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Results tagged with surfaces
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user 1463
A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.
5
votes
Accepted
Covering spaces of surfaces
Given any covering map $\Sigma_h\to\Sigma_g$ between two surfaces, is there some kind of a ``standard" covering $M\to \Sigma_g$, which factors through $\Sigma_h$? …
- 25.2k
6
votes
Accepted
Find a simple closed curve in $S$ which represents a commutator in $\pi_1 S$
The free subgroup generated by $a_1,b_1$ is carried by a one-holed torus $T$ embedded in the surface. Let $\gamma$ be a simple closed curve embedded in $T$, and consider $T\smallsetminus \gamma$. A …
- 25.2k
7
votes
Generate $\mathrm{Mod}(S_g)$ by two Dehn twists
This is addressed in §3.5.2 of Farb and Margalit's Primer on Mapping Class Groups. The subgroups of mapping class groups generated by two Dehn twists $T_a,T_b$ are one of:
$\mathbb{Z}$ if the curves …
- 25.2k
2
votes
Identifying a curve on a closed surface of genus 4
As mentioned in comments, your picture is not entirely accurate. But perhaps this is what you're looking for?
(Note that, if you had chosen a different gluing pattern for your once-punctured genus-tw …
- 25.2k