Algebraic & Geometric Topology

A finite-dimensional approach to the strong Novikov conjecture

Daniel Ramras, Rufus Willett, and Guoliang Yu

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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.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 4 (2013), 2283-2316.

Dates
Received: 18 October 2012
Revised: 29 January 2013
Accepted: 19 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715639

Digital Object Identifier
doi:10.2140/agt.2013.13.2283

Mathematical Reviews number (MathSciNet)
MR3073917

Zentralblatt MATH identifier
1277.19002

Subjects
Primary: 19K56: Index theory [See also 58J20, 58J22] 19L99: None of the above, but in this section 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 57R20: Characteristic classes and numbers
Secondary: 20C99: None of the above, but in this section 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 46L85: Noncommutative topology [See also 58B32, 58B34, 58J22]

Keywords
Baum–Connes conjecture $K$–homology deformation $K$–theory index theory

Citation

Ramras, Daniel; Willett, Rufus; Yu, Guoliang. A finite-dimensional approach to the strong Novikov conjecture. Algebr. Geom. Topol. 13 (2013), no. 4, 2283--2316. doi:10.2140/agt.2013.13.2283. https://projecteuclid.org/euclid.agt/1513715639


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