Let $G$ be a compact Lie group, $X$ be a (compact, oriented) smooth manifold, with $G$ acts on $X$ smoothly. Then we can talk about the $G$-equivariant homology and cohomology.


My question: In what sense can we have a duality between the equivariant homology and cohomology, in analogue with the Poincare duality between the ordinary homology and cohomology of $X$? In particular, the degree of a nontrivial equivariant (co)homology class could exceed the dimension of the manifold $X$. Then in such case, what does the dual mean, geometrically?



Poincare dual in equivariant (co)homology?
There is a paper by Brion: "Poincaré duality and equivariant (co)homology," *Michigan Math. J.* 48 (2000). As mentioned in one of the comments, he uses Borel-Moore as the homology theory.


That said, one should always be careful about the word "duality" in this context -- often people just mean there's a canonical *isomorphism* $H\_G^k X \rightarrow H^G\_{n-k}X$. (Which does exist using equivariant Borel-Moore homology, when $X$ is smooth.) The geometric meaning is the same as in the ordinary case: if $V$ is a $G$-invariant sub(manifold/variety/cycle/etc) of codimension $k$, it defines a fundamental class in $H^G\_{n-k} X$, which is identified with $H\_G^kX$ by means of the isomorphism.


Duality could also refer to a pairing $H\_G^k X \otimes H\_G^\ell X \rightarrow H\_G^{k+\ell-n}(point)$, given by the equivariant pushforward (integral): $$\alpha\otimes\beta \mapsto \int \alpha\cdot\beta .$$ Using this pairing, and assuming that $H\_G^\*X$ is free as a module over $H\_G^\*(point)$, any $H\_G^\*(point)$-module basis for $H\_G^\*X$ has a dual basis, in the usual sense of linear algebra. (Arguably, this duality pairing is what should really be called "Poincaré duality".) Again, the geometric meaning is similar to the ordinary case: if you're lucky enough to have invariant subvarieties whose classes form a basis for $H\_G^\*X$, the Poincaré dual basis is given by classes of subvarieties intersecting the original ones transversally (if they exist). However, the existence of geometrically defined dual bases is a stronger statement, because the equivariant integral is generally nonzero on classes of degree greater than $\dim X$.


(I should say all this is prejudiced toward the equivariant cohomology rings that usually show up in algebraic geometry, e.g., $H\_G^\*X$ is a free module over $H\_G^\*(point)$, so that it makes sense to talk about dual bases.)


PS: Defining equivariant Borel-Moore homology requires a little more care, since the spaces $EG\times^G X \rightarrow BG$ are infinite-dimensional fiber bundles. But they have finite-dimensional approximations $EG\_m \times^G X \to BG\_m$, so it makes sense to define $$H^G\_k X = H\_{k+\dim BG\_m}(EG\_m\times^G X)$$
for $m\gg0$. The equivariant Poincaré isomorphism is just the ordinary one for these approximation spaces.


