Sticky Cantor Sets in ${\mathbb R}^d$

TL;DR

This paper constructs sticky wild Cantor sets in Euclidean spaces of dimension four and higher, demonstrating a new type of topological complexity that cannot be removed by small isotopies.

Contribution

It introduces the concept of sticky wild Cantor sets and provides explicit constructions in dimensions four and above.

Findings

Existence of sticky wild Cantor sets in ${ m R}^d$ for all $d \\geq 4$

These sets cannot be isotoped off of themselves by small ambient isotopies

Advances understanding of topological rigidity in higher dimensions

Abstract

A subset of ${\mathbb R}^d$ is called "sticky" if it cannot be isotoped off of itself by a small ambient isotopy. Sticky wild Cantor sets are constructed in ${\mathbb R}^d$ for each $d\geq 4$.