Limits of manifolds in the Gromov-Hausdorff metric space

Hegenbarth, Friedrich (avtor), Repovš, Dušan (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: 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 $\mathbb{L}$
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: PEF - Pedagoška fakulteta; FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Recenzirani rokopis
• Leto izida: 2023
• Št. strani: 11 str.
• Številčenje: Vol. 20, iss. 1, art. 47
• PID: 20.500.12556/RUL-143766
• UDK: 515.14:514.7
• ISSN pri članku: 1660-5446
• DOI: 10.1007/s00009-022-02250-9
• COBISS.SI-ID: 135986179
• Datum objave v RUL: 11.01.2023

Gradivo je del revije

• Naslov: Mediterranean journal of mathematics
• Skrajšan naslov: Mediterr. j. math.
• Založnik: Springer Nature, Birkhäuser
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433