Locally Lipschitz contractibility of Alexandrov spaces and its applications
type: Item
instance of: scholarly article
title: Locally Lipschitz contractibility of Alexandrov spaces and its applications
zbMATH Open document ID: 1306.53030
author: Ayato Mitsuishi
author: Takao Yamaguchi
published in: Pacific Journal of Mathematics
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1303.0655
reviewed by: Eleutherius Symeonidis
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 6361409
zbMATH Keywords: Alexandrov space
zbMATH Keywords: Lipschitz contractibility
zbMATH Keywords: Plateau problem
MaRDI profile type: Publication
OpenAlex ID: W3105418268
arXiv ID: 1303.0655
DOI: 10.2140/PJM.2014.270.393
review text: The authors introduce the notion of a strongly locally Lipschitz contractible metric space as follows.NEWLINENEWLINE``A metric space \(X\) is \textit{strongly locally Lipschitz contractible}, or SLLC, if for every point \(p\in X\), there exists \(r>0\) and a map NEWLINE\[NEWLINE h:U(p,r)\times[0,1]\rightarrow U(p,r) NEWLINE\]NEWLINE such that \(h\) is a homotopy from \(h(\cdot,0)=\text{id}_{U(p,r)}\) to \(h(\cdot,1)=p\) and \(h\) is Lipschitz (i.e., there exists \(C,C^\prime >0\) such that NEWLINE\[NEWLINE d(h(x,s),h(y,t))\leq Cd(x,y)+C^\prime|s-t| NEWLINE\]NEWLINE for every \(x,y\in U(p,r)\) and \(s,t\in[0,1]\)) and such that for every \(r^\prime <r\), the image of \(h\) restricted to \(U(p,r^\prime)\times[0,1]\) is \(U(p,r^\prime)\).'' Recall that the (weak) locally Lipschitz contractibility, as introduced in [the second author, Kyushu J. Math. 51, No. 2, 273--296 (1997; Zbl 0914.53028)], merely requires that for every \(p\in X\) and \(\varepsilon>0\) there exists \(r\in(0,\varepsilon]\) and a Lipschitz map \(h:U(p,r)\times[0,1]\rightarrow U(p,\varepsilon)\) such that \(h(\cdot,0)=\text{id}_{U(p,r)}\) and \(h(x,1)=p\) for every \(x\in U(p,r)\).NEWLINENEWLINENEWLINEThe main result of the authors is that every finite-dimensional Alexandrov space is strongly locally Lipschitz contractible. The proof uses the gradient flow of the distance function from a metric sphere, which function is shown to be regular on a smaller concentric punctured ball. NEWLINENEWLINENEWLINEAfter that, the authors show that a continuous map from a Lipschitz simplicial complex to a strongly locally Lipschitz contractible metric space \(X\) is homotopic to a Lipschitz one. Here, a Lipschitz simplicial complex is, roughly speaking, a metric space having a triangulation such that every simplex is a bi-Lipschitz image of a simplex in a Euclidean space. It follows that for an Alexandrov space \(X\), the \(k\)-th Lipschitz homotopy group \(\pi_k^{\text{Lip}}(X,x_0)\) and the (standard) \(k\)-th homo…
description: scientific article; zbMATH DE number 6361409