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Strong collapsibility of the arc complexes of orientable and non-orientable crowns

Pallavi Panda

Abstract

We prove that the arc complex of a polygon with a marked point in its interior is a strongly collapsible combinatorial ball. We also show that the arc complex of a Möbius strip, with finitely many marked points on its boundary, is a simplicially collapsible combinatorial ball but is not strongly collapsible.

Strong collapsibility of the arc complexes of orientable and non-orientable crowns

Abstract

We prove that the arc complex of a polygon with a marked point in its interior is a strongly collapsible combinatorial ball. We also show that the arc complex of a Möbius strip, with finitely many marked points on its boundary, is a simplicially collapsible combinatorial ball but is not strongly collapsible.
Paper Structure (28 sections, 21 theorems, 16 equations, 16 figures)

This paper contains 28 sections, 21 theorems, 16 equations, 16 figures.

Table of Contents

  1. Abstract.
  2. Introduction
  3. Collapsibility.
  4. Collapsible manifolds.
  5. Strong collapses.
  6. Arc complexes.
  7. Contributions.
  8. Plan of the paper
  9. Acknowledgments
  10. Simplicial Topology
  11. Combinatorial manifolds.
  12. Simplicial collapses.
  13. Strong collapses.
  14. Vocabulary
  15. The surfaces
  16. ...and 13 more sections

Key Result

Corollary 1

For $n\geq 1$, the full arc complex $\mathcal{A}({\mathcal{P}_{n} ^{\circledcirc}})$ of a crown $\mathcal{P}_{n} ^{\circledcirc}$ is a combinatorial ball of dimension $n-1$.

Figures (16)

  • Figure 1: A crown with three vertices.
  • Figure 2: A non-orientable crown with three vertices.
  • Figure 3: The arc complexes of $\mathcal{P}_{1} ^{\circledcirc}, \mathcal{P}_{2} ^{\circledcirc}$ and $\mathcal{P}_{3} ^{\circledcirc}$
  • Figure 4: The arc complexes of $\mathcal{M}_{1}, \mathcal{M}_{2}$ and $\mathcal{M}_{3}$
  • Figure 5: The arc complexes $\mathcal{A}_{2,3}$, $\mathcal{A}_{4,1}$ are strongly collapsible.
  • ...and 11 more figures

Theorems & Definitions (55)

  • Corollary
  • Theorem
  • Corollary
  • Proposition 2.1
  • Theorem 2.2
  • Example 2.3
  • Theorem 2.4
  • Definition 3.1
  • Definition 3.2
  • Remark 3.1
  • ...and 45 more