Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On spaces of embeddings of circles in surfaces

Ryan C. Gelnett

TL;DR

This work studies the topology of Jordan configurations, i.e., collections of circles on a non-positively curved surface that bound disks. By encoding nesting via finite rooted trees, the authors prove that each tree-indexed component is aspherical and serves as a classifying space for a braided automorphism group, with fundamental groups described as semidirect products involving surface braid groups. A central technical achievement is a Con(R^2)-equivariant strong deformation retraction from the space of Jordan configurations to the subspace of round Jordan configurations, first in the plane and then on general non-positive curvature surfaces via universal covers and the exponential map. The results connect configuration spaces of circles to braided automorphism groups, provide a CW-type (K(π,1)) structure, and offer a framework for understanding mapping-class-type actions on Jordan configurations in surfaces.

Abstract

We consider the space of embeddings of finitely many circles that bound disks in non-positively curved surfaces. We index the connected components of this space with finite rooted trees and show that the connected components are classifying spaces of the ``braided" automorphism groups of the associated trees. An intermediate step to proving these results is to construct a strong deformation retract onto the subspace of geometric circles; moreover, this strong deformation retraction is equivariant with respect to transformations of the surface.

On spaces of embeddings of circles in surfaces

TL;DR

This work studies the topology of Jordan configurations, i.e., collections of circles on a non-positively curved surface that bound disks. By encoding nesting via finite rooted trees, the authors prove that each tree-indexed component is aspherical and serves as a classifying space for a braided automorphism group, with fundamental groups described as semidirect products involving surface braid groups. A central technical achievement is a Con(R^2)-equivariant strong deformation retraction from the space of Jordan configurations to the subspace of round Jordan configurations, first in the plane and then on general non-positive curvature surfaces via universal covers and the exponential map. The results connect configuration spaces of circles to braided automorphism groups, provide a CW-type (K(π,1)) structure, and offer a framework for understanding mapping-class-type actions on Jordan configurations in surfaces.

Abstract

We consider the space of embeddings of finitely many circles that bound disks in non-positively curved surfaces. We index the connected components of this space with finite rooted trees and show that the connected components are classifying spaces of the ``braided" automorphism groups of the associated trees. An intermediate step to proving these results is to construct a strong deformation retract onto the subspace of geometric circles; moreover, this strong deformation retraction is equivariant with respect to transformations of the surface.
Paper Structure (11 sections, 31 theorems, 38 equations, 8 figures)

This paper contains 11 sections, 31 theorems, 38 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The space of Jordan configurations
  3. Definitions and Notations
  4. Initial Facts
  5. Equivariant centers and parametrizations of their domains
  6. Trees and path connected components
  7. Rounding Jordan configurations
  8. Convex curves in the plane
  9. Jordan configurations in the plane
  10. Jordan configurations in non-positively curved surfaces
  11. Homotopy groups

Key Result

Theorem A

There is a one-to-one correspondence between the collection of finite rooted trees with $n$ non-root vertices and the connected components of $\mathop{\mathrm{UJ}}\nolimits_n(X)$.

Figures (8)

  • Figure 1:
  • Figure 2: Conformal parametrization of the interior of a curve $[f]$ with center $p$ given by Carathéodory's Theorem.
  • Figure 3: Three labeled trees, $T_1$, $T_2$, and $T_3$, where edges are directed upwards. As labeled trees, $T_1$ and $T_2$ are equal, and $T_3$ is not equal to either of them. The underlying unlabeled trees of $T_1$, $T_2$, and $T_3$ are equal. As planar labeled trees, $T_1$ and $T_2$ are isomorphic but not equal, and $T_3$ is not isomorphic to either of them. The underlying planar unlabeled tree of $T_3$ is equal to that of $T_2$, and is isomorphic but not equal to that of $T_1$.
  • Figure 4: A labeled configuration $\kappa$ of circles in $\mathop{\mathrm{Conf}}\nolimits_7(S^1,\mathbb{R}^2)$, and the associated abstract labeled tree $T(\kappa)$. (Reprinted from curry2024configurationspacescirclesplane)
  • Figure 5: A labeled Jordan configuration $f$ in $\mathop{\mathrm{J}}\nolimits_4(\mathbb{R}^2)$, and the associated abstract labeled tree $T(f)$.
  • ...and 3 more figures

Theorems & Definitions (76)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition A: Configuration space of points
  • Definition B: Configuration space of circles
  • Definition C: Embeddings of circles
  • Definition D
  • Definition E: Jordan configurations
  • Lemma F
  • proof
  • ...and 66 more