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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

##### 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