Talk:Equivariant homology (Ex)

Latest revision as of 13:19, 2 September 2013

We can simply use the Borel-construction for equivariant homology. This takes an unequivariant homology theory $<wikitex>; We can simply use the Borel-construction for equivariant homology. This takes an unequivariant homology theory $\mathcal{K}$ and defines for a group $G$ and any $G$-space $X$ the $G$-equivariant homology to be $$ \mathcal{K}^G_*(X) = \mathcal{K}_*(EG\times_G X) $$ i.e., the equivariant homology of $X$ is the unequivariant homology of the homotopy orbits of the $G$-action on $X$. Of course we then get $$ \mathcal{K}^G_*(G/H) = \mathcal{K}_*(EG\times_G G/H) = \mathcal{K}_*(EG/H) =\mathcal{K}_*(BH) $$ as $EG$ is a contractible space on which $H \subset G$ acts freely, and hence $EG$ is a model for $EH$. This also comes from a functor from groupoids to spectra as follows: Given a groupoid $\mathcal{G}$ we associate to it the spectrum $$ | \mathcal{N}( \mathcal{G}) | _{+} \wedge \mathcal{K}$$ where we view $\mathcal{K}$ as the spectrum associated to the unequivariant homology theory $\mathcal{K}$. To see what the $G$-homology theory $\mathcal{K}^G$ is, it suffices to understand the composite functor $$\xymatrix{ Or(G) \ar[r] & \text{groupoids} \ar[r] & \text{Sp} } $$ where the functor $Or(G) \to \text{groupoids}$ takes a homogenous space $G/H$ to its transport groupoid $\overline{G/H}$, which is equivalent to the group $H$ viewed as groupoid. Hence we see that $$| \mathcal{N}(\overline{G/H})|_+ \wedge \mathcal{K} \simeq |\mathcal{N}(H)|_+ \mathcal{K} \simeq BH_+\wedge \mathcal{K}.$$ 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. </wikitex>\mathcal{K}$ and defines for a group $G$ and any $G$-space $X$ the $G$-equivariant homology to be

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

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

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

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

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

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

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

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

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

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

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

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.