Minor nomenclature question:
What is the standard name for the ring structure induced on the homology of an H-space by its multiplication? I've seen "homology ring" and "Pontrjagin ring."
Hopefully deeper questions:
In this paper (on pp. 1117 & 1134), Samelson states (as best I can understand through Google Translate) that
if a connected subgroup $U$ of a compact Lie group $G$ is not nullhomologous, then
the inclusion $U \hookrightarrow G$ induces an injection in rational homology, and
the rational homology rings of $G$ and the product manifold $U \times (G/U)$ are isomorphic.
It occurs to me that one could use this to understand the (co)homology of homogeneous spaces $G/U$ (or alternately, assuming one knows something about homogeneous spaces, to understand the homology of $G$ in terms of homogeneous spaces and tori). How feasible is this strategy?
When are subgroups nullhomologous? I've never really thought of that before. I'm especially concerned about tori. My thinking is that the cohomology ring of a compact Lie group is an exterior algebra; by Poincaré duality, I would expect tori to generally be nullhomologous, then, because $\dim H_1(G)$ will tend to be too small to support large tori. Is that right?
Are there illuminating examples? How should I think about this?