Limits of manifolds in the Gromov-Hausdorff metric space

Friedrich Hegenbarth (Author), Dušan Repovš (Author)

Abstract

We apply the Gromov-Hausdorff metric ▫$d_G$▫ for characterization of certain generalized manifolds. Previously, we have proven that with respect to the metric ▫$d_G$▫, generalized ▫$n$▫-manifolds are limits of spaces which are obtained by gluing two topological ▫$n$▫-manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the present paper, we consider the so-called manifold-like generalized ▫$n$▫-manifolds ▫$X^{n}$▫, introduced in 1966 by Mardeić and Segal, which are characterized by the existence of ▫$\delta$▫-mappings ▫$f_{\delta }$▫ of ▫$X^{n}$▫ onto closed manifolds ▫$M^{n}_{\delta }$▫, for arbitrary small ▫$\delta >0$▫, i.e., there exist onto maps ▫$f_{\delta }:X^{n}\rightarrow M^{n}_{\delta}$▫ such that for every ▫$M^{n}_{\delta }$▫, ▫$f^{-1}_{\delta }(u)$▫ has diameter less than ▫$\delta$▫. We prove that with respect to the metric ▫$d_G$▫, manifold-like generalized ▫$n$▫-manifolds ▫$X^{n}$▫ are limits of topological ▫$n$▫-manifolds ▫$M^{n}_{i}$▫. Moreover, if topological ▫$n$▫-manifolds ▫$M^{n}_{i}$▫ satisfy a certain local contractibility condition ▫${\mathcal {M}}(\varrho, n)$▫, we prove that generalized ▫$n$▫-manifold ▫$X^{n}$▫ is resolvable.

Keywords

Gromov-Hausdorff metric;Gromov topological moduli space;manifold-like generalized manifold;absolute neighborhood;retract;cell-like map;▫$\delta$▫-map;structure map;controlled surgery sequence;▫$\varepsilon$▫-homotopy;2-patch space;

Data

Language:	English
Year of publishing:	2023
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	515.14:514.7
COBISS:	135986179
ISSN:	1660-5446
Views:	133
Downloads:	24
Average score:	0 (0 votes)
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Other data

Type (COBISS):	Article
Embargo end date (OpenAIRE):	2024-02-01
Pages:	art: 47 (11 str.)
Volume:	ǂVol. ǂ20
Issue:	ǂiss. ǂ1
Chronology:	Feb. 2023
DOI:	10.1007/s00009-022-02250-9
ID:	17675899