Chern-Weil map for principal bundles over groupoids

2004/01/29 by Camille Laurent-Gengoux, Laurent-Gengoux, Camille, Jean-Louis Tu +3
Mathematics · #58F05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.math/0401420

openalex publication_date 2004/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 → H^*(BG) when the manifold is a point.

Citations

• The non-commutative Weil algebra

Related

• Connections on Lie groupoids and Chern-Weil theory
• Chern-Weil forms and abstract homotopy theory
• About extending the Chern-Weil classification to the simplicial principal bundles
• The Homotopy Type of Spaces of Flat Connections for Classical Lie Groups
• Principal bundles with groupoid structure: local vs. global theory and nonabelian ucech cohomology