Scrawny Cantor sets are not definable by tori

Abstract:

We define a Cantor set $C$ in ${{\mathbf {R}}^3}$ to be scrawny if for each $p \in C$ and each $\varepsilon > 0$ there is a $\delta > 0$ such that for each map $f:{S^1} \to \operatorname {Int} B(p,\delta ) - C$ there is a map $F:{D^2} \to \operatorname {Int}{\mkern 1mu} B(p,\varepsilon )$ such that $F|\partial {D^2} = f$ and ${F^{ - 1}}(C)$ is finite. We show the existence and explore some of the properties of wild scrawny Cantor sets in ${{\mathbf {R}}^3}$. We prove, among other things, that wild scrawny Cantor sets in ${{\mathbf {R}}^3}$ are not definable by solid tori.