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

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 $\alpha <{\omega }_1$.

15 pages. Dedicated to D. P. Bellamy on the occasion of his 60th birthday. Research partially supported by MNiSW Grant Nr. N201 034 31/2717