\(L^ 2\)-topological invariants of 3-manifolds
type: Item
instance of: scholarly article
title: \(L^ 2\)-topological invariants of 3-manifolds
zbMATH Open document ID: 0876.57050
DOI: 10.1007/BF01241121
author: John Lott
author: Wolfgang Lück
published in: Inventiones Mathematicae
publication date: 1995-04-20T00:00:00Z
full work available at URL: https://eudml.org/doc/144268
review text: For a large class of compact 3-manifolds \(M\), the authors compute the \(L^2\)-Betti numbers \(b_k(M)\) and give estimation for the Novikov-Shubin invariants \(\alpha_k(M)\). It turns out that in these cases the \(b_k(M)\) are rational and the \(\alpha_k(M)\) are positive. Thus, basic conjectures concerning these invariants are confirmed in dimension 3. In the first half of the paper, the authors study properties of the \(b_k\) and the \(\alpha_k\) in arbitrary dimensions. They work in the general setting of Hilbert modules over finite von Neumann algebras. The main result in this part is a set of inequalities for the \(\alpha_k\) in situations of Mayer-Vietoris type. These estimates are then repeatedly used, namely to study connected sums, to extend results known for closed manifolds to compact manifolds \(M\) with boundary by looking at the double of \(M\), and to study decompositions of 3-manifolds. The paper closes with a very useful section, where important conjectures in the field are stated and commented upon.
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 746068
zbMATH Keywords: \(L^ 2\)-Betti numbers
zbMATH Keywords: Novikov-Shubin invariants
zbMATH Keywords: 3-manifolds
reviewed by: Wilhelm Singhof
MaRDI profile type: Publication
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Wikidata QID: Q124850995
description: scientific article; zbMATH DE number 746068