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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A "length function" on measured lamination
I am a little confused about Corollary 4.13 in paper of Leininger and Aramayona. Let's keep everything simple that we denote by $S,\tau(S),\mathcal{S}(S),ML(S)$ closed orientable surface of genus $g\...
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How can I map the fractional part of all real numbers onto a circle of unit circumference? [closed]
The example and solution below are from a real analysis textbook. The example and solution are as follows:
"Suppose we assign all real numbers with the same fractional
part to the same class. ...
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Basic question on Dehn twists
Let $\Sigma$ be a topological surface (compact and orientable, if necessary). Let $C$ be a subspace of $\Sigma$ homeomorphic to the annulus $A := [0,1] \times S^1$, together with a homeomorphism $h: A ...
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The action of Mod$(T)$ on Teich($T$)
This is Proposition 7.1.5 in Martelli's book which I am getting confused.
The action of MCG(T) on $\tau(T)$ is the following action of $SL_2(\mathbb{Z})$ on $\mathbb{H}^2$ as Mobius transformations: $...
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Relation of the mapping class group and the symplectic group.
Generally, I am currently investigating automorphisms of Riemann surfaces and their non-trivial action on the first homology.
I wonder whether one can make explicit statements on the relationship ...
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Isotopic diffeomorphisms of surface are diffeotopic?
Due to Epstain, if two homeomorphisms $f_0, f_1$ of a closed two-dimensional manifold $S$ are homotopic then they are isotopic. I've heard, that if $f_0, f_1$ are diffeomorphisms then the isotopy can ...
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Relationship between braid group $ B_n $ and mapping class group of punctured spheres
The braid group on $ n $ strands, $ B_n $, is the mapping class group of the disk with $ n $ punctures.
Let $ \Sigma_{g,n} $ denote an oriented surface of genus $ g $ with $ n $ punctures. Let $ M_{g,...
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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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Relationship between braid group and $ MCG(\Sigma_g) $
Let $ B_n $ be the braid group on $ n $ strands. Let $ MCG(\Sigma_g) $ be the mapping class group of a surface of genus $ g $.
Is there a relationship between $ B_{2g+1} $ and $ MCG(\Sigma_g) $?
For $ ...
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Is the mapping class group of a surface virtually torsion-free?
Question:
Is the mapping class group of a surface virtually torsion-free?
Context: The mapping class group of the $ n $ punctured disk is the braid group $ B_n $. The braid group is torsion-free and ...
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Example of periodic mapping class expressed by Dehn twists
I know the mapping class group is finitely generated by some Dehn-twists. But is there an easy example to construct a periodic mapping class by hands from some Dehn-twists?
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Sequence of simple closed geodesics converging to a lamination
This question is from here
at the time around 44'.
I wonder why would a sequence of simple closed geodesics converging to a set of disjoint curves (lamination). Intuitively, wouldn't a sequence of ...
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Pure mapping classes
I am a new guy in mapping class groups and I am very confused about pure mapping classes tonight. From the Primer by Farb and Margalit, I learned the definition of pure mapping class groups PMod$(S_{g,...
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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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Half-twist (Dehn twist) on braids represented as elements of the mapping class group on an n-punctures disk
Let $\mathbb{D}_n$ denote a disk with $n$ marked points or punctures. The mapping class group of $\mathbb{D}_n$ is isomorphic to the braid group $B_n$.
Elements of the mapping class group of $\mathbb{...
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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 ...
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Homotopy classes of simple closed curves in punctured torus are in correspondence with those in torus
I am reading the book "A Primer on Mapping class groups" by Farb and Margalit and I have two questions about homotopy classes of simple closed curves in $S_{1,1}$:
Why is it true that ...
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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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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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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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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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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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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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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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"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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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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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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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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