Chern-Weil map for principal bundles over groupoids

Details Chern-Weil map for principal bundles over groupoids Chern-Weil map for principal bundles over groupoids

2004-01-29

arxiv.org/abs/math/0401420v3

Mathematics

Differential Geometry

43 pages. References added, typos corrected

Scientific paper

The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this paper, we study the differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern-Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application, we recover the equivariant Chern-Weil map of Bott-Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott-Shulman map $S({\mathfrak g}^*)^G \to H^*(BG)$ when the manifold is a point.

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