Limits of manifolds in the Gromov-Hausdorff metric space

Abstract

We apply the Gromov-Hausdorff metric $d_G$ for characterization of certain generalized manifolds. Previously, we have proved 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 {\sl manifold-like} generalized $n$-manifolds $X^n,$ introduced in 1966 by Mardešić and Segal, which are characterized by the existence of $\delta$-mappings $f_{\delta }$ of $X^n$ onto closed manifolds $M_{\delta }^n,$ for arbitrary small $\delta >0$, i.e. there exist onto maps $f_{\delta }:X^n\to M_{\delta }^n$ such that for every $u\in M_{\delta }^n$, $f_{\delta }^{-1}(u)$ 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 $M_i^n$. Moreover, if topological $n$-manifolds $M_i^n$ satisfy a certain local contractibility condition $\mathcal{M}(\varrho ,n)$, we prove that generalized $n$-manifold $X^n$ is resolvable.

References in corpus (2)

• On Steenrod $\mathbb{L}$-homology, generalized manifolds, and surgery
• Generalized manifolds, normal invariants, and $\mathbb{L}$-homology