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

TL;DR

This paper establishes the equality of various dimensions of the mapping class group of non-orientable surfaces, providing explicit models for their classifying spaces and extending results to surfaces with boundaries and punctures.

Contribution

It proves the equality of proper cohomological, geometric, and virtual cohomological dimensions for non-orientable surface mapping class groups, and constructs explicit classifying space models.

Findings

Dimensions are equal for all non-orientable surfaces except g=4,5.

Explicit model for classifying space of dimension 2g-5.

Results extend to surfaces with boundaries and punctures.

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 $\mathrm{vcd}(\mathcal{N}_g)=2g-5$. 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.