Questions tagged [mapping-class-groups]

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

Filter by
Sorted by
Tagged with
4 votes
1 answer
129 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^...
  • 43
-5 votes
0 answers
161 views

Why we define subgroup of Grothendieck-Teichmüller group?

$\newcommand{\GT}{\mathrm{GT}}$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., we have: ...
  • 117
0 votes
1 answer
176 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 ...
  • 117
3 votes
1 answer
118 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
561 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 $\...
  • 161
21 votes
2 answers
746 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-...
  • 303
4 votes
1 answer
239 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 ...
  • 117
0 votes
1 answer
201 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 ...
  • 117
0 votes
0 answers
83 views

About two surgeries on a collection of circles

in this paper (A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed orientable surface) page 4 we have : There are five essentially distinct possibilities, corresponding to ...
  • 117
13 votes
2 answers
505 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 ...
  • 37.5k
2 votes
1 answer
91 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 \...
  • 381
0 votes
0 answers
72 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; ...
  • 117
0 votes
1 answer
268 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. ...
  • 117
3 votes
0 answers
158 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 ...
  • 1,150
0 votes
0 answers
94 views

What is mean of action of group on tower?

What is mean of action of group on tower ?in the page 17 of following paper (action on teichmuller tower ) In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class ...
  • 117
0 votes
0 answers
176 views

What is "inn" in this paper?

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.,in the page 18 we have: $$ \begin{aligned} ...
  • 117
6 votes
1 answer
105 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
125 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 ...
's user avatar
1 vote
1 answer
71 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
142 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 $...
's user avatar
5 votes
2 answers
829 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 $\partial S$....
's user avatar
11 votes
2 answers
437 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
150 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
188 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
75 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
325 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
73 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}(...
  • 117
2 votes
1 answer
106 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 $...
  • 117
1 vote
0 answers
114 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 ...
  • 117
3 votes
0 answers
64 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(...
  • 2,226
2 votes
1 answer
182 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 ...
  • 2,226
0 votes
0 answers
119 views

why Alexander method gives us a finite combinatorial problem?

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 ...
  • 117
2 votes
0 answers
79 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 ...
  • 173
4 votes
1 answer
178 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 $G$-...
  • 381
2 votes
0 answers
92 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
138 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 $\...
  • 4,207
4 votes
1 answer
102 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 $G$...
3 votes
1 answer
55 views

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 ...
8 votes
2 answers
241 views

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 &...
  • 5,251
2 votes
0 answers
86 views

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 ...
  • 327
4 votes
1 answer
128 views

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 ...
  • 309
1 vote
0 answers
55 views

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 ...
  • 455
1 vote
0 answers
74 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 ...
  • 455
10 votes
2 answers
315 views

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 ...
  • 4,207
3 votes
0 answers
178 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 ...
7 votes
2 answers
278 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, ...
2 votes
2 answers
138 views

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 ...
8 votes
2 answers
486 views

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 ...
  • 6,418
6 votes
1 answer
313 views

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 ...
  • 163

1
2 3 4 5 6