# Non-removability of Sierpinski spaces

## TL;DR

This paper proves that all Sierpiński spaces in higher-dimensional spheres are non-removable for (quasi)conformal maps, showing they can be mapped to positive measure sets by homeomorphisms conformal outside them, thus providing the first examples of such non-removable sets in higher dimensions.

## Contribution

It establishes the non-removability of all Sierpiński spaces in ${f S}^n$, $n \\geq 2$, for (quasi)conformal maps, generalizing previous results and constructing explicit non-removable examples.

## Key findings

- All Sierpiński spaces in ${f S}^n$ are non-removable for (quasi)conformal maps.
- Existence of homeomorphisms conformal outside the Sierpiński space that map it to a positive measure set.
- First known class of non-removable sets in higher-dimensional spheres.

## Abstract

We prove that all Sierpi\'nski spaces in ${\mathbb{S}}^n$, $n\geq 2$, are non-removable for (quasi)conformal maps, generalizing the result of the first named author arXiv:1809.05605. More precisely, we show that for any Sierpi\'nski space $X\subset \mathbb{S}^n$ there exists a homeomorphism $f\colon \mathbb{S}^n\to \mathbb{S}^n$, conformal in $\mathbb{S}^n\setminus X$, that maps $X$ to a set of positive measure and is not globally (quasi)conformal. This is the first class of examples of non-removable sets in higher dimensions.