Equivariant cohomology of slice groupoids

Abstract

Let $G$ be a compact Lie group, let $M$ be a smooth manifold with a $G$-action. Then all the data of this model is contained in the action groupoid $G\ltimes M$. If $U_y$ is a small enough neighbourhood of $y\in M/G$, the slice theorem says that \begin{equation*} \pi^{-1}(U_y)=S_{x}\times_{G_{x}} G \end{equation*} where $x$ is a point in the $y$ orbit, $S_x$ is the slice of $x$ and $G_x$ is the isotropy group of $x$. An alternative approach to describe group actions on spaces is through the language of groupoids. Local properties of Lie groupoids are often studied via linearization theorems. One can compute the equivariant cohomology $H_G^*(\pi^{-1}(U_y))$ of $\pi^{-1}(U_y)$ using the Weil model or the Cartan model. Also by the homotopy theory, the equivariant cohomologies $H_G^*(\pi^{-1}(U_y))$ and $H_{G_x}^*(S_x)$ are isomorphic. In this paper, we explicitly construct a natural chain map between the Weil (and respectively between the Cartan) models of $(\pi^{-1}(U_y), G)$ and $(S_x, G_x)$, and prove that it induces an isomorphism in equivariant cohomology. We then introduce the notion of slice (or locally linearizable) groupoids, which are locally modeled on Lie group actions on manifolds with gluing data, several examples and applications are discussed. In the last section, we generalize the equivariant theory to these groupoids using sheaf-theoretic methods. We further show that when a slice groupoid is equivalent to an action Lie groupoid, the equivariant cohomology of the slice groupoid coincide with the equivariant cohomology of the Lie group action on a manifold.