On dimensionally exotic maps

TL;DR

This paper investigates the existence of maps without dimensionally regular values on Boltyanskii compacta, revealing that high-dimensional examples always admit such maps, while some lower-dimensional cases do not.

Contribution

It proves that all Boltyanskii compacta of dimension at least 6 admit maps without dimensionally regular values, and provides a counterexample in dimension 4.

Findings

High-dimensional Boltyanskii compacta admit maps without regular values.

A 4-dimensional Boltyanskii compactum exists where all maps have regular values.

The results connect the structure of compacta with the existence of special maps.

Abstract

We call a value $y=f(x)$ of a map $f:X\to Y$ dimensionally regular if $\dim X\le \dim(Y\times f^{-1}(y))$. It was shown in \cite{first-exotic} that if a map $f:X\to Y$ between compact metric spaces does not have dimensionally regular values, then $X$ is a Boltyanskii compactum, i.e. a compactum satisfying the equality $\dim(X\times X)=2\dim X-1$. In this paper we prove that every Boltyanskii compactum $X$ of dimension $\dim X \geq 6$ admits a map $f:X\to Y$ without dimensionally regular values. Also we exhibit a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value.