Questions tagged [mapping-class-groups]

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

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Arc complex characterization of braids with trivial closure

A braid $\beta\in B_n,$ the braid group with $n$ strands, viewed as the mapping class group $\mathrm{Mod}(\mathbb{D}_n)$ of the disk with $n$ punctures is trivial if and only if $\beta$ acts trivially ...
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Manifolds with trivial mapping class group and large $H^1$?

Are there smooth closed manifolds $M^n$ in every dimension $n \geq 3$ with trivial mapping class groups and with $H^1(M^n;\mathbb{Z}/2\mathbb{Z})$ arbitrarily large? I am under the impression that &...
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Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?

As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms. There are a relationship between the mapping class ...
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1answer
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Mapping class groups of Haken Seifert 3-manifolds (not small)

This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $M$, in particular how to systematically compute it. I want to ...
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Classification of genus 1 Lefschetz fibration over a disk with 6 singular fibers

For a genus 1 Lefschetz fibration over a sphere, there is a good classification in B. Moishezon, Complex surfaces and connected sums of complex projective planes. In this case, the number $K$ of ...
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The category of connected ribbon graph and its connected component

Let $RG$ be a category of connected ribbon graph, the morphisms are admissible epimorphism or finite composition of contraction. By a ribbon graph we mean a connected graph $\Gamma$ with fixed cyclic ...
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63 views

Do different decompositions of Dehn twists induce the same cobordisms between mapping tori?

Suppose $Y=T^2\times I/f$ is a mapping torus, where $f$ is a diffeomorphism of $T^2$. Suppose $c$ is a curve on $T^2\times \{1\}$ with surface framing and suppose $D_c$ is the positive Dehn twist ...
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Presentations of mapping class groups in dimension $3$

For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then ...
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165 views

Mapping class group of a twisted I-bundle over $RP^2$

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Homeo{Homeo}$Let $\Mod(M)=\pi_0(\Homeo(M))$ be the mapping class group of a manifold, possibly with boundary (I'm including the orientation reversing ...
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248 views

Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, ...
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Can a hyperbolic three-manifold have 𝑛 toric boundary components?

I'm wondering if I can construct a class of hyperbolic three-manifolds with $n$ toric boundaries. My idea is to take a bordered hyperbolic Riemann surface $\Sigma_{g,n}$, of genus $g$ with geodesic ...
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Braid groups and Kazhdan's property (T)

In Nica's dissertation Group actions on median spaces, we can read the following assertion: Braid groups do not contain infinite subgroups satisfying Kazhdan's property (T). This is used in order to ...
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Why is the mapping class group of a surface with nonempty boundary torsion-free?

On page 201 of Farb and Margalit's Primer on Mapping Class Groups, they explain why the mapping class group $\mathrm{Mod}(S)$ is torsion-free when $\partial S \neq \varnothing$. Here is my ...
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Subgroups of Mod(S) generated by Dehn twists depend only on intersection numbers?

$\DeclareMathOperator\Mod{Mod}$Let $S$ be a closed surface and $\Mod(S)$ be its mapping class group. It is a well known fact, proved in the Primer on Mapping class groups for example, that the ...
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1answer
256 views

What does it matter if a group has a non-elementary hyperbolic quotient?

My adviser recently shared a problem with me that seeks to establish non-elementary* hyperbolic quotients for mapping class groups. They told me that this could be useful for establishing results on ...
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1answer
257 views

Homotopy equivalence preserving all geometric intersection numbers

This question again might be silly, like the last post(deleted). Let me know I will delete it. Problem: Let $\Sigma$ be a surface without boundary and $f:\Sigma\to \Sigma$ be a proper homotopy ...
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1answer
568 views

All non-compact simply connected $2$-manifolds with boundary

There are two corresponding posts MSE and MSE by me without any answers. Problem: Let $\Sigma$ be a non-compact simply-connected $2$-dimensional manifold, with boundary. Then, up to homeomorphism $\...
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Can a path-connected domain be completely surrounded by 4 translates?

Question: Does there exist a compact path-connected set $A\subseteq\mathbb C$ such that: $A\cap(A+1)=A\cap(A+i)=\emptyset$, $A\cap(A+1+i)\neq\emptyset$, and $A\cap(A+1-i)\neq\emptyset$? Remarks: If ...
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1answer
228 views

Generalized Birman exact sequence for surfaces with boundaries

Let $S_g^n$ be a surface of genus g with n boundaries and let $Mod(S_g^n)$ be its mapping class group. We will also denote by $S_{g,m}^n$ a surface of genus g with n boundaries and m punctures. The ...
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flips on labelled fatgraphs and mapping classes

A fatgraph $G$ is a graph with a cyclic ordering of the edges at each vertex. A labelled fatgraph $(G,L)$ is a fatgraph together with a labelling $L$ of each edge. A labelled fatgraph spine $(G,L,e)$ ...
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Representation of the mapping class group in terms of flips on triangulations

$\DeclareMathOperator{\MCG}{\operatorname{MCG}}$Consider a bordered, punctured, orientable surface $S$. Associated to it there is its mapping class group $\MCG(S)$. One way to concretely think about ...
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Minimal number of (Dehn twists) generators of the mapping class group of a marked sphere

Let $\Gamma_{g,n}$ denote the mapping class group of an oriented surface of genus $g$ and with $n$ marked points. We assume that elements of $\Gamma_{g,n}$ are not allowed to permute the marked points....
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Reducible finite order mappings classes and their action on the Thurston boundary

Let $S$ denote the closed oriented surface of genus $g\geq 2$, and $\text{Mod}(S)$ be the mapping class group of $S$. Let $f\in \text{Mod}(S)$ be a finite order reducible element i.e. $f$ has a ...
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Center of symplectic derivation Lie algebra

Morita–Sakasai–Suzuki studied the graded Lie algebra $\mathfrak{h}_{g,1}$ of symplectic derivations, as well as variations $\mathfrak{h}_{g,\ast}$ and $\mathfrak{h}_g$. This is the Lie algebra of ...
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1answer
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Pants decomposition and moduli space of $\Sigma_g$ for $g>1$

By Section 8.3.1 of the book: A primer on mapping class groups by Farb and Margalit, a pair of pants is a compact surface of genus 0 with three boundary components. Let $S$ be a compact surface with $\...
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Representing relative homology classes orientable surfaces with boundary

Let $S$ be compact oriented surface without boundary. Then it is a classical result that a primitive class $\gamma \in H_1(S; \mathbb{Z})$ is always represented by a simple closed curve. It implies ...
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119 views

Are all mapping classes also Dehn twists?

Let $X$ be a Riemann surface and $\Gamma$ its (pure) Mapping Class Group, then $\Gamma$ is generated by Dehn twists along simple closed curves. Is \emph{any} element of the mapping class group also a ...
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Borel conjecture and arbitrary surface

Before starting my question I want to write something that I already know. Borel Conjecture: Any homotopy equivalence between two closed aspherical manifolds is homotopic to a homeomorphism. Now, my ...
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1answer
116 views

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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1answer
372 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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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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1answer
264 views

Mapping class group of torus, why is $(ST)^3=S^2$?

In the context of topological quantum field theories, I am interested in the mapping class group of a torus. Here I can consider the torus as a square with identified edges and also decorated with ...
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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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1answer
277 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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701 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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1answer
104 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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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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642 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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Abelianization of mapping class groups $\Gamma_{g,n}$

Let $S_{g,n}$ be a Riemann surface of genus $g$, with $n$ points removed. The mapping class group of $S_{g,n}$ is denoted by $\Gamma_{g,n}$. Is there a reference where the abelianization of $\Gamma_{...
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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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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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149 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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1answer
173 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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1answer
369 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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1answer
282 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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210 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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