# $G$-equivariant intersection theory using differential topology?

For $$G$$-CW complexes, in the sense that the local building blocks are $$G \times_{H} V$$ where $$H < G$$ and $$V$$ is a $$G$$-representation, one can define $$RO(G)$$-graded cohomology theories. Standard constructions from algebraic topology, including Poincare duality, Thom isomorphisms, cup and cap products etc, could be constructed in such theory. See the reference Equivariant ordinary homology and cohomology. (Perhaps not the most canonical reference, but it includes the necessary backgrounds).

Here is the question: is there a way to construct an analogue of intersection product on the "middle-dimensional" cohomology? Is it possible to recover such product by studying intersection behavior of $$2$$ middle-dimensional $$G$$-CW subcomplexes/$$G$$-invariant submanifolds in general position?

Remark: It is well-known that equivariant transversality is hard to achieve. Let's just assume that we can indeed put things in general positions.

It should work essentially the same as it does nonequivariantly, if you assume, as you say, that you start with things in general position. Part of why Stefan and I wrote that book was to point out that all the necessary machinery is there for this sort of thing, including the Thom isomorphism and Poincaré duality. So if you have, say a $$2V$$-dimensional manifold $$M$$ and within it two $$V$$-dimensional manifolds in general position, you can take the cup product of their dual classes, evaluate on the fundamental class $$[M]$$, and get an element of $$H_0^G(M)$$ corresponding to the intersection of the manifolds. You can push this into the Burnside ring $$A(G)$$ if you prefer. You do have to be careful about what you mean by $$V$$-dimensional manifold and you will have to think about whether you want the final answer to lie in $$H_0^G(M)$$ or $$A(G)$$, depending on what you're trying to do, but there shouldn't be any technical issue with doing all this (other than lack of transversality in general, of course).

You may want to take a look at

Klein, J.R., Williams, B. Homotopical intersection theory, II: equivariance. Math. Z. 264(2010),849–880.

An arXiv version appears here:

https://arxiv.org/abs/0803.0017

In this paper, we set up an intersection theory in the equivariant case (avoiding the need for transversality).

• Things were worked out for finite groups in this paper. Is there any fundamental difficulty generalizing the arguments to general compact Lie groups? Dec 24 '20 at 16:56
• I have not thought about that. Some of it could possibly generalize, but I doubt all. Dec 25 '20 at 14:34