On the fundamental groups of perforated surfaces

TL;DR

This paper classifies perforated surfaces, explores their fundamental groups, and demonstrates that these groups are large and non-Hopfian, extending surface topology concepts to complex, dense subsets.

Contribution

It provides a classification theorem for perforated surfaces and analyzes the properties of their fundamental groups, including largeness and non-Hopfian nature.

Findings

Perforated surfaces are classified using surface classification theorems.

Fundamental groups of perforated surfaces are large.

Fundamental groups of the Sierpiński and Menger curves are not Hopfian.

Abstract

A perforated surface is the complement $\mathring\Sigma:=\Sigma\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\Sigma$. It is known that the topological type of $\Sigma\setminus A$ is independent of the choice of $A$. Any perforated surface is one-dimensional, connected, locally path connected, and is not semi-locally simply connected at any of its points. In this paper we obtain a classification theorem for perforated surfaces, using the classification theorem for surfaces. We show that any connected covering of a perforated surface $\mathring \Sigma$ arises from a covering of a surface $\Sigma'$ such that $\mathring\Sigma\cong \mathring\Sigma'$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring \Sigma$, the Sierpi\'nski curve and the Menger curve are not Hopfian.