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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.

139 questions
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General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid group $B_n$, namely as the mapping class group of the…
anon
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Progress on a conjecture of Burnside...

Given a group $G $, the set of automorphisms of $G $ also forms a group, $\rm {Aut}(G) $,with composition as the operation (recall that an automorphism of a group is a bijective endomorphism) . An inner automorphism is one determined by conjugation…
user403337
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Contradiction in Computation of Homology Groups of the Mapping Class Group of a Surface?

One of the two main results of a paper by Nathalie Wahl on homological stability of the mapping class group of a surface is the following: Theorem 1.2 The map $H_*(\delta_g) : H_*(\Gamma_{g,1};\mathbb{Z}) \to H_*(\Gamma_{g,0};\mathbb{Z})$ is…
jasnee
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Diffeomorphisms of Spheres and Real Projective Spaces

In the comments to Mapping torus of orientation reversing isometry of the sphere it was stated that there are only two $ S^n $ bundles over $ S^1 $ up to diffeomorphism. The conversation related to this led me to wonder several things: Is every $…
Ian Gershon Teixeira
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Seeking an intuitive explanation of the Mapping Class Group

For a surface $S$ the mapping class group $MCG(S)$ of $S$ is defined as the group of isotopy classes of orientation preserving diffeomorphisms of $S$: $$MCG(S)=Diff^+(S)/Diff_0(S).$$ I understand this definition as well as all of its component…
Nathan Lopez
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Why is the Dehn Twist well-defined?

I am reading the third chapter on "Dehn Twists" from "A Primer on Mapping class group" by Benson Farb and Dan Margalit. I am stuck on the definition of Dehn Twist. The book defines the Dehn Twist as follows. First, define the twist map on the…
Dwaipayan Sharma
  • 604
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Mapping class group of surfaces, free products, and trees

Let $ \Sigma $ be a surface, possibly with boundary. Let $ MCG(\Sigma) $ denote the mapping class group. Is it true that $ MCG(\Sigma) $ has a quotient which is a nontrivial free product $ A \ast B $ if and only if that quotient is $ A \ast B \cong…
Ian Gershon Teixeira
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Mapping Class Group

$\newcommand{\MCG}{\mbox{MCG}}$Let $\alpha$, $\beta$ be non-isotopic, non-separating curves on a surface $S$ (meaning that "cutting along " them will not disconnect the surface). How do we show that the Dehn twists $D_{\alpha} , D_{\beta}$ are…
Name
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Diffeotopy group, Mapping Class group, Isometry group

There are several closely related concepts on the symmetries or symmetry groups of the space. My apology, but some vague imprecise definitions may be as: Mapping class group (MCG) is an important algebraic invariant of a topological space. Briefly,…
wonderich
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Motivations for mapping class group representations

I am new to mapping class groups for surfaces and representation theory. I would like to know why people care about representations of mapping class groups. I think in general representation theory reduces a problem into a problem in linear algebra…
user65175
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Showing that Mg, the Mapping Class Group of the 1-Torus, is $SL(2,\mathbb Z)$

All: I am trying to figure out the mapping class groupof the torus ; more accurately, I am trying to show that it is equal to $SL(2,\mathbb Z)$. The method: every homeomorphism h: $ T^2 \rightarrow T^2 $ gives rise to, aka, induces, an isomorphism…
gary
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Why is the center of the Mapping class group of a genus $g \geq 3$ surface trivial

I am studying about the center of the mapping class group of a genus $g$ surface. I am having some difficulties in understanding the proof for the triviality of the group $Z(Mod(S_{g}))$. Here is the proof: Proof. Any central element $f$ of…
Dwaipayan Sharma
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The analogy between surfaces and vector space

When I am reading the first chapter of "A Primer on Mapping Class Groups" by Farb and Margalit, there is a beautiful analogy between surfaces and vector spaces. I have interpreted as the following and please correct me if anything is…
quuuuuin
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Mapping Torus of Klein bottle

The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold. The mapping class group of the Klein bottle $ K $ is the Klein four group $ C_2 \times C_2 $. See proposition 20 of https://arxiv.org/abs/1410.1123 There are exactly four…
Ian Gershon Teixeira
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Is every Nil manifold a nilmanifold?

First of all this is an absolutely superlative answer it almost brought me to tears: https://math.stackexchange.com/a/3791368/758507 I am fairly certain that all compact 3 dimensional Nil manifolds (manifolds admitting Nil geometry) are nilmanifolds…
Ian Gershon Teixeira
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