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
–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
–theory
and cohomology of representation spaces.
Keywords
Baum–Connes conjecture, $K$–homology, deformation
$K$–theory, index theory
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
USA and Shanghai Center for Mathematical Sciences
Fudan University
Shanghai
China