Equivariant Cohomology for Slice Groupoids and Its Local Linear Models

Let G be a compact Lie group acting smoothly on a manifold M. The local structure of such actions is described by the slice theorem, which reduces the study of G-spaces to smaller isotropy group actions. In this paper, we investigate the equivariant cohomology of these local models from the viewpoint of Weil and Cartan complexes. We construct explicit reduction morphisms between the equivariant cohomology complexes associated with a slice neighborhood G × GxSx and the slice Sx with the isotropy group Gx. We prove that these morphisms induce isomorphisms in equivariant cohomology. Based on this local description, we introduce slice groupoids, a class of groupoids locally modeled on Lie group actions, and define their equivariant cohomology using sheaf-theoretic methods. Finally, we show that for slice groupoids arising from action groupoids, the resulting cohomology agrees with the classical equivariant cohomology of the underlying group action.