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Cohomologie équivariante délocalisée. (Delocalized equivariant cohomology). (French) Zbl 0589.55003
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

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