Limits of manifolds in the Gromov-Hausdorff metric space

Hegenbarth, Friedrich (author), Repovš, Dušan (author)

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 $u \in M^{n}_{\delta }$, $f^{-1}_{\delta }(u)$ 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.

• Language: English
• 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, $\varepsilon$-homotopy equivalence, periodic surgery spectrum $\mathbb{L}$
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Author Accepted Manuscript
• Year: 2023
• Number of pages: 11 str.
• Numbering: Vol. 20, iss. 1, art. 47
• PID: 20.500.12556/RUL-143766
• UDC: 515.14:514.7
• ISSN on article: 1660-5446
• DOI: 10.1007/s00009-022-02250-9
• COBISS.SI-ID: 135986179
• Publication date in RUL: 11.01.2023

Record is a part of a journal

• Title: Mediterranean journal of mathematics
• Shortened title: Mediterr. j. math.
• Publisher: Springer Nature, Birkhäuser
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433