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  "body": "\n## Definition\n\nLet $G$ be a topological group and let $X$ be a space with a continuous left [[action|0WCY42EV9V]] of $G$.\nLet $EG \\to BG$ be a [[universal principal $G$-bundle|1HDH716Z24]]: $EG$ is a contractible space with a free right action of $G$, and $BG = EG / G$ is the [[classifying space|WSZ40H5E4T]].\nThe *Borel construction* (or homotopy quotient) is\n$$X_G = EG \\times_G X = (EG \\times X) / \\sim, \\qquad (e g, x) \\sim (e, g x) \\text{ for } g \\in G.$$\nFor a commutative ring $R$, the *equivariant cohomology* of $X$ is\n$$H^\\ast_G(X; R) = H^\\ast(X_G; R),$$\nthe [[singular cohomology|6N7BC70CX0]] of the Borel construction.\n\n## Remarks\n\nUp to isomorphism, $H^\\ast_G(X; R)$ does not depend on the choice of $EG$.\nThe projection $X_G \\to BG$ makes $H^\\ast_G(X; R)$ an algebra over $H^\\ast(BG; R) = H^\\ast_G(\\mathrm{pt}; R)$.\n\nExample: let $G = S^1$ act on a point.\nOne can take $EG = S^\\infty$, the unit sphere in $\\C^\\infty$, so $X_G = BS^1 = \\C P^\\infty$ and $H^\\ast_{S^1}(\\mathrm{pt}; \\Z) \\cong \\Z[u]$ with $u$ of degree $2$.\n\nNon-example: the ordinary cohomology of the orbit space $X / G$ is not the equivariant cohomology in general.\nFor $S^1$ acting on a point, $X / G$ is a point with cohomology $\\Z$ in degree $0$ only, unlike $\\Z[u]$.\nIf $G$ is a compact Lie group acting freely on $X$, then the map $X_G \\to X/G$ is a fibration with contractible fibre $EG$, and $H^\\ast_G(X; R) \\cong H^\\ast(X/G; R)$.\n",
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