Periodic Cyclic Homology and Equivariant Gerbes

This paper is our first step in establishing a de Rham model for equivariant twisted $K$-theory using machinery from noncommutative geometry. Let $G$ be a compact Lie group, $M$ a compact manifold on which $G$ acts smoothly. For any $α\in H^3_G (M, {\mathbb Z})$ we introduce a notion of localized equivariant twisted cohomology $H^\bullet ({\barΩ}^\bullet (M, G, L)_g, d^α_{G^g})$, indexed by $g\in G$. We prove that there exists a natural family of chain maps, indexed by $g\in G$, inducing a family of morphisms from the equivariant periodic cyclic homology $HP^G_\bullet ( C^\infty (M, α) )$, where $C^\infty (M, α)$ is a certain smooth algebra constructed from an equivariant bundle gerbe defined by $α\in H^3_G (M,{\mathbb Z} )$, to $H^\bullet ( {\barΩ}^\bullet (M, G, L)_g, d^α_{G^g})$. We formulate a conjecture of Atiyah-Hirzebruch type theorem for equivariant twisted $K$-theory.

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