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

J. W. Cannon, and G. R. Conner. "The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊²." Fundamenta Mathematicae 197.1 (2007): 35-66. <http://eudml.org/doc/283099>.

@article{J2007,
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?},
author = {J. W. Cannon, G. R. Conner},
journal = {Fundamenta Mathematicae},
keywords = {codiscrete set; dimension; homotopy dimension; Peano continuum},
language = {eng},
number = {1},
pages = {35-66},
title = {The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊²},
url = {http://eudml.org/doc/283099},
volume = {197},
year = {2007},
}

TY - JOUR
AU - J. W. Cannon
AU - G. R. Conner
TI - The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊²
JO - Fundamenta Mathematicae
PY - 2007
VL - 197
IS - 1
SP - 35
EP - 66
AB - 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?
LA - eng
KW - codiscrete set; dimension; homotopy dimension; Peano continuum
UR - http://eudml.org/doc/283099
ER -