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Reference: Atiyah & Bott, The moment map and Equivariant cohomology

Question: What could be the motivation and the intuition behind the construction of the space $M_G=EG\times _G M$? When I am studying the equivariant cohomology, I have no idea about why we would like to define the equivariant cohomology as the ordinary cohomology of $M_G$.

A simple case is, if $G$ is a compact Lie group and acts freely on $M$, then $M_G$ is simply $M/G$, the space of all $G$-orbits.

But, in general cases, how to understand the picture of $M_G$? And, how does $M_G$ relate to $M/G$?

For example, take $G=S^1\times S^1$ and $M=\mathbb P^1$, and the action is given by $(t_0,t_1): [x,y]\mapsto [t_0x, t_1y]$. Then this action is not free and it has two fixed point $[1,0]$ and $[0,1]$. In this case, what is the difference between $M/G$ and $M_G$?

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    $\begingroup$ $EG$ is nonequivariantly homotopy equivalent to a point. Naively, we can expect that replacing a space by its homotopy equivalent one doesn't change anything significant, thus we identify $M$ with $EG \times M$. But now we have converted an arbitrary $G$-action into a free one, where we understand what "G-equivariant" means - just functions (sheaves etc) on the $G$-factor. This logic works fine for ordinary equivariant cohomology, but breaks down for more complex cohomology theories (e.g. K-theory or bordism), the reason being that $EG \ne \ast$ as $G$-spaces. $\endgroup$
    – Anton Fetisov
    Commented Jan 27, 2017 at 23:18
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    $\begingroup$ In general we need to develop the full $G$-equivariant homotopy theory, with $G$-spaces, $G$-spectra etc, and it can be quite nonobvious what should constitute an equivariant analogue of some classic cohomology theory: for K-theory we have to work with equivariant bundles, for elliptic cohomology the construction is quite intricate, and for an arbitrary cohomology theory there is no reason to suspect equivariant analogue. However, for ordinary cohomology the situation is much simpler: the homotopy quotient construction reduces the equivariant theory to a nonequivariant one uniquely. $\endgroup$
    – Anton Fetisov
    Commented Jan 27, 2017 at 23:21

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In homotopy theory and in homological algebra, there a general idea that for any given operation that you might want to perform, there's a class of "nice" objects for that operation. It is a good idea to replace your object by another one which is "nice" and "equivalent" in some appropriate sense before performing the operation.

If the operation is $-/G$ for some group $G$, then the class of "nice" objects is $G$-spaces with a free action. It's a good idea to replace $M$ by $M\times EG$ before performing $-/G$.

If the operation is $\times_BX$ for some map $X\to B$, the the class of "nice" objects are the spaces equipped with a map to $B$ which is a fibration.

If the operation is $-\otimes_RM$ for some ring $R$ and some $R$-module $M$, then the class of "nice" objects are the complexes of projective modules.

If the operation is $Hom_R(M,-)$ for some ring $R$ and some $R$-module $M$, then the class of "nice" objects are the complexes of injective modules.

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