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?
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-3,
author = {J. W. Cannon and G. R. Conner},
title = {The homotopy dimension of codiscrete subsets of the 2-sphere S2},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {35-66},
zbl = {1148.54017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-3}
}
J. W. Cannon; G. R. Conner. The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊². Fundamenta Mathematicae, Tome 193 (2007) pp. 35-66. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-3/