Cyclic homology and the Atiyah-Patodi-Singer index theorem

Abstract. We apply the boundary pseudodifferential calculus of Melrose to study the Chern character in entire cyclic homology of the Dirac operator of a manifold with boundary. Recently, Melrose has proved the Atiyah-Patodi-Singer index theorem using a calculus of pseudodifferential operators for manifolds with boundary [11]. In this article, we apply his method to study the Chern character of the Dirac operator of a manifold with boundary, working in the setting of entire cyclic cohomology. Recall that if M is an n-dimensional spin-manifold with boundary, its Dirac operator D defines an element [D] ∈ Kn(M,∂M) of the K-homology of the pair (M,∂M). (There is a more general construction using Clifford modules, but we prefer in this introduction to restrict ourselves to the simplest case.) It has been proved by Baum, Douglas and Taylor [1] and Melrose and Piazza [12] that in the long exact homology sequence − → Kn(∂M) − → Kn(M) − → Kn(M,∂M) ∂− → Kn−1(∂M) − →, [D] ∈ Kn(M,∂M) maps to the Dirac operator of the boundary [D ∂ ] ∈ Kn−1(∂M). The homology Chern character Ch ∗ maps this sequence into the long exact sequence − → Hn(∂M) − → Hn(M) − → Hn(M,∂M) ∂− → Hn−1(∂M) − →. In this article, we will prove an extension of the Atiyah-Patodi-Singer index theorem to entire cyclic cohomology, which realizes the formula ∂ Ch∗[D] = Ch∗[D ∂ ] at the level of cyclic cochains instead of cohomology classes. Let us describe Melrose’s proof in outline. He introduces a calculus of pseudo-differential operators Ψ∗b(M), generated by the vector fields tangential to the bound-ary. Such vector fields are sections of a bundle bTM naturally associated to the manifold with boundary, called the b-tangent bundle. (This bundle is isomorphi