Talk:Equivariant homology (Ex)

We can simply use the Borel-construction for equivariant homology. This takes an unequivariant homology theory $\mathcal{K}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\mathcal{K}$ and defines for a group $G$$G$ and any $G$$G$-space $X$$X$ the $G$$G$-equivariant homology to be

$\displaystyle \mathcal{K}^G_*(X) = \mathcal{K}_*(EG\times_G X)$ $$\displaystyle \mathcal{K}^G_*(X) = \mathcal{K}_*(EG\times_G X)$$

i.e., the equivariant homology of $X$$X$ is the unequivariant homology of the homotopy orbits of the $G$$G$-action on $X$$X$. Of course we then get

$\displaystyle \mathcal{K}^G_*(G/H) = \mathcal{K}_*(EG\times_G G/H) = \mathcal{K}_*(EG/H) =\mathcal{K}_*(BH)$ $$\displaystyle \mathcal{K}^G_*(G/H) = \mathcal{K}_*(EG\times_G G/H) = \mathcal{K}_*(EG/H) =\mathcal{K}_*(BH)$$

as $EG$$EG$ is a contractible space on which $H \subset G$$H \subset G$ acts freely, and hence $EG$$EG$ is a model for $EH$$EH$.

This also comes from a functor from groupoids to spectra as follows:

Given a groupoid $\mathcal{G}$$\mathcal{G}$ we associate to it the spectrum

$\displaystyle | \mathcal{N}( \mathcal{G}) | _{+} \wedge \mathcal{K}$ $$\displaystyle | \mathcal{N}( \mathcal{G}) | _{+} \wedge \mathcal{K}$$

where we view $\mathcal{K}$$\mathcal{K}$ as the spectrum associated to the unequivariant homology theory $\mathcal{K}$$\mathcal{K}$.

To see what the $G$$G$-homology theory $\mathcal{K}^G$$\mathcal{K}^G$ is, it suffices to understand the composite functor

$\displaystyle \xymatrix{ Or(G) \ar[r] & \text{groupoids} \ar[r] & \text{Sp} }$

where the functor $Or(G) \to \text{groupoids}$$Or(G) \to \text{groupoids}$ takes a homogenous space $G/H$$G/H$ to its transport groupoid $\overline{G/H}$$\overline{G/H}$, which is equivalent to the group $H$$H$ viewed as groupoid.

Hence we see that

$\displaystyle | \mathcal{N}(\overline{G/H})|_+ \wedge \mathcal{K} \simeq |\mathcal{N}(H)|_+ \mathcal{K} \simeq BH_+\wedge \mathcal{K}.$ $$\displaystyle | \mathcal{N}(\overline{G/H})|_+ \wedge \mathcal{K} \simeq |\mathcal{N}(H)|_+ \mathcal{K} \simeq BH_+\wedge \mathcal{K}.$$

Thus we see that we have the correct coefficients as stated in the exercise and a little computation in coends also shows that the equivariant homology theory associated to this groupoid-spectrum is given by the above Borel-construction.