Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A spine for the decorated Teichmüller space of a punctured non-orientable surface

Nestor Colin, Rita Jiménez Rolland, Porfirio L. León Álvarez, Luis Jorge Sánchez Saldaña

TL;DR

The article constructs an equivariant spine for the decorated Teichmüller space of a punctured non-orientable surface by passing to the orientable double cover and restricting Harer’s spine to the $\sigma$-fixed locus, yielding a minimal-dimension model for $\underline{E}\mathrm{PMod}(N_{g+1}^{n})$. It provides an explicit dimension formula: $\dim \mathcal{Z}_{g+1}^{n,\ell} = \mathrm{vcd}(\mathrm{PMod}(N_{g+1}^{n})) + \ell$ for $\ell < n$ and $\mathrm{vcd}(\mathrm{PMod}(N_{g+1}^{n})) + (\ell-1)$ for $\ell = n$, with $1\le \ell \le n$. The construction hinges on $\sigma$-invariant arc systems and the equivariant deformation retraction of Harer’s spine, yielding a spine that is also a model for the classifying space for proper actions. Consequently, the authors show that the proper geometric dimension equals the virtual cohomological dimension in these cases and, in particular, obtain minimal-dimension models for punctured non-orientable mapping class groups.

Abstract

Building on work of Harer \cite{Ha86}, we construct a spine for the decorated Teichmüller space of a non-orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the deformation retraction onto this spine is equivariant with respect to the pure mapping class group of the non-orientable surface. As a consequence, we obtain a model for the classifying space for proper actions of the pure mapping class group of a punctured non-orientable surface, which is of minimal dimension in the case there is a single puncture.

A spine for the decorated Teichmüller space of a punctured non-orientable surface

TL;DR

The article constructs an equivariant spine for the decorated Teichmüller space of a punctured non-orientable surface by passing to the orientable double cover and restricting Harer’s spine to the -fixed locus, yielding a minimal-dimension model for . It provides an explicit dimension formula: for and for , with . The construction hinges on -invariant arc systems and the equivariant deformation retraction of Harer’s spine, yielding a spine that is also a model for the classifying space for proper actions. Consequently, the authors show that the proper geometric dimension equals the virtual cohomological dimension in these cases and, in particular, obtain minimal-dimension models for punctured non-orientable mapping class groups.

Abstract

Building on work of Harer \cite{Ha86}, we construct a spine for the decorated Teichmüller space of a non-orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the deformation retraction onto this spine is equivariant with respect to the pure mapping class group of the non-orientable surface. As a consequence, we obtain a model for the classifying space for proper actions of the pure mapping class group of a punctured non-orientable surface, which is of minimal dimension in the case there is a single puncture.

Paper Structure

This paper contains 13 sections, 19 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Decorated Teichmüller spaces and the orientable double cover
  3. Decorated Teichmüller spaces for orientable and non-orientable surfaces
  4. The orientable double cover
  5. Mapping class groups, decorated Teichmüller spaces and the orientable double cover
  6. Ideal triangulations and spines for decorated Teichmüller spaces
  7. Arc systems and Harer's complex of arcs
  8. Ideal triangulations of decorated Teichmüller spaces.
  9. Spines for decorated Teichmüller spaces
  10. The dimension of the spine
  11. Properties of sigma-invariant arc systems
  12. Computation of the dimension
  13. Decorated Teichmüller spaces as classifying spaces for proper actions

Key Result

Theorem 1.1

Let $1 \leq \ell \leq n$ and $g +n> 1$. There exists a $\mathop{\mathrm{PMod}}\nolimits(N_{g+1}^n)$-equivariant spine $\mathcal{Z}=\mathcal{Z}_{g+1}^{n,\ell}$ for the decorated Teichmüller space $\mathscr T(N_{g+1}^n;\pi(\Delta))$ of dimension Moreover, the spine $\mathcal{Z}$ gives a model for the classifying space for proper actions of $\mathop{\mathrm{PMod}}\nolimits(N_{g+1}^n)$.

Figures (5)

  • Figure 1: Model surface $F_0=F_g-P$ with sets of punctures$P$ and of marked points$\Delta$
  • Figure 2: A $\sigma$-invariant arc system ${\mathbf B}$ with $2g+2$ arcs
  • Figure 3: Polygons obtained by cutting $F_0$ along the arc system ${\mathbf B}$.
  • Figure 4: A $\sigma$-invariant arc system ${\mathbf B}_{max}$ of maximal rank for a surface of even genus
  • Figure 5: A minimal $\sigma$-invariant arc system ${\mathbf B}_{min}$ that fills up $F_0$ of even genus

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2: A spine for Teichmüller space of a non-orientable surface with punctures
  • Corollary 1.3
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.5
  • Proposition 2.6
  • Remark 3.1
  • Definition 3.2: Harer's complex of arcs
  • Remark 3.3
  • ...and 28 more