Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [mapping-class-group]

The tag has no usage guidance.

0
votes
1answer
25 views

Fundamental group of group of homeomorphism of a compact surface

I'm reading "A Primer on Mapping Class Group", and there is something I don't understand in the proof of Theorem 4.6. Define $\mathrm{Homeo}^+(S)$ to be the group of orientation-preserving ...
4
votes
0answers
36 views

Diffeotopy group, Mapping Class group, Isometry group

There are several closely related concepts on the symmetries or symmetry groups of the space. My apology, but some vague imprecise definitions may be as: Mapping class group (MCG) is an important ...
3
votes
1answer
53 views

Extended mapping class group of $S^p \times S^q$

If I understand correctly, (1) the extended mapping class group of $S^2 \times S^1$ is $$ \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2, $$ how do I understand the third generator that ...
1
vote
1answer
18 views

$\mathrm{LMod}(X)$ has finite index in $\mathrm{Mod}(X)$

I am reading about the Birman-Hilden conjecture, specifically the thesis of Rebbeca Winarski and there is some basic point I am not getting. We consider a covering space $p: Y \rightarrow X$ between ...
1
vote
1answer
45 views

What maps descend to homeomorphisms

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, where the restriction of $M$ on $...
0
votes
1answer
50 views

What algorithm to use to find the non-linear mapping function between 2D shapes generated from biosignals attractors?

I have two biosignals recording the same phenomenon with different methods. Target signal is a reference signal and I would like to find some non-linear mapping from the original signal to the ...
2
votes
1answer
43 views

Is this topological transformation group locally path connected?

A surface is an oriented connected sum of $g\geq 0$ tori, with $b \geq 0$ open disks removed, and $n \geq 0$ punctures in its interior. Let Aut$^+(S,\partial S)$ denote the group (under composition) ...
0
votes
0answers
30 views

Mapping $\Re(z) > 1$ Across The Complex Plane

Question: Map $\Re(z) > 1$ under $f(z) = z^2 + 2z + 1$ What I've Tried: Using algebraic manipulation, I've gotten the equations: $(w)^2 - 1 = z$ and by extension, $(u + iv)^{1/2} -1 = x + iy$ (...
3
votes
2answers
113 views

Homeomorphisms of the 2-sphere $S^2$ fixing a set of points.

I was reading the book by Benson Farb and Dan Margalit titled A Primer on Mapping Class Groups. In chapter 2, Proposition 2.3, it is given that The action given by $$\text{Mod}(S_{0,3}) \...
2
votes
1answer
46 views

Commutativity in Mapping Class Groups

I am trying to understand the behavior of finite order mapping classes for surfaces of genus g>=2. After fiddling for a while I started to think that no finite order mapping class commutes with any ...
2
votes
1answer
63 views

spin mapping class group of circles

The MCG of the circle is $\mathbb{Z}/2$, generated by an orientation-reversing diffeomorphism. The circle has two spin structures, a periodic one and an anti-periodic one. Each has a nontrivial ...
0
votes
0answers
57 views

Dehn Surgery Presentation of the Figure Eight Knot Complement

If $K$ is a figure eight knot how can I realize $S^3-K$ as a Dehn filling on a genus $g$ handle-body? I had the simplistic thought that a genus 5 handle-body and a (5,1) Dehn filling would do the ...
2
votes
1answer
54 views

Is there a solvable subgroup with finite index and finite type in the mapping class group of a surface?

I want to find a subgroup $H$ of the (orientation preserving) mapping class group $G=MCG(g,n)$ of a surface with genus $g$ and $n$ boundary components that satisfies the following properties: $[G:H]&...
3
votes
1answer
131 views

Mapping class group of $S^p \times S^q$

If I understand correctly, (1) the mapping class group of $S^2 \times S^1$ is $$ \mathbb{Z}_2 \times \mathbb{Z}_2, $$ how do I understand these two generators? (2) What is the mapping class ...
1
vote
1answer
74 views

Mapping Class Group acts properly discontinuous; Alexander method

Let $S$ be a closed surface of genus $g$. The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups" (p.59) (roughly) states that if $c_1,c_2$ are two filling curves in minimal ...
0
votes
1answer
36 views

Any orientation-preserving automorphism of the annulus is isotopic to the identity

How can I prove that an orientation-preserving self-homeomorphism of the annulus $[0,1]\times S^1$ that preserves each boundary component is isotopic to the identity?
3
votes
1answer
45 views

Fixed points of finite order mapping classes

I am currently working myself through the proof that every finite order element of the mapping class group $\mathrm{MCG}(S_g)=\mathrm{Hom}^+(S_g) / \mathrm{Hom}_0(S_g)$ of a closed, hyperbolic, genus $...
0
votes
2answers
46 views

Given the transformation $T:\Bbb R^5 \to\Bbb R^2$ where $T(x) = Ax$, how many rows and columns does matrix $A$ have? [closed]

Given the transformation $T:\Bbb R^5 \to\Bbb R^2$ where $T(x) = Ax$, how many rows and columns does matrix $A$ have? Please tell me the procedure to solve this question.
0
votes
1answer
83 views

generators for the mapping class group of a neighborhood of curves?

Let $S$ be a compact, connected, orientable surface, let $a_1,...,a_k \subset S$ be simple closed curves and let $N = N(a_1 \cup \cdots \cup a_k)$ be a regular neighborhood of $a_1 \cup \cdots \cup ...
2
votes
1answer
127 views

Dehn twist generators for mapping class group of a punctured disc

Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(D_n,\partial D_n)$, mapping class group of a $n$-punctured disc? (for the problem I am working on, $n=3,...
1
vote
1answer
139 views

mapping class group of 4-punctured sphere vs. that of the torus

I am trying to figure out how to obtain generators for the mapping class group of a $4$-punctured sphere and I ran across this discussion in Farb and Margalit's A primer on Mapping Class Groups, which ...
1
vote
1answer
267 views

mapping class group of a 4-punctured sphere

I am reading Farb and Margalit's A primer on Mapping Class groups which can be found in the following link: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf. My question is regarding the ...
1
vote
0answers
78 views

Mapping class group of a 4-punctured disc

We want to look at the Dehn twists $t_a$, $t_b$ and $t_c$ about the curves $a$, $b$ and $c$ in $D_4$ (a $4$-punctured disc). Are either of the following statements true in the mapping class group of $...
0
votes
0answers
35 views

abut curves on a surface

Suppose we have two simple closed curves $\alpha$ and $\beta$ on an $n$-times punctured disc $D_n$ which bound the same boundary components. Is it true that there is a $\phi\in MCG(D_n)$(mapping ...
2
votes
0answers
108 views

Existence of mapping class group subgroups isomorphic to the whole group?

For a surface $S$, we denote the mapping class group of $S$ as $\operatorname{Map}(S)$. Question: Are there surfaces $S$ such that there exists some subgroup $H<\operatorname{Map}(S)$ such that ...
0
votes
1answer
43 views

Mapping Class Group of Disc Proof

I am reading Farb's book on Mapping Class Groups. For the proof of disc's mapping class group, I have a question. The fact that $Mod(D^2)$ is trivial (for Mod, the definition is orientation preserving ...
1
vote
1answer
38 views

Do these curves fill the surface?

We say a collection of closed curves $\{\alpha_1,...,\alpha_k\}$ on a surface $S_{g,n}$ (a surface of genus $g$ with $n$ boundary components) fill the surface $S_{g,n}$, if $S_{g,n}-\{\alpha_1,...,\...
2
votes
1answer
66 views

Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface

Assume that $\mathcal{R}$ is a hyperbolic surface with $m$ geodesic boundary components and $n$ punctures. If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is ...
1
vote
1answer
77 views

Lift of self-diffeomorphism of punctured disk is quasi-isometry

I'm considering the unit disk $\mathbb{D}\subset\mathbb{C}$, on which I put a hyperbolic metric such that the boundary $S^1$ is a geodesic, and $n$ equally spaced points $P$ on the x-axis are 'pushed ...
0
votes
1answer
117 views

Self-homeomorphism of punctured unit disk which induces identity-automorphism of fundamental group

The mapping class group of the closed unit disk $D_n\subset\mathbb{C}$ with $n$ equally spaced points $P$ on the x-axis (0,1) removed is defined as the set of isotopy classes of self-homeomorphisms $...
2
votes
1answer
188 views

Why does Dehn twist about the inner circle of of an annulus $A$ act trivially on the arc complex $\mathcal{A}(A)$?

I'm reading the book A Primer of Mapping Class Group by Benson Farb and Dan Margalit. In order to show that the mapping class group of a surface $S$ is finitely presentable. They make use of an ...
0
votes
0answers
136 views

Why orientation-preserving homeomorphisms in definition of Mapping Class Group

I am currently studying mapping class groups, and would like to know why one uses orientation-preserving homeomorphisms in their definition. I read in 'A Primer on Mapping Class Groups' by Benson and ...
2
votes
0answers
345 views

General definition of orientation-preserving (continuous) map of surfaces

Everybody seems to just use these maps but there is never a formal definition given. I looked at this question: Orientation preserving homeomorphisms but no answer is given and I can't make sense of ...
2
votes
1answer
160 views

Change of coordinates principle for mapping class groups

So I am reading Farb and Margalit's text, and his statement for the change of coordinates principle doesn't seem complete. Namely, he states that there is an orientation preserving homeomorphism of a ...
3
votes
0answers
64 views

Johnson homomorphism image of a Dehn twist about a separating simple closed curve

So this is from Benson Farb's text and I need clarification on his explanation. QUESTION: So my first question is why is $\gamma \in \Gamma'$, where $\Gamma=\pi_1(S_g^1)$. My second question is why ...
1
vote
1answer
140 views

mapping class group of $S_g$, a genus $g$ surface, preserves symplectic inner product

So in Benson Farb's text on Mapping Class Groups he discusses how we can construct a map from $\mathcal{M}(S_g)$, the mapping class group, to $Sp(2g;\mathbb{Z})$. In doing so he discusses how the ...
0
votes
1answer
37 views

Prove that for any $x,y \in$ Int $D^n$, $\exists$ homeomorphism $\phi: D^n \rightarrow D^n$ such that $\phi(a)=b$ for all $n \geq 1$, fixing boundary.

I know a geometric proof, is there any algebraic proof? My approach, say for $n=1$, say for any $a,b \in (0,1)$,(say $a < b$)We need to find a homeomorphism $\phi$ that fixes the boundary, that is $...
2
votes
1answer
70 views

Mapping Class group of 3-manifold $S^1\times S^1\times D^1$

Is Mapping Class group of $S^1\times S^1\times D^1$ trivial? $D^1$ stands for 1-dimensional disc. Thanks
2
votes
1answer
54 views

mapping class group and system of disks

Let $S_g$ be the closed, compact, orientable surface of genus g, which is defined up to homeomorphism. Let $MCG(S_g )$ be the mapping class group of $S_g$, which is defined as the isotopy classes of ...
2
votes
1answer
204 views

Is the boundary Dehn twist central in the mapping class group of surface or not? [resolved]

Please help me find a mistake and resolve a paradox: Let $S$ be an orientable surface of genus $g\ge 2$ with $n\ge 1$ boundary components. Consider $T_\partial$, the Dehn twist about a curve parallel ...
7
votes
4answers
185 views

Seeking an intuitive explanation of the Mapping Class Group

For a surface $S$ the mapping class group $MCG(S)$ of $S$ is defined as the group of isotopy classes of orientation preserving diffeomorphisms of $S$: $$MCG(S)=Diff^+(S)/Diff_0(S).$$ I understand ...
2
votes
0answers
29 views

Hyper-elliptic curves and mapping class groups

What is the relation between hyper-elliptic mapping class groups and hyper-elliptic curves? I have seen the tag hyper-elliptic curves under a few articles about hyper-elliptic mapping class group ...
5
votes
0answers
116 views

Explicit Dehn twist for $S^n\times S^n$

Fix $n$ odd and let $M=S^n\times S^n$. The diffeomorphism group Diff($M$) acts on the homology group $H_n(M)\simeq \mathbb Z^2$ inducing a surjection $d: \text{Diff}(M) \rightarrow \text{SL}(2,\mathbb ...
0
votes
0answers
542 views

Orientation preserving homeomorphisms

I am looking at orientation preserving (self-)homeomorphisms of surfaces and I was wondering about how to show that whether a homeomorphism is orientation preserving or orientation reversing. In ...
2
votes
1answer
240 views

Centralizers in mapping class groups

According to Nielsen-Thurston classification, given a closed surface $S$, the elements of the mapping class group $\mathrm{MCG}(S)$ lie in three categories: periodic, reducible and pseudo-Anosov. ...
0
votes
1answer
89 views

A proof of the braid relation that is satisfied by the Dehn twist

I'm studying the mapping class group of a closed compact surface and I'm trying to prove the braid relation among the Dehn twists but the only reference I have do that with a draw. How can I prove it ...
4
votes
1answer
309 views

Motivations for mapping class group representations

I am new to mapping class groups for surfaces and representation theory. I would like to know why people care about representations of mapping class groups. I think in general representation theory ...
14
votes
1answer
353 views

General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ...
3
votes
1answer
129 views

“Uniqueness” of the Multi- Dehn-twist

I'm trying to writing down a proof for the following claim about Dehn-Twists: Let $\{a_1,...,a_m\}$ be a collection of distinct nontrivial isotopy classes of simple closed curves in a surface $S$ , ...
7
votes
1answer
218 views

Mapping Class Group

$\newcommand{\MCG}{\mbox{MCG}}$Let $\alpha$, $\beta$ be non-isotopic, non-separating curves on a surface $S$ (meaning that "cutting along " them will not disconnect the surface). How do we show that ...