Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

On the dimension of the mapping class groups of a non-orientable surface

Pages: 347 – 372

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a17

Authors

Cristhian E. Hidber (Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, Mexico)

Luis Jorge Sánchez Saldaña (Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad de México, Mexico)

Alejandra Trujillo-Negrete (Centro de Investigación en Matemáticas (CMAT), Guanajuato, Gto., Mexico)

Abstract

Let $\mathcal{N}_g$ be the mapping class group of a non-orientable closed surface. We prove that the proper cohomological dimension, the proper geometric dimension, and the virtual cohomological dimension of $\mathcal{N}_g$ are equal whenever $g \neq 4,5$. In particular, there exists a model for the classifying space of $\mathcal{N}_g$ for proper actions of dimension $\operatorname{vcd}(\mathcal{N}_g)=2g-5$. Similar results are obtained for the mapping class group of a non-orientable surface with boundaries and possibly punctures, and for the pure mapping class group of a non-orientable surface with punctures and without boundaries.

Keywords

mapping class group, non-orientable surface, virtual cohomological dimension, proper cohomological dimension, proper geometric dimension

2010 Mathematics Subject Classification

20F34, 20F65, 20J05

Received 6 October 2020

Received revised 3 May 2021

Accepted 17 June 2021

Published 18 May 2022