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

TL;DR

This paper constructs a geometric version of the analytic surgery group for group actions, establishing its algebraic properties and connecting it to invariants like eta, advancing geometric understanding of assembly maps.

Contribution

It introduces a geometric model of the Higson-Roe analytic surgery group, linking it to assembly maps and eta-type invariants for group actions.

Findings

The geometric group fits into a six-term exact sequence with the assembly map.

Defined a map analogous to eta-invariants for representations of the same rank.

Established connections between geometric models and analytic invariants.

Abstract

We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of $\eta$-type invariants from the geometric group (with respect to assembly for the full group $C^*$-algebra) to the real numbers.