# Questions tagged [mapping-class-groups]

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

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### Spines of Teichmuller space of a non-orientable surface

Let $S_{g,n}$ be the orientable surface of genus $g$ and $n$ punctures. Denote $\Gamma_{g,n}$ be the mapping class group of $S_{g,n}$ and $\mathcal T_{g,n}$ the Teichmüller space of $S_{g,n}$.
In http:...

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### Embedded surfaces in pseudo-Anosov mapping tori

Given a genus $g$ (where $g \geq 2$) closed surface $S$, and a pseudo-Anosov map $\varphi: S \to S$, one can construct the mapping torus, which turns out to be a hyperbolic $3$-manifold, which we'll ...

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346 views

### The largest group acting on a non-orientable surface of genus 5

Let $N_5$ denote the non-orientable surface of genus 5.
In Conder's database https://www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByGenus.txt we can see that the biggest finite group $...

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### Mapping class groups of $T^2 \times [0, 1]$ and $T^2 \times S^1$

Are the mapping class groups of $T^2 \times [0, 1$] and $T^2 \times S^1$ explicitly known?

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### Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...

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101 views

### Purely analytic proof of the Nielsen-Thurston classification theorem

I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory
and Applications to ...

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### A version of Hurwitz' theorem in terms of Euler characteristic

Page 203 of Farb and Margalit's Primer on Mapping Class Groups contains the result:
Let $g ≥ 2$. The order of any finite subgroup of $MCG(S_g)$ is at most $84(g − 1)$.
I've been told by my ...

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265 views

### Permuting $n$ points in a $2$-manifold

Given $n$ points on a connected $2$-manifold $M$, I'd like to consider the homotopy classes of paths that "permute" these points.
Edit (Clarifying what I mean by this):
Given a set of $n$ ...

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589 views

### Conjugacy classes of the mapping class group

I am not sure if this is a well known problem, but I was not able to find anything online that answered my question.
Is it known how to tell whether two elements of the mapping class group of a ...

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97 views

### Arcs and elements of the mapping class group

Is there any way to represent every element of the mapping class group of a surface as an arc on that surface?

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### Is the mapping class group of $\Bbb{CP}^n$ known?

In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an ...

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125 views

### J. F. Adams Proof of Cellular Approximation Theorem

In Ronald Brown's discussion of the proof of The Cellular Approximation Theorem in Topology and Groupoids Sec. 7.6 he writes that, "the elegant formulation of the proof is due to J. F. Adams." Does ...

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599 views

### Are these local systems on $\mathscr{M}_{g,1}$ motivic?

Let $(\Sigma_g, x)$ be a pointed topological surface of genus $g$, and let $MCG(g,1)$ be the mapping class group of this pointed surface. Then $MCG(g,1)$ has a natural action on $\pi_1(\Sigma_g, x)$ $$...

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125 views

### Is there an equivariant simplicial deformation retract of Teichmüller space?

Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...

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114 views

### Are these two arguments incompatible?

I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong.
First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...

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137 views

### Symmetries of MCG in terms of Humphries generators?

The Riemann-Hurwitz formula gives $84(g-1)$ as the upper bound to the order of a finite group acting faithfully on a closed genus g surface. Famously the bound is realized when $g=3$ by a simple ...

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142 views

### Group completion of $E_k$-algebras

Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is ...

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344 views

### Is there a Morita cocycle for the mapping class group Mod(g,n) when n > 1?

Write Mod(g,n) for the mapping class group of a genus-$g$ surface $\Sigma$ with $n$ boundary components. When $n=0,1$ we define the Torelli group $T$ to be the subgroup of Mod(g,n) which acts ...

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150 views

### Does WRT invariant detect hyperelliptic involution on the genus 2 surface?

The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\...

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199 views

### What does it mean for two natural numbers to be *approximately equal*?

This is related to this other question of mine about a paper of Colin and Honda.
I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the ...

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92 views

### Mapping class group orbits of principal bundles

Suppose $M$ is a manifold (I would be happy with low-dimensional examples like surfaces, but let me ask more generally).
Then for any discrete group $G$ (again, I would be happy with a finite group) ...

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391 views

### On a corollary of a paper by Colin and Honda

The question is about the last sentence of the last corollary of Stabilizing the monodromy of an open book decomposition by Vicent Colin and Ko Honda. This question is also related to this other ...

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327 views

### Orbit of an irreducible representation of a surface group under that action of the mapping class group

Let $F$ be a closed oriented surface of negative Euler characteristic. Let $X^i(F)$ be the subset of the $SL_2\mathbb{C}$-character variety of the fundamental group of $F$ corresponding to irreducible ...

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### Order of separating Dehn twists in the image of Johnson homomorphisms

Let $S$ be a closed surface, $\Gamma=\pi_1 S$ and $\Gamma_i$ be the lower central series defined by $\Gamma_0=\Gamma$ and $\Gamma_i=[\Gamma,\Gamma_{i-1}].$
The Johnson filtration $\lbrace \mathcal{K}...

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### Stability for mapping class groups, spaces of sections, and polynomial coefficient systems

Let $X$ be a simply connected space. Cohen and Madsen https://arxiv.org/abs/math/0601750 proved that the functor sending a surface with boundary M to $H_i(Map(M,X))$ has polynomial degree $\leq i$. ...

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143 views

### Stabilizing an open book with Anosov piece

It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...

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87 views

### Deck transformations of Teichmuller space as a universal cover of Torelli space

I'm reading the article by Geoffrey Mess The Torelli groups for genus 2 and 3 surfaces (pp. 785 - 786), and I'm trying to understand the part that concerns genus 3. We've got a map $UT(S) \to T_3/\...

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### Mapping classes as Lefschetz fibrations over surfaces with positive genus

Let $\Sigma_{g,r}$ be the surface of genus $g$ and $r$ boundary components. It is known that, from a positive factorization of a mapping class $\phi$ in the mapping class group $MCG(\Sigma_{g,r}, \...

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219 views

### Generators of the mapping class group for surfaces with punctures and boundaries

Let $\Gamma_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.
It is clear that in general, ...

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116 views

### 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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327 views

### 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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456 views

### 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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363 views

### 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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152 views

### 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 ...