On Equivariant Poincar\'e Duality, Gysin Morphisms and Euler Classes
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Publication:6283121
arXiv1702.03889MaRDI QIDQ6283121
Publication date: 13 February 2017
Abstract: 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 $$mathop{
m Extgr}
olimits_{H_G}(H_{G,
m c} (M),H_G)Rightarrow H_{G}(M)[d_{M}],.$$ The equivariant Gysin functor $(_)_!:=Omega_{G}(_)inmathcal D^{+}(mathord{
m DGM}(H_{G}))$ (resp. $(_)_{*}:=Omega_{G,
m c}(_)$) 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 $Omega_{G,
m c}(_)$ (resp. $Omega_{G}(_)$) with the duality functor $Imkern-4.5muR,{
m Hom}_{H_{G}}^{�ullet}(_,H_{G})$ and the equivariant Poincar'e adjunction $Imkern-4.5muD_{G} (M):Omega_{G} (M)[d_{M}] o Imkern-4.5muR,{
m Hom}_{H_{G}}^{�ullet}(Omega_{G,
m c} (M),H_{G} )$ (resp. $Imkern-4.5muD_{G}' (M):Omega_{G,
m c} (M)[d_{M}] o Imkern-4.5muR,{
m Hom}_{H_{G}}^{�ullet}(Omega_{G} (M),H_{G} )$). Equivariant Euler classes are next introduced for any closed embedding $i:Nsubseteq M$ as ${
m Eu}_{G}(N,M):=i^{*}i_{!}(1)$ where $i^{*}i_{!}:H_{G}(N) o H_{G}(N)$ 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.
Equivariant homology and cohomology in algebraic topology (55N91) Derived categories and associative algebras (16E35)
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