Recall the notion of the classifying space $|G|$ of a group $G$.
Is it possible to relate the classifying spaces $|A\oplus B|$ and $|A\otimes_{\mathbb{Z}}B|$ of a tensor product or a direct sum of abelian groups $A$ and $B$ with the classifying spaces $|A|$ and $|B|$ of its factors?
In particular, how does group cohomology (i.e. cohomology of classifying spaces of groups) behave under direct sums and tensor products?