# 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.

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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 $A \ast B$ ...
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### Isometric embedding of Teichmüller spaces

Let $S_{g,k}$ be a genus $g$ surface with $k$ punctures. Let $\mathrm{Mod}(S)$ be the extended mapping class group of a surface, defined as the isotopy classes of the self-homeomorphisms of surface $S$...
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### Convert continuous variables to discrete and back? But with reduced continuous input dimension.

I'm trying to think of a way to map continuous variables (can be a number or a vector but if it's a vector preferably smaller than 3 in length) to discrete ones and back. The only way I can think of ...
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### What is the mapping class group of an annulus with two marked points?

Let $A$ be an annulus, and consider ${\rm Mod}(A, \{x,y\})$, the group of connected components of homeomorphisms of $A$ that fix its two boundary components pointwise, and preserve the set $\{x, y\}$. ...
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### What is the Jacobian of det???

In some questions of submanifold, I want to use determinant as a $C^{\infty}$Map, and show that, for example, $det^{-1}\{-1\}$ is a submanifold of $M_{2n}(\mathbb{R})$(this may be not true). However, ...
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### what is definition of homotopy of homeomorphisms and isotopy of homeomorphisms?

I read "A Primer on Mapping Class Groups" By Benson Farb, Dan Margalit. I can't find definition of homotopy of homeomorphisms and isotopy of homeomorphisms in this book but in this book we ...
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### what is mean of invariant?

in the A Primer on Mapping Class Groups we have : Geometric intersection number is a useful invariant but, as we will see, it is more difficult to compute than the algebraic intersection number. what ...
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### Primer on Mapping Class Groups Chapter 1: Arbitrarily short loops around Punctures

Dear fellow Mathematicians, This is the first question I ask in this forum, so please excuse any formal mistakes, which I am, of course, trying to avoid. I am currently briefly revisiting hyperbolic ...
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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://...
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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 (...
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### what is mean of pointwise in this definition?

Definition. Let $\Sigma_{g, n}^{m}$ denote a topological surface with genus $g \geqq 0, n \geqq 0$ punctures, and $m \geqq 0$ boundary components, i.e. such that filling in the $n$ punctures gives a ...
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### understanding the definition of subgroup of the Grothendieck-Teichmuller group

Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...
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In Segal's paper on Mapping Configuration spaces to moduli spaces, I'm not understanding what the map $\Phi$ is, explicitly. Also in section 2, he goes on to say $M_{g,2} \simeq BHomeo^{+}(F_{g,2}; \... 2 votes 1 answer 92 views ### "Simply connected" like condition for isotopy I'm looking for literature regarding a locally compact Hausdorff topological space$X$where any two embeddings$f,g:[0,1]\to X$with fixed boundary points$f(0)=g(0),f(1)=g(1)$are isotopic. If we ... 3 votes 0 answers 90 views ### What exactly does the "change of coordinates principle" say? I'm currently reading the "Primer on mapping class groups" by Farb-Magalit. Something that almost always comes up in the proofs is the "change of coordinates principle". The first ... 2 votes 1 answer 71 views ### What is meant by "side of a curve"? I'm currently reading "A primer on mapping class groups" by Farb-Magalit. A notion that often turns up is that of a "side of a curve". For example in the proof of Proposition 3.2 ... 2 votes 2 answers 168 views ### Maximal rank abelian subgroups of mapping class group I'm currently getting involved with mapping class groups of surfaces ($MCG$). So for a genus g surface with b boundary components$S_{g,b}$it is well-known that$MCG(S_{g,b})$contains not only a lot ... 1 vote 1 answer 322 views ### Every homotopy equivalence of closed surface is homotopic to a homeomorphism I am reading the first proof of$\text{Theorem 8.9:}$If$g ≥ 2$, then any homotopy equivalence$S_g → S_g$is homotopic to a homeomorphism from the book A Primer on Mapping Class Groups. The proof ... 11 votes 5 answers 606 views ### Progress on a conjecture of Burnside... Given a group$G $, the set of automorphisms of$G $also forms a group,$\rm {Aut}(G) $,with composition as the operation. An inner automorphism is one determined by conjugation by some element$g\in ... 142 views

### Doubt in the Proof on Dehn-Nielsen-Baer theorem

I'm reading the proof of Dehn-Nielsen-Baer theorem given in the book 'A primer on mapping class groups' by Margalit and Farb. I'm having trouble understanding the following line from the proof of ...
TLDR; Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group? Let $S$ be a compact surface of negative Euler characteristic and let $f :S\to S$ be a homeomorphism. ...