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arXiv · 2007.12542

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

Cristhian E. Hidber·Luis Jorge Sánchez Saldaña·Alejandra Trujillo-Negrete
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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.

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Cristhian E. Hidber, Luis Jorge Sánchez Saldaña, Alejandra Trujillo-Negrete. 2020-07-24. On the dimension of the mapping class groups of a non-orientable surface. https://arxiv.org/abs/2007.12542

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