The homotopy dimension of codiscrete subsets of the 2-sphere

Details The homotopy dimension of codiscrete subsets of the 2-sphere The homotopy dimension of codiscrete subsets of the 2-sphere

2006-03-03

arxiv.org/abs/math/0603069v1

Mathematics

Geometric Topology

18 pages, 2 figures

Scientific paper

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?

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