The topological Atiyah–Segal map

Associated to each finite dimensional linear representation of a group G G , there is a vector bundle over the classifying space B G 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).