The non-existence of common models for some classes of higher-dimensional hereditarily indecomposable continua

TL;DR

This paper proves that certain classes of complex, high-dimensional hereditarily indecomposable continua do not have a universal model from which all members of these classes can be obtained as continuous images.

Contribution

It establishes the non-existence of common models for various classes of high-dimensional hereditarily indecomposable continua, extending understanding of their structural complexity.

Findings

No common model exists for strongly chaotic hereditarily indecomposable n-dimensional Cantor manifolds.

No common model exists for strongly chaotic hereditarily indecomposable hereditarily strongly infinite-dimensional Cantor manifolds.

No common model exists for hereditarily indecomposable continua with transfinite dimension.

Abstract

A continuum $K$ is a common model for the family ${\mathcal K}$ of continua if every member of ${\mathcal K}$ is a continuous image of $K$. We show that none of the following classes of spaces has a common model: 1) the class of strongly chaotic hereditarily indecomposable $n$-dimensional Cantor manifolds, for any given natural number $n$, 2) the class of strongly chaotic hereditarily indecomposable hereditarily strongly infinite-dimensional Cantor manifolds, 3) the class of strongly chaotic hereditarily indecomposable continua with transfinite dimension (small or large) equal to $\alpha$, for any given ordinal number $\alpha < \omega_{1}$.