# Questions tagged [mapping-class-groups]

Topology of groups of automorphisms of surfaces, and high dimensional analogues.

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### Mapping Class Group and Triangulations

I am a physicist who's getting started with Mapping Class Group for Riemann surfaces, pants decompositions and triangulations so I apologise in advance if the following is a stupid question/wrong.
I ...

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### Mapping class group of $\mathbb{S}^3$

If I recall correctly from a lecture I attended the last year we have that
$MCG(\mathbb{S^2})\simeq\frac{\mathbb{Z}}{2\mathbb{Z}}$ by Smale in the 60' and $MCG(\mathbb{S^3})\simeq\frac{\mathbb{Z}}{2\...

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### Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy

Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...

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### Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...

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### Symplectic Lefschetz fibrations in terms of factorization in symplectic mapping class group

There is a well-known theorem stating that there is a bijection between diffeomorphism classes of Lefschetz fibrations over $S^2$ whose general fiber is a closed orientable surface $\Sigma_g$ of genus ...

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### Automorphism groups of cocompact Fuchsian groups as mapping class groups

Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...

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### Measures on the space of projective measured foliations

I have questions about the action of the mapping class group $Mod(S_g)$ on the Thurston bounday $PMF$ that is given by the space of projective measured foliations.
Is this action an uniquely ergodic ?...

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### Are isotopic and conjugate homeomorphisms, conjugate by an element in $\mathrm{Homeo}_0(M)$?

An answer to this question would also answer Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
Let $M$ be a topological manifold and let $f,g$ be two orientation ...

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### Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures

Let $S=S_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x_1,\ldots , x_n\}$ be a set of $n$ distinguished points in the ...

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### framed n component link on a genus n surface determines the self-homeomorphism?

$S$ is the boundary of a genus $n$ handlebody in $S^3$. $\{m_1, m_2,..., m_n\}$ is the collection of the meridian circles of $S$; $\{l_1,l_2,...,l_n\}$ is the collection of the longitude circles on $S$...

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### Minimum dimension of faithful representation of mapping class groups?

Let $\Sigma_{g}$ be a closed orientable surface of genus $g$. Let $d_g$ denote the minimum dimension of a faithful representation of the mapping class group of $\Sigma_g$. For $g=1$, the mapping class ...

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### Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms

Let $\Sigma$ be an oriented compact surface with non-empty boundary that is not a disk or a cylinder (i.e. negative Euler characteristic). Let $\phi, \psi: \Sigma \to \Sigma$ be two orientation ...

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### Preimages of certain subgroups of $\bigwedge^{3} H $ under Johnson's homomorphism

We have Johnson's homomorphism $\tau_{2}\colon \mathcal{I}_{g,1} \to \bigwedge^{3}H$, where $\mathcal{I}_{g,1}$ - is a Torelli group, i.e. it consists of all elements of a mapping class group that ...

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### Is it true that every mapping class in $\mathrm{Mod}(\Sigma_3)$ commutes with some hyperelliptic involution?

Two questions.
First, let $\Sigma_3$ be the closed genus 3 surface and let $\rm Mod(\Sigma_3)$ be its mapping class group. Is it true that for any mapping class $g\in\rm Mod(\Sigma_3)$ there is some ...

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### Generalizations of Dehn-Nielsen-Baer for topological branched cover?

For any manifolds $M$, a homotopy class of diffeomorphism gives rise to an automorphism of $\pi_1(M)$ (up to conjugacy since we are dealing with free homotopies). Moreover, in the specific case of ...

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### generators for the handlebody group of genus two

Is the handlebody group of genus two surface generated by Dehn twists along properly embedded disks and annuli?
Are there alternative ways to describe a set of generators that are conceptually simple ...

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### Nielsen-Thurston decomposition from the product of Dehn twists

Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is
Given an element as a product of Dehn twist, is it possible ...

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### How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism? [duplicate]

How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an ...

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### Dehn twist generators for mapping class group of a genus zero surface with boundary

Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(S_{0,n})$, the mapping class group of a genus $0$ surface with $n$ boundary components, fixing the ...

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### a question about mapping class groups

According to Thurston's construction, which can be found for instance in Farb-Margalit's A Primer on Mapping Class Groups, theorem 14.1 (here is a link to the version I am using: http://www.maths.ed....

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### Lutz twist and open book decompositions

Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...

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### Length of a simple closed curve under Pseudo-Anosov maps

Let $S$ be a fixed hyperbolic surface with genus $g$ and $n$ punctures. Given any pseudo-Anosov map $f$ on $S$ (with stretch factor $\lambda$) with stable and unstable measured foliations $\mu^s$ and $...

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### Topological entropy and pseudo-Anosov dilatation for punctured surface

Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ punctures. Assume $2-2g-n<0$. Let $f$ be a pseudo-Anosov mapping class with dilatation $\lambda_f$. In the introduction (1st page) of the ...

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### Are pseudo-Anosov foliations dense?

A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of ...

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### Well definedness of square roots of separating Dehn Twists

Let $c_1,c_2,c_3,c_4,c_5$ be a five chain of circles on a genus 2 surface (i.e $i(c_k,c_{k+1})=1$ and zero otherwise). Then $(T_{c_1} T_{c_2})^6 = (T_{c_4} T_{c_5})^6 = T_c$ where $c$ is a separating ...

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### Symplectic mapping class group and the “Lagrangian sphere complex”

For a genus $g$ surface $\Sigma_g$, the mapping class group $\mathrm{Mod}(\Sigma_g)$ acts on the curve complex $\mathcal C(\Sigma_g)$: vertices being isotopy classes of essential, nonseparating, ...

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### What is known about mapping class groups of 4-manifolds?

I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+...

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### Generators for the mapping class group of pointed curves

I am looking for a suitable set of Dehn-twists generators for the mapping class group of a curve of genus $g$ with $n$ marked points (i.e. the mapping class group of $\mathcal M_{g,n}$).
For $\...

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### Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface

If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups:
$$\text{Mod}(\mathcal{R}')\longrightarrow \...

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### Reference request: Mapping class group action on homology of surface with boundary

This is a request for a reference to a proof of a result. The result is not very hard, but I'd rather cite than reprove.
I'm looking for a generalization of the following result (Farb and Margalit, ...

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### The action of the mapping class group of a punctured disk on the boundary at infinity of the universal cover

Let $\mathbb{D}\subset\mathbb{C}$ be the unit disk, and remove $n\geq 2$ of its points $P$. The resulting object will be called the punctured disk $\mathbb{D}_n$ in the following. I am interested in ...

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### Relation between point pushing pseudo-Anosov map and the minimum length

Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space.
By Birman exact sequence we get the point pushing map $Push:\pi_1(S,...

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### Translation discreteness of pseudo-Anosov elements

Consider the action of mapping class groups on the curve graphs. A pseudo-Anosov element $g$ acts by a loxodromic isometry on the hyperbolic curve graphs. The translation length of $g$ is the limit $\...

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### Mapping class group and representation of fundamental group of Riemann surfaces

Let $S$ be a Riemann surface with genus $g>0$. Let $M$ be the mapping class group of $S$. $Hom(\pi_1(S),Gl(n, \mathbb{C}))$ is the representation space of fundamental group of $S$
Question: Is ...

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### What's the meaning of the Johnson filtration in terms of configuration spaces?

This question is inspired of course by the remarkable paper of Tetsuhiro Moriyama from 2008.
Let $\Sigma$ be a genus $g \geq 3$ closed surface.
Let $\phi : \Sigma \to \Sigma$ be an orientation ...

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### Making diffeomorphism of submanifolds boring

This is probably very well known in surgery theory... I'm looking for a modern reference on the following questions (the only one I know is Browder's "Diffeomorphism of 1-connected manifolds" from ...

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### Mapping class group of certain 3-manifolds

Let $\xi : M^3 \to F$ be an orientable circle bundle over a closed orientable surface $F$ of genus $g \geq 2$. I am mostly interested to the case where the bundle $\xi$ is non-trivial. My question is ...

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### Goldman Lie algebra of a bordered surface vs. a closed surface?

How are the Goldman Lie algebra of a closed surface $\overline{S}$ and the bordered surface $S$ obtained by taking $\overline{S}$ and removing an open disc (or more generally, $n$ disjoint discs) ...

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### Finiteness properties of mapping class groups

Question: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses:
1) Does every finite generating set give us a finite ...

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### actions of orientation-preserving maps on homology group

Let $M$ be a orientable, connected, closed n-manifold with all of its homology group $H_{*}(M;Z)$ torsion-free. $f: M \rightarrow M $ is an orientation-preserving homeomorphism, and $f_{*,k}: H_{k}(M;...

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### Higher genus Cohen-Jones-Segal's conjecture?

Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism
$$(ev,\Phi):\...

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### Explicit description (=pictures!) of elements in $Mod_g[k]$?

This question is probably obvious to experts but I couldn't find the answer in the literature...
Background: Consider the mapping class group $Mod_g$ of the closed genus $g$ surface. There are many ...

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### What is the fundamental group of Kontsevich's space of stable maps?

... at least in the case where the target is a rationally connected variety.
This question is a follow-up to question
Constructing embedded families of curves with general moduli
and Jason Starr's ...

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### The fundamental group of torus bundle

A torus bundle is labeled by an element $M$ of $SL(2,\mathbb{Z})$ -- the mapping class group of a torus. How to compute the fundamental group of a torus bundle from the 2-by-2 matrix $M \in SL(2,\...

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### Does the Torelli space appear “in nature”?

What I mean by the (slightly facetious) title is:
The classical theory of algebraic curves from the 19th century was split in two in the 20th century (much like the theory of groups): the theory of ...

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### Cycles in the Chow ring of the moduli of curves coming from Hurwitz theory

Are there interesting cycles (other then the famous ones such as: Harris and Mumford's gonality divisor, the Gieseker-Petri divisor [which can be realized as the branch locus of a forgetful map from ...

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### Categorical mapping class group action

Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A_\infty$-) category which admits a faithful action
$$ MCG(\Sigma) \to Auteq(\mathcal{C})$$
by ...

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### Compact manifolds with big mapping class group

I was wondering if compact surfaces were the only compact manifolds with a "big" or "complicated" mapping class group.
Are there higher dimensional manifolds (which are not in some way reducible to ...

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### If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?

Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...

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### Geometric intersection number for product of elements of the fundamental group

Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...