The aim of these notes, originally intended as an appendix to a book on the foundations of equivariant cohomology, is to set up the formalism of the $G$-equivariant Poincar\'e duality for oriented $G$-manifolds, for any connected compact Lie group $G$, following the work of J.-L. Brylinski leading to the spectral sequence $$\mathop{ \rm Extgr}\nolimits_{ H_G}(H_{ G,\rm c} (M),H_G)\Rightarrow H_{ G}(M)[d_{ M}]\,.$$ The equivariant Gysin functor $(\_)_!:=\Omega_{ G}(\_)\in\mathcal D^{ +}(\mathord{ \rm DGM}(H_{ G}))$ (resp. $(\_)_{ *}:=\Omega_{ G,\rm c}(\_)$) is then defined in the category of oriented $G$-manifolds and proper maps (resp. unrestricted maps) with values in the derived category of the category of differential graded modules over $H_{ G}$, as the composition of the Cartan complex of equivariant differential forms functor $\Omega_{ G,\rm c}(\_)$ (resp. $\Omega_{ G}(\_)$) with the duality functor $I\mkern-4.5muR\,{ \rm Hom}_{ H_{ G}}^{ \bullet}(\_,H_{ G})$ and the equivariant Poincar\'e adjunction $I\mkern-4.5muD_{ G} (M):\Omega_{ G} (M)[d_{ M}]\to I\mkern-4.5muR\,{ \rm Hom}_{ H_{ G}}^{ \bullet}(\Omega_{ G,\rm c} (M),H_{ G} )$ (resp. $I\mkern-4.5muD_{ G}' (M):\Omega_{ G,\rm c} (M)[d_{ M}]\to I\mkern-4.5muR\,{ \rm Hom}_{ H_{ G}}^{ \bullet}(\Omega_{ G} (M),H_{ G} )$). Equivariant Euler classes are next introduced for any closed embedding $i:N\subseteq M$ as ${ \rm Eu}_{ G}(N,M):=i^{ *}i_{ !}(1)$ where $i^{ *}i_{ !}:H_{ G}(N)\to H_{ G}(N)$ is the push-pull operator. Some localization and fixed point theorems finish the notes. The idea of introducing Gysin morphisms through an equivariant Poincar\'e duality formalism \`a la Grothendieck-Verdier has many theoretical advantages and is somewhat uncommon in the equivariant setting, warranting publication of these notes.
Categories: math.AT math.GR