Analytic Realization of Relative K-Homology on Manifolds with Boundary and Pairing with K-Cohomology

Abstract

Although K-homology can be defined abstractly as a dual of the ordinary K-theory of spaces, its use in index theory and analysis comes about through the concrete realization where elliptic operators are cycles. This is well understood for manifolds without boundary but not when a boundary is present. In this note we provide a realization for relative cycles using elliptic operators on the doubled manifold and explore pairings of these cycles with K-cohomology in terms of Fredholm indexes and spectral flow.