A construction of N\"obeling manifolds of arbitrary weight

TL;DR

This paper constructs N"obeling manifolds of arbitrary weight with specific topological properties, providing a new class of universal, fractal-like metric spaces that generalize classical N"obeling manifolds.

Contribution

It introduces a method to construct N"obeling manifolds of any given weight with universal and homotopy properties, extending the classical separable case.

Findings

Spaces are complete metric n-dimensional with specified weight.

Spaces are absolute neighborhood extensors in dimension n.

Spaces are strongly universal in their class.

Abstract

For each cardinal $\kappa$, each natural number $n$ and each 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 metric $n$-dimensional 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 $n$-dimensional complete metric spaces of weight~$\kappa$. 4. $\pi$ is an $n$-homotopy equivalence. For $\kappa = \omega$ the constructed spaces are $n$-dimensional separable N\"obeling manifolds. The constructed spaces have very interesting fractal-like internal structure that allows for easy construction, subdivision, and surgery of brick partitions.