The homotopy dimension of codiscrete subsets of the 2-sphere S²

J. W. Cannon¹, G. R. Conner²
Fund. Math. 197 (2007), 35-66
doi:10.4064/fm197-0-3

Abstract: Andreas Zastrow conjectured, and Cannon–Conner–Zastrow proved, that filling one hole in the Sierpiński curve with a disk results in a planar Peano continuum that is not homotopy equivalent to a 1-dimensional set. Zastrow's example is the motivation for this paper, where we characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopy equivalent to 1-dimensional compacta, 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?

MSC (2000): Primary 57N05; Secondary 54F45, 54F50, 55M10, 55P10.
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Department of Mathematics
Brigham Young University
Provo, UT 84602, U.S.A.
Department of Mathematics
Brigham Young University
Provo, UT 84602, U.S.A.