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:

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