Questions tagged [mapping-class-group]

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Function class consisting of gradients of real-valued convex functions

Denote $\mathcal F$ as the function class consisting of gradients of all real-valued convex functions in $\mathbb R^d$, that is, $\mathcal F = \{ \nabla \phi ~|~ \phi: \mathbb R^d \to \mathbb R \text{ ...
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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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A Primer on mapping class groups: Question on Proof of Lemma 3.16

Dear fellow Mathematicians, I have a question concerning the proof of the - quote - "simple" lemma 3.16 in the "Primer on Mapping Class Groups" by Farb and Margalit. At first ...
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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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Redundacy in gluing map of a Heegaard Splitting

I am reading about the handlebody group which is the subgroup of all those mapping classes that extend to the handlebody from here. On page 12 the author defines an equivalence relation on mapping ...
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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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Pure mapping class group of the sphere with marked points

Consider Pure Mapping Class Group of the sphere $\mathbb{S}^2$ with finitely many marked points $P$, i.e. $$ \mathrm{PMCG}(\mathbb{S}^2, P) = \{\varphi \in \mathrm{Homeo}^+(\mathbb{S}^2) : \varphi|_{P}...
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prove the quotient $R^{\prime}=$ $R /\langle\phi\rangle$ is an annulus.

Let $\alpha$ and $\beta$ be two essential simple closed curves in a hyperbolic surface $S$.Choose lifts $\widetilde{\alpha}$ and $\widetilde{\beta}$ of $\alpha$ and $\beta$ that have the same ...
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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 ...
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when two curves are **transverse**?

in the "A Primer on Mapping Class Groups by Benson Farb and Dan Margalit" we define intersection number as follow : Let $\alpha$ and $\beta$ be a pair of transverse, oriented, simple closed ...
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what is level curves? what is non-critical level curves?

in this paper we have non-critical level curves. "On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps at Besançon " We have :...
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what is definition of generic function?

what is definition of generic function in following paper ? i need a reference for definition generic function . "A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed ...
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why Alexander method gives us a finite combinatorial problem?

The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups" For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...
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The Mapping Class Group of the Annulus

I was reading "A primer on Mapping Class Groups" by B. Farb and D. Margalit and I'm stuck at a point in the proof of Mod$(A)\cong \mathbb{Z}$ where $A$ is the annulus. Proposition 2.4 $\...
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Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

Let $B_{k}(S_{g}),$ $MCG(S_{g};k)$ and $MCG(S_{g}))$ are Braid group, Mapping class group (relative to $k$) and Mapping class group of orientable surface $S_{g}$, respectively. For $g\geq3,$ we have ...
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First examples of mapping classes.

In the page 46 of the book A Primer on Mapping Class Groups by Farb and Margalit we have: As a first example of a nontrivial element of $\operatorname{Mod}\left(S_{g}\right)$, one can take the order ...
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Why does the Lehmer's conjecture imply the Short Geodesic Conjecture?

I need some translation help. In this article (https://homeweb.unifr.ch/kellerha/pub/IML-2013summer4-01.pdf), on page 15 of the pdf, it says that the "short geodesic conjecture" is a ...
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Different definitions for the Teichmüller space of puctured spheres

My question is about understanding of equivalence of two different definition for the Teichmüller space of hyperbolic surfaces $\mathbb{S}^2 \setminus P$, where $|P| \geq 3$. The first definition for ...
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Define the mapping that models the number of players that are going to continue or stop a game

We have a multistage game where it is played by $I<\infty$ players. This game is played for $T$ rounds. If we suppose that $K$ players where $K<I$ are not going to continue playing in the game ...
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Mapping Class Group of the Real Plane

Claim: The group of orientation-preserving diffeomorphisms of the plane modulo isotopy is trivial (that is, any such diffeomorphism is isotopic to the identity). Proof: If we have such diffeomorphism $...
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Prove the conditions 1 and 1' is equivalent.

Notation:Let $F_2$ be the free group on generators $x,y$. Definition. Let $\Sigma_{g, n}^{m}$ denote a topological surface with genus $g \geqq 0, n \geqq 0$ punctures, and $m \geqq 0$ boundary ...
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equivalent definition of Grothendieck-Teichmüller group

Notation:Let $F_2$ be the free group on generators $x,y$. Denote the profinite completion of group $G$ by $\hat{G} = \underset{N}{\varprojlim} G/N$. For any group homomorphism $$ \begin{aligned} &\...
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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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3 votes
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Configuration spaces to moduli spaces

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}; \...
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Generators of the Torelli group for surfaces with punctures

Are there any references for the generators of the Torelli group of surfaces with punctures? Johnson only assumes that the surface has boundary components. See Lemma 10 and Lemma 11 here (page 253). ...
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"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 ...
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Nielsen Realization for punctured surfaces

I am trying to find a reference to a proof of the Nielsen Realization theorem for surfaces with punctures (of negative Euler characteristic). In Farb-Margalit's "A Primer on Mapping Class Groups&...
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3 votes
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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 ...
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2 votes
1 answer
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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 ...
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2 votes
2 answers
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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 ...
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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 ...
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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. An inner automorphism is one determined by conjugation by some element $g\in ...
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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 ...
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Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group?

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. ...
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What does "subgroup of $F_n$ consisting of all words of exponent sum $k$" mean?

I am currently reading the following paper by Birman and Hilden, https://www.jstor.org/stable/1970830?casa_token=7y0vMR6x_lsAAAAA:...
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Homotopy coincides with isotopy in dimension two

I am currently reading Mapping Class Groups from the lecture note https://massuyea.perso.math.cnrs.fr/notes/formality.pdf. Now, on page 22, I got this highlighted line. Is there any reference to this ...
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Is mapping class group topological property?

Is mapping class group topological property ? This means that if $X$ and $Y$ are homeomorphic topological spaces then $MCG(X) \cong MCG(Y) $. is this true ? If no what is counterexample? When ...
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center of a finite index subgroup of a mapping class group is trivial?

(Sorry for the long read, you may skip to the very end, to see the question there.) I am reading an article and trying to understand the proof, but cannot work out a crucial detail. Here's the ...
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1 vote
2 answers
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About mapping class group.

What is best book for self learning mapping class group? I read "A Primer on Mapping Class Groups" By Benson Farb, Dan Margalit. Is there a topological space $X$ where we don't know $\...
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Conjugation Classes inside the Orbit of close curve in Hyperbolic Surface under $Mod(S)$

I've heard the statement: Let $S$ be a hyperbolic surface with boundary and $\left[ \alpha \right] \in \pi_1 \left( S \right)$ a non-trivial element that is not conjecture to boundary element, then ...
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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 ...
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Surface homeomorphism transitively permutes the boundary curves

Let $h: S\to S$ be a surface homeomorphism. What does it mean to say that $h$ transitively permutes components of $\partial S$, and where does this terminology come from? Is this somehow related to a ...
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Mapping class group of a torus

Seems like i have some false understanding that need to be fixed. I have check varies sources saying that the mapping class group of a torus is isomorphic to $GL(2,\mathbb Z)$, however my false ...
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The monodromy representation $\pi_1(E)\to Mod(F)$

Given a fiber bundle $F\to X\to E$, with $F,E$ are Riemann surfaces. I know the monodromy gives a permutation of the fiber. But how to see we have the monodromy representatio: $\pi_1(E)\to Mod(F)$? ...
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Mapping class group of $S^1 \times S^1 \times I$

I am interested in the computation of the mapping class group of the manifold $M=S^1 \times S^1 \times I$. One can visualize $M$ as a "torus cross $I$", or as an "annulus cross $S^1$". $M$ is ...
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The monodromy of a Lefschetz fibration as right-handed Dehn twists

A fact that one can find in many books is that the monodromy of a Lefschetz fibration is the product of right-handed/positive Dehn twists (one for each vanishing cycle). The only proof I could find ...
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4 votes
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Mapping Class Group of Pants with a Hole

It is known that the mapping class group of the torus $\mathbb{T}^2$ is $\text{Mod}(\mathbb{T}^2) \cong \text{SL}_2(\mathbb{Z})$. We also know that for a pair of pants $P$ (a sphere with three ...
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