Let $X,Y$ be CW complexes. By Kunneth formula, we have a group isomorphim
$$ H^n(X\times Y;G) \cong \oplus\_{p+q=n} H^p(X;H^q(Y;G))$$
Is there a natural map realizing this isomorphism?



Naturality of a Kunneth formula for cohomology
I came across this question in my 1961 DPhil Thesis; this was written up in two papers which are available from my [Publication List](http://pages.bangor.ac.uk/~mas010/publicfull.htm), 


[3]. ``Cohomology with chains as coefficients'', *Proc. London Math. Soc*. (3) 14 (1964), 545-565. 


[4]. ``On K\"{u}nneth suspensions'', *Proc. Camb. Phil. Soc.* 60 (1964) 713-720. 


It is shown in the first paper that the isomorphism could be chosen to be natural with respect to maps of $X$ but **not** with respect to maps of either $Y$ or of $G$. The naturality was important for the second paper, which was poorly titled: it should have been something like "$k$-invariants of function spaces". Paper [3] also contains as an Appendix formulae for this isomorphism in some special cases


The aim of this work was to investigate the Postnikov system of $Y^X$ by induction on the Postnikov system of $Y$. Michael Barratt suggested this as another way of determining in some cases the extensions involved in his work on exact sequences of track groups; such a determination in his paper "Track Groups II" used Whitney tube systems! 


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