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A finite-dimensional approach to the strong Novikov conjecture

Daniel Ramras, Rufus Willett and Guoliang Yu

Algebraic & Geometric Topology 13 (2013) 2283–2316

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.

Baum–Connes conjecture, $K$–homology, deformation $K$–theory, index theory
Mathematical Subject Classification 2010
Primary: 19K56, 19L99, 55N15, 57R20
Secondary: 20C99, 46L80, 46L85
Received: 18 October 2012
Revised: 29 January 2013
Accepted: 19 March 2013
Published: 20 June 2013
Daniel Ramras
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003
Rufus Willett
Department of Mathematics
University of Hawai‘i at Mānoa
Honolulu, HI 96822
Guoliang Yu
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
USA and Shanghai Center for Mathematical Sciences
Fudan University