On a theorem of Henri Cartan concerning the equivaraint cohomology

2000/05/07 by Liviu I. Nicolaescu, Nicolaescu, Liviu I.
Mathematics · #57R91 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0005068

openalex publication_date 2000/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose G is a compact Lie group and N is a closed normal subgroup of G acting freely on a smooth manifold X. The Cartan theorem alluded to in the title postulates the existence of a natural isomorphism between the G-equivariant cohomology X and the G/N-equivariant cohomology of X/N. In this note we use J. Kalkman's explicit isomorphism between the Cartan and Weil models of equivariant cohomology to show that 1) Cartan's theorem is a simple consequence of Chern-Weil's transgression formula and 2) explicitly describe this isomorphism at the cochain level.

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