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 Marde\v{s}i\'{c} 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 $f_{\delta}\colon X^{n}\to M^{n}_{\delta}$ 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.