The topological Atiyah-Segal map

TL;DR

This paper develops a homotopy theoretical framework for the Atiyah-Segal map for infinite discrete groups, connecting it to the Novikov conjecture and applications in flat bundle spaces.

Contribution

It introduces a novel homotopy theoretical approach to the Atiyah-Segal map for infinite groups, incorporating the topology of representation spaces.

Findings

Framework relates to the Novikov conjecture

Applications to flat connections on the Heisenberg manifold

Analysis of flat bundles over classifying spaces of groups with property (T)

Abstract

Associated to each finite dimensional linear representation of a group G, there is a vector bundle over the classifying space BG. This construction was studied extensively for compact groups by Atiyah and Segal. We introduce a homotopy theoretical framework for studying the Atiyah-Segal construction in the context of infinite discrete groups, taking into account the topology of representation spaces. We explain how this framework relates to the Novikov conjecture, and we consider applications to spaces of flat connections on the over the 3-dimensional Heisenberg manifold and families of flat bundles over classifying spaces of groups satisfying Kazhdan's property (T).