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Questions tagged [mapping-class-group]

For questions related to mapping class group. The mapping class group is a certain discrete group corresponding to symmetries of the space.

The term mapping class group has a flexible usage. Most often it is used in the context of a manifold $M$. The mapping class group of $M$ is interpreted as the group of isotopy classes of automorphisms of $M$. So if $M$ is a topological manifold, the mapping class group is the group of isotopy classes of homeomorphisms of $M$. If $M$ is a smooth manifold, the mapping class group is the group of isotopy classes of diffeomorphisms of $M$.

For more, check this link.

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Why does Dehn twist about the inner circle of of an annulus $A$ act trivially on the arc complex $\mathcal{A}(A)$?

I'm reading the book A Primer of Mapping Class Group by Benson Farb and Dan Margalit. In order to show that the mapping class group of a surface $S$ is finitely presentable. They make use of an abstract simplicial complex called the arc-complex…
user135520
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Is the boundary Dehn twist central in the mapping class group of surface or not? [resolved]

Please help me find a mistake and resolve a paradox: Let $S$ be an orientable surface of genus $g\ge 2$ with $n\ge 1$ boundary components. Consider $T_\partial$, the Dehn twist about a curve parallel to some boundary component $\partial$. There are…
mathreader
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mapping class group of the real projective plane

In most literature I've read about the mapping class group, I found that many authors have stated without any explanation that any homeomorphism of a real projective 2-space to itself is isotopic to the identity. I'm guessing it is obvious but I…
leahG
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Do these curves fill the surface?

We say a collection of closed curves $\{\alpha_1,...,\alpha_k\}$ on a surface $S_{g,n}$ (a surface of genus $g$ with $n$ boundary components) fill the surface $S_{g,n}$, if $S_{g,n}-\{\alpha_1,...,\alpha_k\}$ is a collection of disks and annuli.…
braid rep
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Any orientation-preserving automorphism of the annulus is isotopic to the identity

How can I prove that an orientation-preserving self-homeomorphism of the annulus $[0,1]\times S^1$ that preserves each boundary component is isotopic to the identity?
Arnaud Maret
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