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Let $X$ be a $G$-space. Are there examples, i.e. conditions or classes of spaces, such that the map induced by the inclusion $X^G\to X$ of the fixed part into the whole space induce a surjection of equivariant homologies $H_{\ast}^G(X^G)\to H_{\ast}^G(X)$ (in all dimensions)?

Now let $X$ be a pointed $G$-space. Define the half smash $EG\ltimes_G X$ as $(EG\times_G X)/(EG\times_G \ast)$. Again, are there examples where the map $H_{\ast}(EG\ltimes_G X^G)\to H_{\ast}(EG\ltimes_G X)$ is a surjection?

Finally, let $Ci(X)$ be the cofibre of the inclusion $i:X\to EG\ltimes_G X$. When is $H_{\ast}(Ci(X^G))\to H_{\ast}(Ci(X))$?

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Incidentally, the definition of equivariant homology as the ordinary homology of the Borel mixing space is not a very nice one, e.g. the module structure over equivariant cohomology is locally nilpotent rather than free. Michel Brion has a (as always very nice) paper defining something that, for compact oriented manifolds, has a Poincar\'e duality to equivariant cohomology.… – Allen Knutson Nov 13 '11 at 13:43
I don't know about this specific cohomology theory, but does it have a long exact sequence property that could answer this? I would expect that this would happen when $X \ X^G$, or something like it, has trivial homology. – Will Sawin Nov 13 '11 at 17:34
I meant to say $X - X^G$. – Will Sawin Nov 13 '11 at 17:35

Here is an example: $X=S^2$, $G=U(1)$, acting by rotation. $X^G$ consists of two points, hence $H_{\ast}(X^G)=H_{\ast} (CP^{\infty} \coprod CP^{\infty})$. The space $EG \times_G X$ is (homotopy equivalent to) the wedge product $CP^{\infty} \vee CP^{\infty}$. The induced map in homology is thus surjective.

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What about the examples for $G=Z_2$? Thanks in advance! – Gao 2Man Nov 14 '11 at 3:39

Is equivariant homology dual to equivariant cohomology ? Because there is a paper of Goresky-Kottwitz-MacPherson that gives condition for the equivariant cohomology of $X$ to inject into that of $X^G$; they even calculate the image in some cases. The paper is "Equivariant cohomology, Koszul duality, and the localization theorem", it's available on Goresky's webpage.

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