A finite-dimensional approach to the strong Novikov conjecture

Abstract

The aim of this paper is to describe an approach to the (strong) Novikov conjecture based on continuous families of finite-dimensional representations: this is partly inspired by ideas of Lusztig related to the Atiyah–Singer families index theorem, and partly by Carlsson’s deformation $K$–theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the $K$–theory and cohomology of representation spaces.