TITLE: Homology versus cohomology of Lie groups
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: Dear Pierre,
I apologize if I seem harsh again, but it is not true that the possible diagonal maps (in cohomology)
are irrelevant for the application you are describing.  One of the very nicest ways to compute $H^*(BG;k)$ is to use the Rothenberg-Steenrod spectral sequence which converges from 
$Ext_{H_*(G;k)}(k,k)$ to $H^*(BG;k)$.  It is precisely the algebra structure on $H_*(G;k)$ that is relevant, hence the coalgebra structure on $H^*(G;k)$, and there is no reason to restrict to $k=\mathbf{Q}$. One answer to the original question is to point to this spectral sequence, which has been in common use since the early 1960's. From the perspective of Lie groups in particular, what is calculationally easy but perhaps conceptually surprising is how often the algebra $H_*(G;k)$ is (graded) commutative even though the product on $G$ is almost never homotopy commutative.  Commutative and cocommutative Hopf algebras over $\mathbf{Q}$ are classified, and in this case the relevant $Ext$ groups are trivial to compute.