The Higson-Roe exact sequence and $\ell^2$ eta invariants

TL;DR

This paper constructs an $ ext{l}^2$ eta invariant morphism on the Higson-Roe structure group, linking geometric analysis with topological invariants, and provides new proofs for $ ext{l}^2$ rigidity theorems.

Contribution

It introduces a group morphism from the Higson-Roe structure group to the reals using the Cheeger-Gromov $ ext{l}^2$ eta invariant, connecting geometric and topological invariants.

Findings

Constructed an $ ext{l}^2$ eta morphism on the Higson-Roe structure group.

Derived the spin $ ext{l}^2$ rho invariant from the morphism.

Provided new proofs for classical $ ext{l}^2$ rigidity theorems.

Abstract

The goal of this paper is to solve the problem of existence of an $\ell^2$ relative eta morphism on the Higson-Roe structure group. Using the Cheeger-Gromov $\ell^2$ eta invariant, we construct a group morphism from the Higson-Roe maximal structure group constructed in [HiRo:10] to the reals. When we apply this morphism to the structure class associated with the spin Dirac operator for a metric of positive scalar curvature, we get the spin $\ell^2$ rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the $\ell^2$ rho invariants of the corresponding signature operators. We thus get new proofs for the classical $\ell^2$ rigidity theorems of Keswani obtained in [Ke:00].