Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Goldman bracket characterizes homeomorphisms between non-compact surfaces

Sumanta Das, Siddhartha Gadgil, Ajay Kumar Nair

TL;DR

The paper provides a sharp criterion for when a homotopy equivalence between non-compact orientable surfaces without boundary is homotopic to a homeomorphism: this holds exactly when the map commutes with the Goldman bracket, excluding the plane and cylinder. The authors develop an exhaustion-based construction that yields a proper map $g$ homotopic to the given $f$, by controlling loop images via fillings and splitting arguments rooted in the Goldman bracket. They then upgrade $g$ to a (proper) homeomorphism using a standard rigidity result for non-compact surfaces, with orientation forcing explained through bracket considerations. The approach ties a Lie-algebraic structure on curves directly to topological rigidity, providing a practical criterion and a robust technique that also yields a related intersection-number characterization.

Abstract

We show that a homotopy equivalence between two non-compact orientable surfaces is homotopic to a homeomorphism if and only if it preserves the Goldman bracket, provided our surfaces are neither the plane nor the punctured plane.

The Goldman bracket characterizes homeomorphisms between non-compact surfaces

TL;DR

The paper provides a sharp criterion for when a homotopy equivalence between non-compact orientable surfaces without boundary is homotopic to a homeomorphism: this holds exactly when the map commutes with the Goldman bracket, excluding the plane and cylinder. The authors develop an exhaustion-based construction that yields a proper map homotopic to the given , by controlling loop images via fillings and splitting arguments rooted in the Goldman bracket. They then upgrade to a (proper) homeomorphism using a standard rigidity result for non-compact surfaces, with orientation forcing explained through bracket considerations. The approach ties a Lie-algebraic structure on curves directly to topological rigidity, providing a practical criterion and a robust technique that also yields a related intersection-number characterization.

Abstract

We show that a homotopy equivalence between two non-compact orientable surfaces is homotopic to a homeomorphism if and only if it preserves the Goldman bracket, provided our surfaces are neither the plane nor the punctured plane.
Paper Structure (18 sections, 10 theorems, 6 equations, 3 figures)

This paper contains 18 sections, 10 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline of the proof of Theorem \ref{['main']}
  3. Background
  4. Subsurfaces
  5. Exhaustions
  6. Intersections and geodesics
  7. Filling curves
  8. Splitting surfaces
  9. Proof of Theorem \ref{['main']}
  10. Constructing the subsurface
  11. Pushing loop images outside subsurfaces
  12. Construction of the map in the base case
  13. Inductive properties
  14. Maps on fundamental groups of components
  15. Maps on complementary components
  16. ...and 3 more sections

Key Result

Theorem 1.1

A homotopy equivalence $f\colon \Sigma'\to \Sigma$ between two non-compact oriented surfaces without boundary is homotopic to an orientation-preserving homeomorphism if and only if it commutes with the Goldman bracket, i.e., for all $x',y'\in\mathbb Z[\widehat{\pi}(\Sigma')]$, we have, where $f_*\colon \mathbb Z[\widehat{\pi}(\Sigma')]\to\mathbb Z[\widehat{\pi}(\Sigma)]$ is the function induced b

Figures (3)

  • Figure 1: $\left\{\alpha_1,...,\alpha_7\right\}$ is a filling system of $K_m$ and $\left\{\alpha_1,...,\alpha_9\right\}$ is an extended filling system for $K_m$ because $I_\Sigma\left(C_1,\alpha_9\right)\neq 0\neq I_\Sigma\left(C_2,\alpha_8\right)$.
  • Figure 2: If $\widehat{V}\neq V$, $f_*([\gamma'])\cdot f_*([\delta])=[\gamma*\theta*\eta*\bar{\theta}]$ is not homotopic to a curve disjoint from $\partial\widehat{V}$, giving a contradiction.
  • Figure 3: Description of $g^{(j)}_m\colon V'=V^{(j)}_m\to V\subset\Sigma\setminus K_{m-1}$. On the top, i.e., in $V'$, the purple, blue, and green portions (arcs and circles) form $\Gamma'$, and blue and black arcs form $T'^{(j)}_m$. At the bottom: the grey and yellow shades indicate $g^{(j)}_m(V')$ and $g^{(j)}_m(C_i'\times [0,1])$, respectively, and red and green loops denote $g^{(j)}_m(C_i'\times \{1\})$ and $g^{(j)}_m(\delta')$, respectively, where $i=1,2$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3: Goldman MR0846929
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 5 more