Baum, Paul; Brylinski, Jean-Luc; MacPherson, Robert Cohomologie équivariante délocalisée. (Delocalized equivariant cohomology). (French) Zbl 0589.55003 C. R. Acad. Sci., Paris, Sér. I 300, 605-608 (1985). Let G be an abelian compact Lie group acting on a \(C^{\infty}\)-manifold X and \(K^*\!_ G(X)\) be the equivariant K-theory defined by Atiyah and Segal. In this note the authors define a de Rham cohomology \(H^*(G,X)\) indexed by \({\mathbb{Z}}/2{\mathbb{Z}}\) by using the complex of basic forms on X and some sheaves on the quotient space \(G\setminus X\) induced in a natural manner by the sheaves on X on fibres of which the corresponding isotropy groups act trivially. A Chern character ch: \(K^*\!_ G(X)\to H^*(G,X)\) is defined; it induces an isomorphism of \(K^*\!_ G(X)\otimes_{{\mathbb{Z}}}{\mathbb{C}}\) with \(H^*(G,X)\). Reviewer: L.Maxim Cited in 1 ReviewCited in 8 Documents MSC: 55N25 Homology with local coefficients, equivariant cohomology 55N15 Topological \(K\)-theory 57S15 Compact Lie groups of differentiable transformations 57R20 Characteristic classes and numbers in differential topology Keywords:actions of compact Lie groups; compact Lie group; equivariant K-theory; de Rham cohomology; basic forms; sheaves on the quotient space; Chern character PDFBibTeX XMLCite \textit{P. Baum} et al., C. R. Acad. Sci., Paris, Sér. I 300, 605--608 (1985; Zbl 0589.55003)