Menger manifolds homeomorphic to their $n$-homotopy kernels

Abstract:

We give a necessary and sufficient condition that an $(n + 1)$-dimensional Menger manifold (${\mu ^{n + 1}}$-manifold) is homeomorphic to its n-homotopy kernel. Such a ${\mu ^{n + 1}}$-manifold is called a $\mu _\infty ^{n + 1}$-manifold. We also prove the following results: (1) Each homeomorphism between two Z-sets in a $\mu _\infty ^{n + 1}$-manifold M extends to an ambient homeomorphism of M onto itself if it is n-homotopic to id in M. (2) An n-homotopy equivalence between two $\mu _\infty ^{n + 1}$-manifolds is n-homotopic to a homeomorphism. (3) Each map from a $\mu _\infty ^{n + 1}$-manifold into a ${\mu ^{n + 1}}$-manifold is n-homotopic to an open embedding.