A construction of Nöbeling manifolds of arbitrary weight
Volume 250 / 2020
Abstract
For each cardinal $\kappa $, each natural number $n$, and each at most $n$-dimensional simplicial complex $K$ we construct a space ${\nu ^n_\kappa (K)}$ and a map $\pi \colon {\nu ^n_\kappa (K)} \to K$ such that the following conditions are satisfied:
(1) ${\nu ^n_\kappa (K)}$ is a complete $n$-dimensional metric space of weight $\kappa $;
(2) ${\nu ^n_\kappa (K)}$ is an absolute neighborhood extensor in dimension $n$;
(3) ${\nu ^n_\kappa (K)}$ is strongly universal in the class of complete $n$-dimensional metric spaces of weight $\kappa $; and
(4) $\pi $ is an $n$-homotopy equivalence.
For $\kappa = \omega $ these spaces are $n$-dimensional separable Nöbeling manifolds. They have very interesting fractal-like internal structure that allows easy construction, subdivision, and surgery on brick partitions.
