Analytic and topological index maps with values in the $K$-theory of mapping cones

Abstract

Index maps taking values in the $K$-theory of a mapping cone are defined and discussed. The resulting index theorem can be viewed in analogy with the Freed–Melrose index theorem. The framework of geometric $K$-homology is used in a fundamental way. In particular, an explicit isomorphism from a geometric model for $K$-homology with coefficients in a mapping cone, $C_{\varphi }$, to $KK(C(X),C_{\varphi })$ is constructed.