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Comparing combinatorial models of moduli space and their compactifications

Daniela Egas Santander, Alexander Kupers

Abstract

We compare two combinatorial models for the moduli space of two-dimensional cobordisms: Bödigheimer's radial slit configurations and Godin's admissible fat graphs, producing an explicit homotopy equivalence using a "critical graph" map. We also discuss natural compactifications of these two models, the unilevel harmonic compactification and Sullivan diagrams respectively, and prove that the homotopy equivalence induces a cellular homeomorphism between these compactifications.

Comparing combinatorial models of moduli space and their compactifications

Abstract

We compare two combinatorial models for the moduli space of two-dimensional cobordisms: Bödigheimer's radial slit configurations and Godin's admissible fat graphs, producing an explicit homotopy equivalence using a "critical graph" map. We also discuss natural compactifications of these two models, the unilevel harmonic compactification and Sullivan diagrams respectively, and prove that the homotopy equivalence induces a cellular homeomorphism between these compactifications.

Paper Structure

This paper contains 28 sections, 34 theorems, 65 equations, 23 figures.

Table of Contents

  1. Introduction
  2. The moduli space of cobordisms
  3. Combinatorial models of moduli space
  4. Applications of these models
  5. Explicit computations of the homology of moduli spaces
  6. Two-dimensional field theories, in particular string topology
  7. Outline of paper
  8. Acknowledgments
  9. Radial slit configurations and the harmonic compactification
  10. The definition
  11. Spaces of radial slit configurations
  12. Cell complexes of radial slit configurations
  13. Relationships
  14. The universal surface bundle
  15. Admissible fat graphs and string diagrams
  16. ...and 13 more sections

Key Result

Theorem \oldthetheorem

We define a space $\textfrak{Rad}^{\sim}$ and maps cor_pi1homeq, cor_pi2homeq and prop_sdubradhomeo such that there is a commutative square \begin{tikzcd} \Rad \arrow{d} & \Radt \lar{\simeq}[swap]{(\ref{cor_pi1homeq})} \rar{(\ref{cor_pi2homeq})}[swap]{\simeq} & \MFatad \arrow[two heads]{dd} \\[-5pt]

Figures (23)

  • Figure 2.1: An example of constructing a cobordism by cutting and glueing slits in annuli. We start with the annulus on the left, cut along the blue lines to obtain the middle figure, and finally glue both the gray sides and the white sides of the cuts to get the cobordism on the right. In this simple example the pairing $\lambda$ and the successor permutation $\omega$ are uniquely determined.
  • Figure 2.2: An example of a radial slit preconfiguration with two slits on the same radial segment; $\zeta_1$ is the shorter blue slit and $\zeta_2$ is the longer red slit. The successor permutation $\omega$ allows us to think of $\zeta_1$ as either infinitesimally clockwise or counterclockwise from $\zeta_2$.
  • Figure 2.3: The configuration of Figure \ref{['figradialglue']} with all its data pointed out.
  • Figure 2.4: Examples of the different types of radial sectors with subsets $\alpha^\pm$ and $\beta^\pm$.
  • Figure 2.5: An example of a radial slit preconfiguration leading to a degenerate surface. The black arc connecting two points on the surface on the right was the line segment between the two red slits.
  • ...and 18 more figures

Theorems & Definitions (135)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
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  • Definition \oldthetheorem
  • Definition \oldthetheorem
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  • ...and 125 more