Construction of Fredholm representations and a modification of the Higson-Roe corona

TL;DR

This paper develops a homology-based approach to Fredholm representations, extending the Higson-Roe corona framework to manifolds with boundary, and explores their applications to homotopy invariants of non-simply connected manifolds.

Contribution

It introduces a homology version of symmetry for Fredholm representations in the context of manifolds with boundary, extending previous work on closed manifolds.

Findings

Constructed a homology version of symmetry for Fredholm representations.

Extended the Higson-Roe corona framework to manifolds with boundary.

Developed applications to homotopy invariants of non-simply connected manifolds.

Abstract

The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula. Earlier a natural family of the Fredholm representations was constructed that lead to a symmetric vector bundle on completion of the fundamental group with a modification of the Higson-Roe corona when the completion is a closed manifold. Here we will discuss a homology version of symmetry in the case when completion with a modification of the Higson-Roe corona is a manifold with boundary. The results were developed during the visit of the first author in Ancona on March, 2007. The second version is supplemented by details of consideration the case of manifolds with boundary.