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

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, $C_ϕ$, to $KK(C(X),C_ϕ)$ is constructed.

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