The topological Atiyah-Segal map
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).
Cite
@article{arxiv.1607.06430,
title = {The topological Atiyah-Segal map},
author = {Daniel A. Ramras},
journal= {arXiv preprint arXiv:1607.06430},
year = {2023}
}
Comments
31 pages. Section 2 from Version 1 was moved into the article arXiv:1807.02613. Version 2 contains some further results on spaces of flat connections (Section 7.4), and has been accepted for publication in a forthcoming volume of Contemporary Mathematics