Almost flat bundles and almost flat structures
type: Item
instance of: scholarly article
title: Almost flat bundles and almost flat structures
zbMATH Open document ID: 1093.19005
DOI: 10.12775/TMNA.2005.025
author: A. S. Mishchenko
author: Nicolae Teleman
published in: Topological Methods in Nonlinear Analysis
publication date: 2006-01-26T00:00:00Z
review text: The Novikov conjecture asserts that, for \(M\) a closed oriented \(n\)-manifold with \(\pi_1 M= G\), \(f: M \to BG\) the classifying map, and \(x \in H^{n -4i} (M, Q)\), \(\langle f^* x \cup L_i M, [M] \rangle\) is a homotopy invariant. A proof of this conjecture for a large class of groups was given by \textit{A. Connes, M. Gromov} and \textit{H. Moscovici} [C. R. Acad. Sci., Paris, Sér. I 310, 273--277 (1990; Zbl 0693.53007)] using \(K\)-cohomology and the notion of an almost flat bundle on a manifold. The paper under review gives an alternative definition of flat bundle which extends to bundles over \(CW\)-complexes. The main tool used is the notion of a quasi-connection, previously defined by the second author. As an application it is shown that an almost flat bundle, \(\alpha\) over \(M\) comes from an arbitrarily large compact subset of \(B \pi_1(M)\) via the classifying map, \(f\). Previously it was only known that some integer multiple of \(\alpha\) has that property.
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 19L41
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 57N65
zbMATH DE Number: 5001923
zbMATH Keywords: almost flat bundle
zbMATH Keywords: quasi-connection
zbMATH Keywords: classifying space
zbMATH Keywords: higher signature
MaRDI profile type: Publication
full work available at URL: tmna.2005.025
OpenAlex ID: W2466237140
description: scientific article; zbMATH DE number 5001923