Calculations of cup products in Bredon cohomology

Question

Let $G$ be a finite group. In [1], Bredon defines an equivariant cohomology theory for $G$-CW complexes $H^*_G(X;M)$. The coefficients are taken in modules over the orbit category of $G$, that is, contravariant functors $M:\mathcal{O}_G\to \mathcal{Ab}$ from the category of finite $G$-sets and $G$-maps to the category of abelian groups. This Bredon cohomology has now been generalized and extended in many ways by many authors. Although this question is manily about the "classical" setting described in Bredon's book, I would be interested also in answers which pertain to more sophisticated versions.

I've recently started trying to do some computations of Bredon cohomology, and have been able to compute the groups $H_G^i(X;M)$ in particular cases. Now I need to compute some cup products $$ H^i_G(X;M)\otimes H^{j}_G(X;N)\to H^{i+j}_G(X;M\otimes N), $$ and find I am lacking in tools, or guiding examples, which are available to me in the non-equivariant case. Therefore I ask:

What are some examples in the literature of calculations of cup products in the Bredon cohomology of specific $G$-spaces?

Perhaps one needs to resort to finding a $G$-equivariant approximation of the diagonal, or extending to an $RO(G)$-graded cohomology theory. Then I would appreciate references where this is done to produce calculations of products in the $\mathbb{Z}$-graded theory.

[1] Glen E. Bredon, MR 214062 Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34 .

Answer 1

Frankly, there aren't many calculations out there. Most of the work I know of is on the calculation of the $RO(G)$-graded cohomology of a point, of a projective space, or of $B_GO(n)$. Here are some references and notes on them:

[1] L. G. Lewis, Jr., The $RO(G)$-graded equivariant ordinary cohomology of complex projective spaces with linear $\mathbb{Z}/p$ actions, in Algebraic topology and transformation groups, Lecture Notes in Math. v. 1361, 1988. (MR 979507)

In an appendix, this has the first published account of Stong's calculation of the $RO(G)$-graded cohomology of a point for $G= \mathbb{Z}/p$, where $p$ is prime. Stong calculated the multiplicative structure for $p = 2$ and $3$, and Lewis extended that to all primes. The coefficient system used is the Burnside ring system, which evaluates to the Burnside ring $A(H)$ at the orbit $G/H$.

The main purpose of the paper, though is the calculation of the cohomology of complex projective spaces, that is, the spaces $\mathbb{C}P(V)$ where $V$ is a finite- or countably infinite-dimensional complex representation of $G$. The calculation includes the multiplicative structure.

[2] W. C. Kronholm, The $RO(G)$-graded Serre spectral sequence, Homology, Homotopy Appl. 12 (2010), pp. 75-92. (MR 2607411)

Kronholm gives a similar calculation for real projective spaces, for $G=\mathbb{Z}/2$ and with constant $\mathbb{Z}/2$ coefficients.

[3] D. Dugger, Bigraded cohomology of $\mathbb{Z}/2$-equivariant Grassmannians, Geom. Topol. 19 (2015), pp.113-170. (MR 2240234)

Dugger uses Kronholm's result to calculate the $RO(G)$-graded cohomology of $B_GO(n)$ for $G = \mathbb{Z}/2$ and constant $\mathbb{Z}/2$ coefficients.

There are a couple of other papers out there with partial calculations of the cohomology of a point, but for solid calculations of the cohomology of interesting spaces, this is all I've got.