The Novikov conjecture for low degree cohomology classes
type: Item
instance of: scholarly article
title: The Novikov conjecture for low degree cohomology classes
zbMATH Open document ID: 1029.19006
DOI: 10.1023/A:1024941020306
author: Varghese Mathai
published in: Geometriae Dedicata
publication date: 2003-09-09T00:00:00Z
review text: Let \(c\) be an \(\mathbb R\)-valued group 2-cocycle on a discrete group \(\Gamma\), \(\sigma^s={\text e}^{2\pi isc}\), \(s\in[0,1]\), \(\sigma^1=\sigma\), and let \(C^*_r(\Gamma,\sigma)\) denote the twisted group \(C^*\)-algebra. The author constructs a \(C^*_r(\Gamma,\sigma)\)-vector bundle \(V_\sigma\to B\Gamma\) over a classifying space \(B\Gamma\) that defines a canonical class \([V_\sigma]\in K_0(C_0(B\Gamma)\otimes C^*_r(\Gamma,\sigma))\). This construction generalizes, for the twisted case, the canonical class of \textit{A. S. Mishchenko} [Izv. Akad. Nauk SSSR Ser. Mat. 34, 501-514 (1970; Zbl 0216.19701)]. Using this, a twisted Kasparov map \(\mu_\sigma:K_*(B\Gamma)\to K_*(C^*_r(\Gamma,\sigma))\) is defined as the intersection product by \([V_\sigma]\). The main result is homotopy invariance of \(\mu_{\sigma^s}(D)\in K_0(C^*_r(\Gamma,\sigma^s))\) for small enough \(s\), where \(D\) denotes the signature operator. This allows the author to give a short proof of the Novikov conjecture for all homotopy classes generated by those of degree \(\leq\) 2, a result that was first established by \textit{A. Connes, M. Gromov} and \textit{H. Moscovici} [Geom. Funct. Anal. 3, 1-78 (1993; Zbl 0789.58069)].
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 1977885
zbMATH Keywords: higher signature
zbMATH Keywords: homotopy invariance
zbMATH Keywords: index theory
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: operator \(K\)-theory
zbMATH Keywords: twisted group \(C^*\)-algebra
reviewed by: Vladimir M. Manuilov
MaRDI profile type: Publication
full work available at URL: a:1024941020306
OpenAlex ID: W187182739
description: scientific article; zbMATH DE number 1977885