Talk:Equivariant homology (Ex)

We can simply use the Borel-construction for equivariant homology. This takes an unequivariant homology theory $\mathcal{K}$$<wikitex>; We can simply use the Borel-construction for equivariant homology. This takes an unequivariant homology theory $\mathcal{K}$ and defines for a group $G$ and any $G$-space $X$ the $G$-equivariant homology to be $$ \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) $$ as $EG$ is a contractible space on which $H \subset G$ acts freely, and hence $EG$ is a model for $EH$. </wikitex>\mathcal{K}$ and defines for a group $G$$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$.