A topological approach to Cheeger-Gromov universal bounds for von Neumann \(\rho\)-invariants
type: Item
instance of: scholarly article
zbMATH Open document ID: 1358.57020
author: Jae Choon Cha
published in: Communications on Pure and Applied Mathematics
publication date: 2016-06-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1405.1805
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 6591533
zbMATH Keywords: rho-invariant
zbMATH Keywords: complexity
zbMATH Keywords: topological manifold
zbMATH Keywords: 3-manifold
zbMATH Keywords: knot
zbMATH Keywords: link
zbMATH Keywords: bordism
zbMATH Keywords: controlled chain homotopy
describes a project that uses: KnotInfo
MaRDI profile type: Publication
OpenAlex ID: W2963081410
arXiv ID: 1405.1805
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DOI: 10.1002/CPA.21597
title: A topological approach to Cheeger-Gromov universal bounds for von Neumann \(\rho\)-invariants
review text: The paper under review presents a topological approach to studying certain \(\rho\)-invariants of \((4k-1)\)-dimensional manifolds.NEWLINENEWLINE\textit{J. Cheeger} and \textit{M. Gromov} [J. Differ. Geom. 21, 1--34 (1985; Zbl 0614.53034)] studied the \(L^{2}\) \(\rho\)-invariant \(\rho^{(2)} (M,\phi) \in \mathbb{R}\) which was defined for a closed smooth \((4k-1)\)-dimensional manifold \(M\) and a homomorphism \(\phi : \pi_{1} (M) \rightarrow G\) for some group \(G\). Using deep analytic methods they showed that for any such \(M\) there exists a constant \(C_{M}\) such that \(|\rho^{(2)} (M,\phi)| \leq C_{M}\) for any \(\phi\) and any \(G\).NEWLINENEWLINEThe main result of the present paper is Theorem 1.3 which says the same as the theorem of Cheeger and Gromov, but the manifold is allowed to be just topological. Naturally the methods of the proof are then also topological.NEWLINENEWLINEThe topological definition of the \(L^2\) \(\rho\)-invariant used here is as the \(L^{2}\)-signature defect of a manifold \(W\) bounding possibly several copies of \(M\) with respect to an enlargement \(\Gamma\) of \(G\), where the \(L^2\)-signature is defined using the dimension function for modules over the von Neumann algebra \(\mathbb{N} \Gamma\) due to \textit{W. Lück} [\(L^2\)-invariants: Theory and applications to geometry and \(K\)-theory. Berlin: Springer (2002; Zbl 1009.55001)]. A key result used in the proof of Theorem 1.3 is the functorial embedding of groups into acyclic groups due to Baumslag-Dyer-Heller [\textit{G. Baumslag} et al., J. Pure Appl. Algebra 16, 1--47 (1980; Zbl 0419.20026)], which enables the author to find \(W\) independent of the choice of \(\phi\).NEWLINENEWLINEThese topological methods also allow the author to obtain relationships of this invariant with various notions of topological complexity of \(3\)-manifolds. In particular \(\rho^{(2)} (M,\phi)\) is estimated from above in terms of the simplical complexity in Theorem 1.5, in terms of the Heegaar…
reviewed by: Tibor Macko
description: scientific article; zbMATH DE number 6591533