arXiv subjects

Nestor Colin

Publications and source records attributed to Nestor Colin.

7 recordsLinked to original sources

arXiv2026

Birman-Hilden theory for big mapping class groups

Let $S$ and $X$ be two connected topological surfaces without boundary, and assume that $S$ is either of infinite type or has negative Euler characteristic. In this paper, we prove that if $p:S\rightarrow X$ is a fully ramified branched covering map, then $p$ satisfies the Birman-Hilden property. This generalizes a theorem of Winarski, and the known results in the literature, to the context of surfaces of infinite type and branched covering maps of infinite degree. As an application, we show that the mapping class group (respectively, the braid group on $k$-strands) of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group (respectively, the braid group on $2k$-strands) of its orientable double cover.

math.GT↗

arXiv2025

A spine for the decorated Teichm\"uller space of a punctured non-orientable surface

Building on work of Harer \cite{Ha86}, we construct a spine for the decorated Teichm\"uller 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.

math.GT↗

arXiv2024

The proper geometric dimension of the mapping class group of an orientable surface with punctures

We show that the {\it full} mapping class group of any orientable closed surface with punctures admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. This was proved for closed orientable surfaces and for {\it pure} mapping class groups by Aramayona and Mart\'inez P\'erez. As a consequence of our result we also obtain the proper geometric dimension of {\it full} spherical braid groups.

math.GT↗

arXiv2024

Farrell cohomology of the pure mapping class group of non-orientable surfaces

For an odd prime $p$, we determine the $p$-primary component of the Farrell cohomology of the pure mapping class groups of a non orientable surface of genus $p$ with $k\geqslant 1$ marked points. To do this, we classify conjugacy classes of subgroups of order $p$ of the pure mapping class group of a non orientable surface of any genus with marked points. This is obtained by extending the notion of topological equivalence for surface kernel epimorphisms of non Euclidean crystallographic groups, adapting it to the setting of surfaces with marked points.

math.GT↗

arXiv2024

On the dimension of Harer's spine for the decorated Teichm\"uller space

In \cite{Ha86} Harer explicitly constructed a spine for the decorated Teichm\"uller space of orientable surfaces with at least one puncture and negative Euler characteristic. In this paper we point out some instances where his computation of the dimension of this spine is off by $1$ and give the correct dimension.

math.GT↗

arXiv2023

Periodicity of the pure mapping class group of non-orientable surfaces

We show that the pure mapping class group $\mathcal{N}_{g}^{k}$ of a non-orientable closed surface of genus $g\geqslant 2$ with $k\geqslant 1$ marked points has $p$-periodic cohomology for each odd prime $p$ for which $\mathcal{N}_{g}^{k}$ has $p$-torsion. Using the Yagita invariant and the cohomology classes obtained by the representation of subgroups of order $p$, we obtain that the $p$-period is less than or equal to $4$ when $g\geqslant 3$ and $k\geqslant 1$. Moreover, combining the Nielsen realization theorem and a characterization of the $p$-period given in terms of normalizers and centralizers of cyclic subgroups of order $p$, we show that the $p$-period of $\mathcal{N}_{g}^{k}$ is bounded below by $4$, whenever $\mathcal{N}_{g}^{k}$ has $p$-periodic cohomology, $g\geqslant 3$ and $k\geqslant 0$. These results provide partial answers to questions proposed by G. Hope and U. Tillmann.

math.AT↗

arXiv2022

The Nielsen Realization Problem for Non-Orientable Surfaces

We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. Also, it is well known that the mapping class group $\text{Mod} (N_g; k)$ of a non-orientable surface can be identified with a subgroup of $\text{Mod} (S_{g-1}; 2k)$, the mapping class group of its orientable double cover. These facts together with the classical Nielsen realization theorem are used to prove that every finite subgroup of $\text{Mod}(N_g; k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\text{Diff}(N_g; k)$. In contrast, we show the projection $\text{Diff}(N_g) \to \text{Mod}(N_g)$ does not admit a section for large $g$.

math.AT↗