Asymptotic Lipschitz cohomology and higher signatures
type: Item
instance of: scholarly article
title: Asymptotic Lipschitz cohomology and higher signatures
zbMATH Open document ID: 0861.57037
DOI: 10.1007/BF02247890
author: Yanyan Li
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-05-12T00:00:00Z
full work available at URL: https://eudml.org/doc/58229
review text: The paper deals with the study of the Novikov conjecture on homotopy invariance of higher signatures. The higher signature of a closed oriented manifold \(M\) with fundamental class \([M]\) is a characteristic number of the form \((L(M)f^*(x))[M]\), where \(L(M)\) denotes the Hirzebruch L-class of \(M\), \(f:M\to K(\Gamma,1)\) is a continuous map and \(x\in H^*(K(\Gamma,1);\mathbb{Q})\). The Novikov conjecture was proved for special classes of groups \(\Gamma\) by several authors. The main result of the present paper is, that the higher signatures are oriented homotopy invariants, if \(\Gamma\) is a torsionfree quasi-geodesic bicombing group. The assumption on the group is a technical one and we will not give the definition here. The author extends in his proof methods of \textit{A. Connes}, \textit{M. Gromov} and \textit{H. Moscovici} [Geom. Funct. Anal. 3, 1-78 (1993; Zbl 0789.58069)]. The basic idea of the author's proof is to approximate \(K(\Gamma,1)\) by finite-dimensional simplicial complexes. In a final step, the proof uses results of \textit{A. S. Mishchenko's} paper [Izv. Akad. Nauk SSSR, Ser. Mat. 38, 81-106 (1974; Zbl 0292.57017)].
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 912456
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: higher signatures
zbMATH Keywords: oriented manifold
zbMATH Keywords: characteristic number
zbMATH Keywords: Hirzebruch L-class
reviewed by: K. H. Mayer
MaRDI profile type: Publication
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OpenAlex ID: W2003328639
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description: scientific article; zbMATH DE number 912456