Concerning Fundamental Groups of Locally Connected Subsets of the Plane

2011/05/02 by Gregory Conner, Gregory R. Conner, Conner, Gregory +2 · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #advanced mathematical theories #math.AT

paper · pdf · doi:10.48550/arxiv.1105.0405

Restructured arguments and corrected typos

openalex publication_date 2011/05/02 · arxiv created 2013/05/17 · arxiv updated 2013/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Higher homotopy wild sets

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• Peano dimension of fundamental groups
• The homotopy dimension of codiscrete subsets of the 2-sphere
• Embedding planar compacta in planar continua with application: homotopic maps between planar Peano continua are characterized by the fundamental group homomorphism
• Nonlocal loss of first homotopy in polyhedral approximations of Peano continua