I would like to know if there is a reference for the fact that the following diagram commutes:
$$
\begin{array}{ccccccccc}
0 & \to & H\_\*(X) \otimes H\_\*(Y) & \to & H\_\*(X\times Y) & \to & \mathrm{Tor}\_1^{\mathbb Z}(H\_\*(X),H\_\*(Y)) & \to & 0 \\
&& \downarrow && \downarrow && \downarrow & \\
0 & \to & H\_\*(Y) \otimes H\_\*(X) & \to & H\_\*(Y\times X) & \to & \mathrm{Tor}\_1^{\mathbb Z}(H\_\*(Y),H\_\*(X)) & \to & 0, \\
\end{array} $$
where both rows are from the Künneth formula (essentially copied from [Wikipedia](http://en.wikipedia.org/wiki/K%C3%BCnneth_theorem#Singular_homology_with_coefficients_in_a_principal_ideal_domain)), and the vertical arrows are the natural isomorphisms obtained by flipping the factors.


$H\_\*$ means singular homology - but actually this part isn't so important; a reference for a version with any homology or cohomology theory is all I want, provided that it is not one where the Tor term trivially vanishes.


I have a proof for K-theory (in fact, C\*-algebra K-theory) but I imagine that it is not the first proof.



Reference request: Flipping the factors in the Künneth formula
Larry Taylor just posted a preprint to the arXiv,


<http://arxiv.org/abs/1506.02063>


called *Splitting the Künneth formula*. Commutativity of your diagram is stated as Corollary 7.2, suggesting that the author is not aware of a published proof.


