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Equivariant cyclic homology and equivariant differential forms. (English) Zbl 0849.55008

For a smooth action of a compact Lie group \(G\) on a compact manifold \(M\), the authors extend in this paper the description which Hochschild-Kostant-Rosenberg gave of the Hochschild cohomology of the algebra \(C^\infty (M)\) in terms of differential forms on \(M\) and which was extended by Connes to cyclic homology. Namely, the description of the equivariant periodic cyclic homology \(HP^G_k(C^\infty (M))\) of \(C^\infty(M)\) as the cohomology of global equivariant differential forms on \(M\) is given; these are defined as the sections of a sheaf over the group \(G\) whose stalk at \(g \in G\) is the space of germs at 0 of maps from the Lie algebra \(g^g\) of the centralizer \(G^g\) of \(g\) to the complex \(\Omega^\bullet (M^g)\) of equivariant differential forms on the fixed-point set \(M^g\), with action of \(G^g\). This description consists in the definition of a quasi-isomorphism of the sheaf \(\Omega^\bullet (M,G)\) of equivariant differential forms (for every \(g \in G\), the stalk of \(\Omega^\bullet (M,G)\) at \(g\) is \(\Omega^\bullet (M,G)_g = \Omega^\bullet_{G^g} (M^g))\) with the sheaf \(C_\bullet (C^\infty(M),G)\) of equivariant \(k\)-chains over \(G\).
Taking the corresponding homology, the authors obtain the isomorphism \(HP^G_\bullet (C^\infty (M)) \simeq H({\mathcal A}^\bullet_G (M), d+\iota)\) where \({\mathcal A}^\bullet_G (M)= \Gamma(G, \Omega^\bullet (M,G))\) is the space of global equivariant forms. In combination with the result relating equivariant \(K\)-theory with equivariant periodic cyclic homology: \(HP^G_k (C^\infty (M)) \simeq K^k_G(M) \otimes_{R(G)} R^\infty (G)\) \((R^\infty (M)\) being the space of smooth functions on \(G\) invariant under adjoint action), a de Rham description of equivariant \(\kappa\)-theory is obtained.

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
18G60 Other (co)homology theories (MSC2010)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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References:

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