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Surfaces and other Peano Continua with no Generic Chains

Gianluca Basso, Alessandro Codenotti, Andrea Vaccaro

Abstract

The space of chains on a compact connected space encodes all the different ways of continuously growing out of a point until exhausting the space. A chain is \emph{generic} if its orbit under the action of the underlying homeomorphism group is comeager. In this paper we show that a large family of topological spaces do not have a generic chain: in addition to all manifolds of dimension at least 3, for which the result was already known, our theorem covers all compact surfaces except for the sphere and the real projective plane - for which the question remains open - as well as all other homogeneous Peano continua, circle excluded. If the spaces are moreover strongly locally homogeneous, which is the case for any closed manifold and the Menger curve, we prove that chains cannot be classified up to homeomorphism by countable structures, and that the underlying homeomorphism groups have non-metrizable universal minimal flows, with all orbits meager, in contrast to the case of 1-dimensional manifolds. The proof of the main result is of combinatorial nature, and it relies on the creation of a dictionary between open sets of chains on one side, and walks on finite connected graphs on the other.

Surfaces and other Peano Continua with no Generic Chains

Abstract

The space of chains on a compact connected space encodes all the different ways of continuously growing out of a point until exhausting the space. A chain is \emph{generic} if its orbit under the action of the underlying homeomorphism group is comeager. In this paper we show that a large family of topological spaces do not have a generic chain: in addition to all manifolds of dimension at least 3, for which the result was already known, our theorem covers all compact surfaces except for the sphere and the real projective plane - for which the question remains open - as well as all other homogeneous Peano continua, circle excluded. If the spaces are moreover strongly locally homogeneous, which is the case for any closed manifold and the Menger curve, we prove that chains cannot be classified up to homeomorphism by countable structures, and that the underlying homeomorphism groups have non-metrizable universal minimal flows, with all orbits meager, in contrast to the case of 1-dimensional manifolds. The proof of the main result is of combinatorial nature, and it relies on the creation of a dictionary between open sets of chains on one side, and walks on finite connected graphs on the other.
Paper Structure (21 sections, 51 theorems, 91 equations, 3 figures)

This paper contains 21 sections, 51 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Peano Continua and Chains
  4. The Space of Maximal Connected Chains
  5. Walks on Graphs
  6. The Winding Number of a Walk
  7. Brick Partitions
  8. Walks on Brick Partitions and their Refinements
  9. A Criterion for the Absence of Generic Chains
  10. Absence of Generic Chains
  11. Walks on Peano continua with no locally separating points
  12. Robust cycles
  13. Locally non-planar open subset
  14. Planar open set with non-locally-separating simple closed curve
  15. Circular covering
  16. ...and 6 more sections

Key Result

Theorem 1.1

If $X$ is a compact surface other than the sphere or the real projective plane, then $X$ has no generic chain.

Figures (3)

  • Figure 1: A sketch of a robust cycle: only the relevant elements of the brick partitions appearing in \ref{['def:robust']} are shown, and the lasso $v' {}^{\frown} c$ is depicted as a non-self-intersecting curve.
  • Figure 2: Domain and codomain of a subchain $\mathcal{I}$.
  • Figure 3: The construction in the proof of \ref{['thm:minimality-sierpinski']}.

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Lemma 3.1: Monotone Lifting
  • proof
  • Definition 3.2
  • ...and 91 more