A construction of Nöbeling manifolds of arbitrary weight

Abstract

For each cardinal $\kappa$, each natural number $n$ and each simplicial complex $K$ we construct a space ${\nu }_{\kappa }^n(K)$ and a map $\pi :{\nu }_{\kappa }^n(K)\to K$ such that the following conditions are satisfied. 1. ${\nu }_{\kappa }^n(K)$ is a complete metric $n$-dimensional space of weight $\kappa$. 2. ${\nu }_{\kappa }^n(K)$ is an absolute neighborhood extensor in dimension $n$. 3. ${\nu }_{\kappa }^n(K)$ 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 Nöbeling manifolds. The constructed spaces have very interesting fractal-like internal structure that allows for easy construction, subdivision, and surgery of brick partitions.