8:47 PM
@ThomasRot You will have tough luck getting a product formula in Borel cohomology. Let $X$ and $Y$ be two spaces with an action by a compact Lie group $G$. Then you are trying to calculate $\text{Tor}_{C_* G}(C_* X, C_* Y)$, using the notation/definitions from Gugenheim-May "differential torsion products", though you can just define this using a bar construction.
They explain that things are best behaved when you equip one of your terms with a "Kunneth resolution", essentially meaning 1) a resolution built out of copies of $C_* G$, such that 2) the homology is flat over the ground ring. In this case, you have a spectral sequence $\text{Tor}_{HG}(HX, HY)$ to your desired output groups.
To obtain something like this, you probably want to put some assumptions on your group (torsion-free homology) and the homology of your orbits $BH$, and then replace $X$ with $X \times EG$. Another good reference that contains many of the results in GM is "6 model structures on dg-algebras", by Barthel-May-Riehl.
A great test case for all of this is to let $G = SU(2)$ and make sure all of your orbits are of type $*, S^2$, or $SU(2)$. Then all cohomology rings involved are polynomial, and things are calculable.
I would write $C_* X$ in some Morse-Bott type way, built out of 'critical orbits' (or a G-CW decomposition), the same for $Y$, and try to understand the resulting way to build up $X \times Y$ out of products of orbits. You get a spectral sequence calculating the equivariant homology coming from the equivariant homologies of products of orbits, which you can determine more easily using the tor-spectral sequence above, especially under the nice assumptions we make on the homologies.
These are basically the same story.
I am not sure if there is anything useful in there, but I suggest tom Dieck's book transformation groups. There might be something around the section on localization theorems, in the case that your groups are sufficiently nice, the intuition being that we mainly need to figure out how to glue together the reducible subspaces.
Addendum to "nice test case" above: Try $S^3$ as the one-point compactification of $\mathfrak{su}(2)$, so a spherical suspension of the $S^3$ action on $S^2$, for one of your spaces. I think it is a fun one.