A central advantage of cohomology theory over homology -- at least in terms of richness of structure and strength as an invariant -- is the additional ring structure from the cup product. Recall that this arises from applying the cohomology functor to the following inclusion map of topological spaces $$X \hookrightarrow X \times X$$ where each $x \in X$ is mapped to $(x,x)$ in the product. The "key" insight here is that homology theory lacks an analogous structure precisely because there is no natural candidate for a continuous map $X \times X \to X$. Fair enough.

But Lie groups provide examples of spaces where there is a *great* candidate for such a map: the group multiplication. I expected that this would make *homology* of Lie groups interesting by imposing some nice multiplicative structure on homology generators inherited from the group multiplication. On the other hand, the cohomology ring would reveal nothing that you couldn't already learn from the cohomology of the underlying manifold independent of group structure.

Is this wrong? Why is the literature full of material on Lie group cohomology whereas Lie group homology is relatively sparse?

I suspect that maybe this product structure is not even well defined on the level of homology, but I'm not sure how one would prove that.

Pontryagin productin homology (under certain conditions). $\endgroup$1more comment