A finite-dimensional approach to the strong Novikov conjecture

Ramras, Daniel¹ ; Willett, Rufus² ; Yu, Guoliang³

¹Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA
²Department of Mathematics, University of Hawai‘i at Mānoa, Honolulu, HI 96822, USA
³Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China

Algebraic and Geometric Topology, Tome 13 (2013) no. 4, pp. 2283-2316

Résumé

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.

Détail

DOI : 10.2140/agt.2013.13.2283

Classification : 19K56, 19L99, 55N15, 57R20, 20C99, 46L80, 46L85
Keywords: Baum–Connes conjecture, $K$–homology, deformation $K$–theory, index theory

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