The homotopy dimension of codiscrete subsets of the 2-sphere

J. W. Cannon; G. R. Conner

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arXiv · math/0603069

The homotopy dimension of codiscrete subsets of the 2-sphere

J. W. Cannon·G. R. Conner

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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?

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J. W. Cannon, G. R. Conner. 2006-03-03. The homotopy dimension of codiscrete subsets of the 2-sphere. https://arxiv.org/abs/math/0603069

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