# Questions tagged [mapping-class-groups]

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

286
questions

2
votes

0
answers

24
views

### Is anything known about the equivariant homotopy theory of surfaces with the action of a finite subgroup of the mapping class group?

The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \...

2
votes

1
answer

84
views

### Conjugacy problem in pure mapping class group of finitely-connected planar domain

Let $D$ be a finitely-connected planar domain, or, even more particularly, a domain obtained from the sphere $S^2$ by removing finitely many disjoint open topological disks. Let $\mathrm{PMCG}(D)$ be ...

5
votes

1
answer

265
views

### Realizing a finite subgroup of $\mathrm{Homeo}^+(S_g)$ as a subgroup of $\mathrm{Isom}^+(S_g)$

Let $G\leq \operatorname{Homeo}^+(S_g)$ be finite, where $S_g$ is a closed, connected, orientable surface of genus at least $2$. Then I have the following questions:
(1) Can $G$ always be realized as ...

1
vote

0
answers

62
views

### Mapping class group interpretation of braid closure

Given a braid (diagram) $\beta\in B_n$, the associated closed braid is the knot/link formed by attaching the ends on which the strings lie. We can also, however, think of $\beta$ as being an element ...

2
votes

1
answer

218
views

### How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?

Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation:
$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...

6
votes

1
answer

252
views

### Reference for a property of Dehn twists

I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.
In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:
Let $\...

9
votes

1
answer

305
views

### Mapping class groups are finitely generated

Let $N$ be a compact smooth manifold. By "mapping class group" I will mean
$$\pi_0 \operatorname{Diff}(N)$$
i.e. the isotopy-classes of diffeomorphisms of $N$.
My presumption is that this ...

4
votes

1
answer

190
views

### Euler class of vertical tangent bundle of the surface bundle over circle

Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...

1
vote

0
answers

18
views

### Aggregation relationship in class diagram

How do you transform an aggregation relatinship between to classes in a class diagram (1 to many or many to many) into relational schema and is it needed/possible. Or is enough if we transform only ...

2
votes

1
answer

173
views

### How to get a presentation of the mapping class group of the $n$-punctured sphere

$\DeclareMathOperator\Mod{Mod}$I would like to compute the mapping class group (homeomorphism preserving orientation modulo those isotopic to the identity) of the sphere $S^2$ minus $n$ points $p_1,\...

8
votes

1
answer

294
views

### If G is a finitely generated group with vcd(G) finite, is vcd(H) finite for H, where H is an automorphism group of G?

$\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Out{Out}$Here I mean $\vcd(G)$ to be the virtual cohomological dimension of $G$. Some ...

6
votes

2
answers

161
views

### Generate $\mathrm{Mod}(S_g)$ by two Dehn twists

Let $S_g$ be a closed orientable surface of genus $g>1$.
How can one prove that its mapping class group $\mathrm{Mod}(S_g)$
is not generated by two Dehn twists?
A pair of simple closed curves in $...

6
votes

1
answer

128
views

### Translation length on annular curve graphs

Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked.
Definition: let the group $G$ act by isometries on a metric space $(X,d)$. ...

4
votes

0
answers

91
views

### "Standard computations" with stable Hopf invariants

I am struggling in understanding the proof of Lemma 10.6 of the paper "Mapping class groups and function spaces" by Bodigheimer, Cohen and Peim http://www.math.uni-bonn.de/people/cfb/...

4
votes

2
answers

115
views

### Action of noncentral mapping classes on curves or arcs on a surface

$\DeclareMathOperator\MCG{MCG}$Let $\Sigma$ be a compact oriented surface, with empty or connected boundary. Let $\mathcal{O}$ the space of orbits of nontrivial simple closed curves on $\Sigma$ under $...

4
votes

0
answers

89
views

### Presentation of handlebody mapping class group

I know some 'nice' infinite presentations of the mapping class group of a surface, such as Gervais' and Luo's. By 'nice' I mean that generators and relations belong to a small number of families.
Is ...

8
votes

1
answer

592
views

### On trivial mapping class group of 3-manifolds

What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...

4
votes

1
answer

173
views

### Stable cohomology of mapping class group with coefficients in $H^{\otimes n}$

Let $\text{Mod}_g$ be the mapping class group of a closed oriented genus-$g$ surface $\Sigma_g$ and let $H = H_1(\Sigma_g;\mathbb{Q})$. Fix some $r \geq 0$. It is known that the cohomology group $H^...

0
votes

1
answer

289
views

### Why do we define the pants complex and the pants decomposition? [closed]

Why do we define the pants complex?
I learned for the first time in A Presentation for the mapping class group of a closed orientable surface (by A. Hatcher and W. Thurston)
that we have definition of ...

3
votes

1
answer

149
views

### 2-orbifolds that I expect to be hyperbolic, but they're nonnegatively curved

I'm considering some complex 1-dimensional/real 2-dimensional orbifolds that I expect to be hyperbolic. However, some of them seem to be Euclidean or spherical. Any thoughts what's going on here? Here ...

16
votes

0
answers

647
views

### What is the current status of the question of whether or not the mapping class group has Kazhdan's Property (T)?

$\DeclareMathOperator\Mod{Mod}$Let $\Mod(S)$ be the mapping class group of a closed oriented surface $S$ of genus at least $3$. My question is easy to state: is it currently known whether or not $\...

21
votes

2
answers

847
views

### Fixed-point free diffeomorphisms of surfaces fixing no homology classes

One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...

6
votes

1
answer

307
views

### Relation between TQFT representations and factorizable sheaves

I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks.
More ...

0
votes

0
answers

45
views

### configurations of three saddles on one level [duplicate]

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) page 13 we have :
There are sixteen ...

0
votes

1
answer

214
views

### Why do we have sixteen possible configurations of three saddles on one level?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) page 13 we have :
There are sixteen ...

13
votes

2
answers

589
views

### Mapping Out(F_n) to the mapping class group

Let $\mathrm{Out}(F_g)$ denote the automorphism group of a free group, and $\mathrm{Mod}_g$ the mapping class group of a closed oriented genus $g$ surface. Is there a map, as indicated with the dashed ...

2
votes

1
answer

122
views

### Isotopic homeomorphisms of surface induces same map on the space of ends

Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \...

0
votes

0
answers

83
views

### definition of isotopy of pants decomposition

What is definition of isotopy of pants decomposition in the following paper ?
when two pants decompositions is isotopic? Also why we say cut system what is definition of system ?
In Hatcher, Allen; ...

0
votes

1
answer

648
views

### What is definition of branched covering?

What is definition of branched covering in the page 10 of following paper ?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. ...

3
votes

0
answers

184
views

### Can Whitehead manifold admit a properly discontinuous cocompact group action?

Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action?
Here "properly discontinuous" doesn't have to be fixed point free, but ...

6
votes

1
answer

164
views

### Positive vs negative Dehn twist monoids

Given a bounded surface $S$ and a mapping class $h$, construct the open book 3-manifold $(S,h)$. If $h$ lies in the positive monoid of the mapping class group, then $(S,h)$ supports a tight contact ...

2
votes

1
answer

207
views

### Mapping class group and pure mapping class group (and their generating sets)

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\PMod{PMod}\DeclareMathOperator\Homeo{Homeo}$I am very confused about the definition of mapping class group and pure mapping class group (and their ...

2
votes

2
answers

171
views

### Measured geodesic laminations have either discrete or Cantor set local cross-sections

I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076.
In section 1, after he defines measured geodesic laminations, he makes the ...

3
votes

1
answer

168
views

### Stabilizer of the action of the mapping class group on a pants decomposition

$\DeclareMathOperator\Mod{Mod}$Let $S$ be a surface and $P=\{a_1,...,a_n\}$ be a pants decomposition of $S$. Denote by $\Mod(S)$ the mapping class group of $S$.
Define the stabilizer of $\Mod(S)$ on $...

5
votes

2
answers

1k
views

### Mapping class group and pure mapping class group

"A Primer on Mapping Class Groups" wrote
Let $\mathrm{Homeo}_+(S, \partial S)$ denote the group of orientation-preserving
homeomorphisms of $S$ that restrict to the identity on $\partial S$....

11
votes

2
answers

551
views

### $ \mathbb{R}P^n $ bundles over the circle

Is every $ \mathbb{R}P^{2n} $ bundle over the circle trivial?
Are there exactly two $ \mathbb{R}P^{2n+1} $ bundles over the circle?
This is a cross-post of (part of) my MSE question
https://math....

0
votes

1
answer

233
views

### About Alexander method in mapping class group

The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups"
For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...

1
vote

1
answer

258
views

### About isotopy of simple close curve

In the Primer mapping class group by farb Margalit. We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...

3
votes

0
answers

114
views

### Proof of homotopic essential simple close curves are isotopic

In the " A Primer on Mapping Class
Groups
Benson Farb and Dan Margalit"
We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\...

4
votes

1
answer

424
views

### About isotopy and homotopy

In the " A Primer on Mapping Class
Groups
Benson Farb and Dan Margalit"
We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\...

1
vote

0
answers

102
views

### what are definitions of born or die (birth-death point) and crossing point?

in this paper we have :
A presentation for the mapping class group of a closed orientable surface.by Hatcher.W.Thurston
...(a) $f_{t_{0}}$ has exactly one degenerate critical point, of the form $f_{t}(...

2
votes

1
answer

117
views

### Example of maximal multicurve complex

in this paper we have :
" On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps."
Definition. The maximal multicurve complex $...

1
vote

0
answers

128
views

### definition of generic function

what is definition of generic function in following paper ? i need a reference for definition generic function .
"A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed ...

3
votes

0
answers

73
views

### Representing Outer Automorphisms by Outer Matrix Conjugation for MCG?

Let $S$ be closed hyperbolic surface. The Dehn-Neilson theorem $\Gamma \approx Out(\pi_1)$ identifies the mapping class group of $S$ with the outer automorphism group of the surface group $\pi_1=\pi_1(...

2
votes

1
answer

204
views

### Is action of MCG on the curve complex computable for closed surfaces? [Yes: Birman Exact Sequence]

$\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are ...

2
votes

0
answers

97
views

### Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\cd{cd}$Let $B_{k}(S_{g}),$ $\MCG(S_{g};k)$ and $\MCG(S_{g}))$ denote the braid group, the mapping class group (relative ...

4
votes

1
answer

184
views

### Inequivalent free $\Bbb Z/n\Bbb Z$-actions on orientable compact bordered surface

Let $S_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S_{g,b})$ is said to be free $G$-...

2
votes

0
answers

100
views

### understanding the definition of subgroup of the Grothendieck-Teichmuller group

Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...

1
vote

0
answers

164
views

### Mapping class group and surgery theory

Given a smooth manifold $M$ of dimension $n$ and a diffeomorphism $\phi: M \to M$, we can construct a smooth cobordism of dimension $(n+1)$ from $M$ to $M$ by gluing $M \times [0,1]$ with itself by $\...

4
votes

1
answer

150
views

### Ideal triangulation of hyperbolic 3-manifold with generic mapping class group

I am from physics background so I apologize in advance if my question is trivial.
Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$...