Concerning Fundamental Groups of Locally Connected Subsets of the Plane

TL;DR

This paper explores the relationship between fundamental groups and topological structures of planar Peano continua, showing how algebraic data can determine local connectivity properties.

Contribution

It establishes that the topological structure of non-semi-locally simply connected points in planar Peano continua is fully determined by their fundamental groups.

Findings

Fundamental groups induce continuous maps up to conjugation.

Topological structure of non-semi-locally simply connected points is algebraically determined.

Reconstruction of topological features from subgroup lattices of fundamental groups.

Abstract

We show that every homomorphism from a one-dimensional Peano continuum to a planar Peano continuum is induced by a continuous map up to conjugation. We then prove that the topological structure of the space of points at which a planar Peano continuum is not semi-locally simply connected is determined solely by algebraic information contained in its fundamental group. Furthermore, we demonstrate how to reconstruct the topological structure of the space of points at which a planar Peano continuum is not semi-locally simply connected using only the subgroup lattice of its fundamental group.