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

• Autorzy: J. W. Cannon , G. R. Conner
• Języki publikacji: EN

Abstrakty

EN

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?

Kategorie tematyczne

• 57N05: Topology of E 2 , 2 -manifolds
• 54F50: Spaces of dimension ≤ 1 ; curves, dendrites
• 55M10: Dimension theory
• 55P10: Homotopy equivalences
• 54F45: Dimension theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2007
• Tom: 197
• Numer: 1
• Strony: 35-66

Daty

wydano

2007

Twórcy

autor

J. W. Cannon

• Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A.

autor

G. R. Conner

• Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A.

Identyfikatory

DOI

10.4064/fm197-0-3