Questions tagged [mapping-class-group]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
58 views

Conjugation Classes inside the Orbit of close curve in Hyperbolic Surface under $Mod(S)$

I've heard the statement: Let $S$ be a hyperbolic surface with boundary and $\left[ \alpha \right] \in \pi_1 \left( S \right)$ a non-trivial element that is not conjecture to boundary element, then ...
0
votes
1answer
27 views

mapping class group of the real projective plane

In most literature I've read about the mapping class group, I found that many authors have stated without any explanation that any homeomorphism of a real projective 2-space to itself is isotopic to ...
0
votes
2answers
18 views

Surface homeomorphism transitively permutes the boundary curves

Let $h: S\to S$ be a surface homeomorphism. What does it mean to say that $h$ transitively permutes components of $\partial S$, and where does this terminology come from? Is this somehow related to a ...
3
votes
0answers
51 views

Mapping class group of a torus

Seems like i have some false understanding that need to be fixed. I have check varies sources saying that the mapping class group of a torus is isomorphic to $GL(2,\mathbb Z)$, however my false ...
0
votes
0answers
22 views

Kernel of the forget map on mapping class groups is the isotopy class of the identity map

For a surface $S$, possibly with punctures (but no marked points) we denote ($S,x$) to be the surface obtained by marking a point $x\in$int$S$. There is a natural homomorphism Forget: Mod$(S,x)\...
1
vote
1answer
99 views

The monodromy representation $\pi_1(X)\to Mod(F)$

Given a fiber bundle $F\to X\to E$, with $F,E$ are Riemann surfaces. I know the monodromy gives a permutation of the fiber. But how to see we have the monodromy representatio: $\pi_1(E)\to Mod(F)$? ...
1
vote
1answer
40 views

Mapping class group of $S^1 \times S^1 \times I$

I am interested in the computation of the mapping class group of the manifold $M=S^1 \times S^1 \times I$. One can visualize $M$ as a "torus cross $I$", or as an "annulus cross $S^1$". $M$ is ...
0
votes
0answers
9 views

The kernel of the action of the mapping class group on homology gives a pure subgroup

(From "A Recipe for Short-Word Pseudo-Anosovs" by J. Mangahas, pg 7) In the mapping class group of a surface, $\text{Mod}(S)=\text{Homeo}^+(S)/\text{Homeo}_0(S)$, the reduction system $C$ of a ...
0
votes
1answer
33 views

The monodromy of a Lefschetz fibration as right-handed Dehn twists

A fact that one can find in many books is that the monodromy of a Lefschetz fibration is the product of right-handed/positive Dehn twists (one for each vanishing cycle). The only proof I could find ...
4
votes
1answer
72 views

Mapping Class Group of Pants with a Hole

It is known that the mapping class group of the torus $\mathbb{T}^2$ is $\text{Mod}(\mathbb{T}^2) \cong \text{SL}_2(\mathbb{Z})$. We also know that for a pair of pants $P$ (a sphere with three ...
0
votes
0answers
20 views

Do we have a conformal mapping from the pentagon to the square?

There is a way that can map a pentagon to a square. I find that it is possible map a pentagon to a disc. How about the square?
0
votes
1answer
38 views

Is first homology (with integer coefficients) always a vector space?

I've run into a bit of an issue while reading through Farb and Margalit's "Primer on Mapping Class Groups," which I'm sure is due to my own incomplete knowledge! In chapter 6, they discuss the ...
0
votes
0answers
29 views

Mapping from square to annulus ring

There are two figures, a unit square and an annulus. I want to find a function that maps from square to annulus ring and also an inverse function that maps back from annulus to unit square. Image of ...
2
votes
0answers
69 views

What is the curve complex of the 5-times punctured sphere?

There are many theorems about connectedness and other such properties of the curve complex of the 5-times punctured sphere. However, I cannot find any explicit descriptions of the curve-complex. (By ...
0
votes
2answers
146 views

Essential closed curves and orientation preserving homeomorphism

Q1. In the book "A primer on mapping class groups", the author gives a definition of essential closed curve as "A closed curve is called essential if it is not homotopic to a point, puncture, or a ...
2
votes
1answer
139 views

Mapping Class Group of Annulus

I studied the construction of mapping class group for an annulus, primarily from "A primer of Mapping class group" (Page 51-52 https://www.maths.ed.ac.uk/~v1ranick/papers/farbmarg.pdf). However, I ...
0
votes
1answer
79 views

Mapping Class Group of $S^2$

I try to give a detialed verification why the mapping class group of $2-$sphere is trivial $$\mbox{MCG}(\mathbb{S}^2) =1.$$ I already know that $$\mbox{MCG}(\mathbb{D}^2) = 1 \ \mbox{and} \ \mbox{MCG}...
0
votes
1answer
69 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 ...
5
votes
0answers
114 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
82 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
23 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 ...
2
votes
1answer
91 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
55 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
58 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
33 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
290 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
53 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
101 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 ...
2
votes
1answer
67 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
314 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
140 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
72 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
83 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
111 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
95 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
253 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
199 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 ...
2
votes
1answer
442 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
127 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
37 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
115 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 $H\...
0
votes
1answer
63 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
55 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
76 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
114 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
217 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
232 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
226 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
575 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
289 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 ...