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Homeomorphisms of surfaces that preserve continuously differentiable curves

Katherine Williams Booth

TL;DR

Addresses the problem of characterizing $Homeo^{1}(S)$, the group of surface homeomorphisms that map every $C^1$ curve to a $C^1$ curve. The authors introduce a local criterion based on the projective tangent bundle $\mathbb{P}TS$ and the notion of transverse sequences, proving a three-part condition that is necessary and sufficient for membership. The main contributions include the Main Theorem, explicit examples showing $Homeo^{1}(S)$ properly contains $\mathrm{Diff}^{1}(S)$, and a constructive recovery of $Homeo^{1}(S)$ from tangent-space data. The work advances understanding of $C^1$-preserving maps on surfaces and connects to automorphisms of the $C^1$-curve graph, with implications for the algebraic structure and topology of these groups.

Abstract

In this paper, we study Homeo$^1(S)$, the group of homeomorphisms of a surface that preserve the set of one-dimensional $C^1$ submanifolds of that surface. The group Homeo$^1(S)$ belongs to a family of similarly defined groups Homeo$^k(S)$ that were recently introduced by the author. In a separate paper, we have shown that for most closed surfaces, Homeo$^k(S)$ is naturally isomorphic to the automorphisms of a smooth fine curve graph. By contrast, the work in this paper gives local conditions that characterize Homeo$^1(S)$. We show that there exists a collection of conditions that are both necessary and sufficient for a homeomorphism of the surface to be an element of this group. These conditions primarily depend upon the structure of the induced map on the projective tangent bundle. Additionally, we provide examples of several types of elements of Homeo$^1(S)$ that are not diffeomorphisms. These include inducing discontinuous maps on the projective tangent bundle and having infinitely many non-differentiable points.

Homeomorphisms of surfaces that preserve continuously differentiable curves

TL;DR

Addresses the problem of characterizing , the group of surface homeomorphisms that map every curve to a curve. The authors introduce a local criterion based on the projective tangent bundle and the notion of transverse sequences, proving a three-part condition that is necessary and sufficient for membership. The main contributions include the Main Theorem, explicit examples showing properly contains , and a constructive recovery of from tangent-space data. The work advances understanding of -preserving maps on surfaces and connects to automorphisms of the -curve graph, with implications for the algebraic structure and topology of these groups.

Abstract

In this paper, we study Homeo, the group of homeomorphisms of a surface that preserve the set of one-dimensional submanifolds of that surface. The group Homeo belongs to a family of similarly defined groups Homeo that were recently introduced by the author. In a separate paper, we have shown that for most closed surfaces, Homeo is naturally isomorphic to the automorphisms of a smooth fine curve graph. By contrast, the work in this paper gives local conditions that characterize Homeo. We show that there exists a collection of conditions that are both necessary and sufficient for a homeomorphism of the surface to be an element of this group. These conditions primarily depend upon the structure of the induced map on the projective tangent bundle. Additionally, we provide examples of several types of elements of Homeo that are not diffeomorphisms. These include inducing discontinuous maps on the projective tangent bundle and having infinitely many non-differentiable points.

Paper Structure

This paper contains 17 sections, 24 theorems, 39 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Statement of our Main Theorem
  3. The projective tangent bundle and transverse sequences
  4. Further questions
  5. Paper structure and overview of the proof
  6. Acknowledgments
  7. Example elements from $\mathbf{\mathop{\mathrm{\textbf{Homeo}}}\nolimits^1(S)}$
  8. Descriptions of the examples
  9. The proof of Proposition \ref{['propexamples']}
  10. Properties of elements of $\mathbf{\mathop{\mathrm{\textbf{Homeo}^{\textbf{1}}}}\nolimits(S)}$
  11. A well-defined map of projective tangent space
  12. Property $\mathbf{(b)}$: Homeomorphism on $\mathbf{\mathbb{P}T_pS}$
  13. Property $\mathbf{(c)}$: Mapping transverse sequences
  14. Recovering $\mathop{\mathrm{\textbf{Homeo}^{\textbf{1}}}}\nolimits(\mathbf{S})$
  15. Well-defined tangent lines
  16. ...and 2 more sections

Key Result

Theorem 1

Fix a smooth surface $S$. A homeomorphism $f$ of $S$ is an element of $\mathop{\mathrm{Homeo^{1}}}\nolimits(S)$ if and only if $f$ has the following three properties:

Figures (12)

  • Figure 1: A transverse sequence of tangent lines from $y=x^2 \sin(1/x)$ with slope 1 that converge to the origin along the $x$-axis
  • Figure 2: The action of $G$ on the vectors $(1, 0)$, $(0, 1)$, and $(1,1)$ based at the origin
  • Figure 3: The action of $H$ on a grid near the origin
  • Figure 4: The action of $H^{-1}$ on a grid near the origin
  • Figure 5: The action of $W$ on lines passing through the origin
  • ...and 7 more figures

Theorems & Definitions (46)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • proof : Proof of Proposition \ref{['propNex']}
  • Proposition 6
  • proof
  • ...and 36 more