A finite dimensional approach to the strong Novikov conjecture

TL;DR

This paper introduces a novel finite-dimensional representation approach to the strong Novikov conjecture, inspired by Atiyah-Singer index theory and deformation K-theory, providing new proofs for specific group classes.

Contribution

It presents a new, accessible method based on continuous finite-dimensional representations to prove the strong Novikov conjecture for certain groups.

Findings

Proves the strong Novikov conjecture for crystallographic groups

Proves the strong Novikov conjecture for surface groups

Provides insights into K-theory and cohomology of representation spaces

Abstract

The aim of this paper is to introduce an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig using 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.