R/Z-valued index theory via geometric K-homology

Abstract

A model of K-homology with coefficients in a mapping cone using the framework of the geometric cycles of Baum and Douglas is developed. In particular, this leads to a geometric realization of K-homology with coefficients in R/Z. In turn, this group is related to the relative η-invariant via index pairings.

Realizing the analytic surgery group of Higson and Roe geometrically, part I: the geometric model

“…The next proposition implies that addition within S geo * (X; L) is compatible with addition within K 0 (W, ∂W ; µ L ). The proof (which is left to the reader) is similar to the proof in [9,Proposition 4.11] (also see [27,Proposition 4.3.2]).…”

“…It is here that the notion of normal bordism is required. The proof is similar to the proof of [8, Theorem 2.20], [9,Theorem 4.19]; it uses ideas from [16] and [27].…”

“…The main goal of this paper is the construction of such a model. As in [8,9], the geometric model for the analytic surgery group is a relative cycle theory. As such, a cycle should consist of a cycle for K geo * (X) that vanishes under the assembly mapping and a precise reason for its vanishing.…”

“…The construction here is based on results in [17] (also see [9]). For a topological space Y and a Banach algebra D we let K * (Y ; D) denote the K-theory of the Banach algebra C 0 (Y, D).…”

“…. It suffices to consider easy cocycles if L is trivial (for example, see the geometric models considered in [9]).…”