Proof of Kunneth theorem

Question

What are different ways to prove Kunneth theorem relating singular homology of product space $X * Y$ in terms of homology of $X$ and $Y$? or reference?I know some ways: use cell homology for cell complex that is homotopy equivalent to original space, or similar to the proof of universal coefficient theorem. Is there any others?

Answer 1

The join $X\ast Y$ is the homotopy pushout of the two projections

$$X\xleftarrow{pr_1} X\times Y\xrightarrow{pr_2} Y.$$

This lets you use a Mayer-Vietoris sequence of the form

$$\dots \xrightarrow\Delta H_{*+1}(X\ast Y)\rightarrow H_*(X\times Y)\xrightarrow{pr_{1*}-pr_{2*}}H_*(X)\oplus H_*(Y)\xrightarrow{j_{X*}+j_{Y*}}H_*(X\ast Y)\xrightarrow\Delta H_{*-1}(X\times Y)\rightarrow\dots$$

Now, inspection shows that the inclusion maps $j_X:X\rightarrow X\ast Y$ and $j_Y:Y\rightarrow X\ast Y$ are null-homotopic, so the homomorphism $j_{X\ast}+j_{Y\ast}$ is trivial and the sequence splits

$$H_*(X\times Y)\cong H_*(X)\oplus H_*(Y)\oplus H_{*+1}(X\ast Y)$$

with $H_{*+1}(X\ast Y)$ mapping onto the equaliser of $pr_{1*},pr_{2*}$.

Note the dimension shift, by the way. This makes sense since there is a homotoyp equivalence $X\ast Y\simeq \Sigma X\wedge Y$.