Questions tagged [mapping-class-groups]
Topology of groups of automorphisms of surfaces, and high dimensional analogues.
296
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Euclidean algorithm for simple closed curves
In the proof of Proposition 6.2 in Farb & Margalit, "A primer on mapping class groups", an analog of the Euclidean algorithm is used to construct a simple, closed representative (...
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1
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Fundamental domain for the action on curve complex
Suppose $S$ is a surface and the Mapping Class Group, $Mod(S)$, of $S$, is the group of self-homeomorphisms of $S$, up to isotopy. This group acts on a graph called "curve graph", denote by $...
2
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0
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Nielsen–Thurston classification and configuration spaces
Viewing the $n$-strand braid group as the mapping class group of an $n$-punctured disk, braids can be classified as periodic, reducible, or pseudo-Anosov. The same group is also the fundamental group ...
4
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1
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Hyperbolic three-manifolds that fiber over the circle
Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...
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1
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188
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Fixed points free automorphisms of Teichmüller spaces
Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
2
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1
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An interior cone condition for Teichmuller spaces
Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
3
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1
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Are the two families of Johnson invariants of the Torelli groups related beyond the first one?
$\newcommand{\sp}{\operatorname{Sp}(H)}$
$\newcommand{\gr}{\operatorname{gr}}$
$\newcommand{\id}{\operatorname{id}}$
$\newcommand{\der}{\operatorname{Der}}$
Johnson has defined two families $\tau_k,\...
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Given a word $w$ in the braid group $B_n$, representing a pure braid, find the image of $w$ in the abelianization of $P_n$
Suppose I have a word $w$ in the standard generators $\sigma_1,\dots,\sigma_{n-1}$ of the braid group $B_n$ representing an element which we know belongs to the pure braid group $P_n$, is there an ...
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0
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What are the finite quotients of the braid group?
Are all known finite quotients of the braid group given by reducing the Burau or Lawrence-Krammer representations mod $p$ and evaluating at some element in $\mathbb{F}_p$? I recently saw a paper ...
3
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What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?
Follow up question, edited in on 12/20 below:
Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
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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
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1
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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
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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 ...
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0
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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
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1
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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:
$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
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Reference for a property of Dehn twists
I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.
In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:
Let $\...
9
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1
answer
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Mapping class groups are finitely generated
Let $N$ be a compact smooth manifold. By "mapping class group" I will mean
$$\pi_0 \operatorname{Diff}(N)$$
i.e. the isotopy-classes of diffeomorphisms of $N$.
My presumption is that this ...
4
votes
1
answer
210
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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 $\Sigma\times I$ ...
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0
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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
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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,\...
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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
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2
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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
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1
answer
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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
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0
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"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
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2
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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
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0
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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
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1
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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
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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
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1
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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
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1
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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
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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 $\...
22
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2
answers
902
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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
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1
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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 ...
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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
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1
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215
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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
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2
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612
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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
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1
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130
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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
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0
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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; ...
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1
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804
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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
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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
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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
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1
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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 ...
4
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2
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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
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1
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175
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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 $...
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2
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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
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2
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$ \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....
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1
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254
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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
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1
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308
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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 ...
5
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1
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Proof of homotopic essential simple close curves are isotopic
In the book by Benson Farb and Dan Margalit A primer on mapping class groups, Princeton Mathematical Series 49. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/...
4
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1
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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 $\...