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Homotopy types of fine curve and fine arc complexes

Ryan Dickmann, Zachary Himes, Alexander Nolte, Roberta Shapiro

TL;DR

The paper shows that the fine curve complex $\\mathcal{C}^ ightarrow(S_{g,b})$ has the same homotopy type as the classical curve complex $\\mathcal{C}(S_{g,b})$ by proving the projection map $f$ has contractible fibers and hence is a homotopy equivalence. It also proves that the fine arc complex $\\mathcal{A}^ ightarrow(S_{g,b})$ is contractible for a broad range of $(g,b)$ via a Hatcher-flow–type argument, with a complementary combinatorial approach for the curve case. Together, these results connect the topology of refined curve/arc complexes to mapping class group techniques, enabling applications akin to Harer–Hatcher stability and related geometric-group-theoretic analyses. The methods blend dimension-theoretic controls of intersection sets, bigon surgery arguments, and Whitehead’s theorem to convert local combinatorics into global homotopy conclusions, with the arc case supplemented by a flow-based contraction argument.

Abstract

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to the curve complex. We also prove that the fine arc complex is contractible.

Homotopy types of fine curve and fine arc complexes

TL;DR

The paper shows that the fine curve complex has the same homotopy type as the classical curve complex by proving the projection map has contractible fibers and hence is a homotopy equivalence. It also proves that the fine arc complex is contractible for a broad range of via a Hatcher-flow–type argument, with a complementary combinatorial approach for the curve case. Together, these results connect the topology of refined curve/arc complexes to mapping class group techniques, enabling applications akin to Harer–Hatcher stability and related geometric-group-theoretic analyses. The methods blend dimension-theoretic controls of intersection sets, bigon surgery arguments, and Whitehead’s theorem to convert local combinatorics into global homotopy conclusions, with the arc case supplemented by a flow-based contraction argument.

Abstract

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose -simplices are collections of disjoint curves. We prove that the fine curve complex is homotopy equivalent to the curve complex. We also prove that the fine arc complex is contractible.
Paper Structure (7 sections, 12 theorems, 2 equations, 3 figures)

This paper contains 7 sections, 12 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Key definitions and a critical proposition
  3. Homotopy type of the fine curve complex
  4. Simplicial complexes
  5. Preimages of simplices are contractible
  6. Proof of Theorem \ref{['thm:cchom']}
  7. Homotopy type of the fine arc complex

Key Result

Theorem 1.1

Let $S_{g,b}$ be an orientable surface with $g\geq 1$ or $b\geq 4$. The map $f:\mathcal{C}^\dagger(S_{g,b})\to \mathcal{C}(S_{g,b})$ that sends a curve to its isotopy class is a homotopy equivalence. In particular, when $(g, b)\neq(0, b)$ with $b\leq 3$, $\mathcal{C}^\dagger(S_{g,b})$ is homotopy eq

Figures (3)

  • Figure 1: Left: a crossing intersection. Right: a touching intersection.
  • Figure 2: Our algorithm involves finding an innermost bigon with $c_n$, such as the pictured shaded one formed by $c_1$, and homotoping across it. An alternative innermost bigon is the one created by $c_2.$
  • Figure 3: Left: $\beta$ appears vertically in black and there are several $\gamma_i$ crossing it. A neighborhood $N$ of the portion of $\beta$ between $\partial S_{g,b}$ and the first point of intersection is shaded in pink. Right: the first intersection is resolved via surgery with $\partial N.$

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2: e.g. HurewiczDimensionTheory
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5: newman1951elements
  • proof : Proof of Proposition \ref{['lemma:keylemma']}
  • ...and 11 more