On Gromov's positive scalar curvature conjecture for duality groups
type: Item
instance of: scholarly article
title: On Gromov's positive scalar curvature conjecture for duality groups
zbMATH Open document ID: 1305.57045
DOI: 10.1142/S1793525314500162
published in: Journal of Topology and Analysis
publication date: 2014-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1312.2857
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 6327952
zbMATH Keywords: scalar curvature
zbMATH Keywords: macroscopic dimension
zbMATH Keywords: essential manifolds
Wikidata QID: Q123326478
author: Alexander Dranishnikov
MaRDI profile type: Publication
arXiv ID: 1312.2857
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OpenAlex ID: W2131505493
review text: Let \(X\) be a metric space. The \textit{macroscopic dimension} \(\dim_{mc}X\) of \(X\) is defined as the minimum \(k\) such that there is a continuous uniformly cobounded map \(f:X\rightarrow N^{k},\) where \(N^k\) is a \(k\)-dimensional simplicial complex, i.e. there is a constant \(C>0\) with \(\mathrm{diam}(f^{-1}(y))<C\) for any \(y\in N^{k}\). \textit{M. Gromov} in [Prog. Math. 132, 1--213 (1996; Zbl 0945.53022)] conjectured that \(\dim_{mc}\tilde{M}\leq n-2,\) the macroscopic dimension of the universal cover \(\tilde{M}\) of a closed \(n\)-manifold \(M\) with a positive scalar curvature metric is at most \(n-2\). In the article under review, the author gives a characterization of macroscopic dimension in terms of coarsely equivariant cohomology. It is proved that \(\dim_{mc}\tilde{M}\leq n-2\) for almost spin \(n\)-manifolds \(M\) with positive scalar curvature whose fundamental group \(\pi_{1}(M)\) is a virtually duality group satisfying the coarse Baum-Connes conjecture.
reviewed by: Shengkui Ye
description: scientific article; zbMATH DE number 6327952