The homotopy dimension of codiscrete subsets of the 2-sphere

2006/03/03 by J. W. Cannon, Cannon, J. W., Gregory R. Conner +2
Computer Science · Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.GT #msc:54F50 #msc:55M10

paper · pdf · doi:10.48550/arxiv.math/0603069

18 pages, 2 figures

arxiv created 2006/03/03 · arxiv updated 2009/12/01

Abstract

We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following question: Is a planar Peano continuum homotopically 1- dimensional if its fundamental group is isomorphic with the fundamental group of a 1-dimensional planar Peano continuum?

Related

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• Peano dimension of fundamental groups
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• Nonlocal loss of first homotopy in polyhedral approximations of Peano continua