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 $\alpha \in H_G^3(M,\mathbb{Z})$ we introduce a notion of localized equivariant twisted cohomology $H^\bullet ({\bar{\Omega}}^\bullet (M, G, L)g, d^\alpha{G^g})$,indexedby$g\in G$. 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 $HP^G_\bullet ( C^\infty (M, \alpha ) )$,where$C^\infty (M, \alpha )$ is a certain smooth algebra constructed from an equivariant bundle gerbe defined by $\alpha \in H_G^3(M,\mathbb{Z})$, to $H^{\bullet }({\bar{\mathrm{\Omega }}}^{\bullet }(M,G,L{)}_g,d_{G^g}^{\alpha })$. We formulate a conjecture of Atiyah-Hirzebruch type theorem for equivariant twisted $K$-theory.