On Equivariant Poincar\'e Duality, Gysin Morphisms and Euler Classes

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.