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  "body": "\n## Summary\n\nEquivariant cohomology is a cohomology theory for spaces with an action of a topological group $G$.\nIn the Borel construction, one replaces the space $X$ by the homotopy quotient $EG \\times_G X$ and takes its ordinary cohomology.\nFor a free action of a compact Lie group this is the cohomology of the orbit space.\nFor the action on a point it is the cohomology of the [[classifying space|WSZ40H5E4T]] $BG$.\nIt is a main tool for localization theorems, such as the Atiyah–Bott formula, and for the study of fixed points.\n\n## Explanations\n\n### L1 human\n\nWhen a group acts on a space, the plain quotient often loses information, for example when points have large stabilisers.\nThe Borel construction first lets the group act freely by pairing each point with a point of a big contractible space on which the group acts freely.\nThen it takes the quotient, which is now well behaved, and computes its cohomology.\nFor the circle group acting on a single point, the result is a polynomial ring in one variable of degree $2$.\nYet the plain quotient is just a point.\n",
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