X), H_l(Y)\big) \right).

$$

Comments and questions?