Questions tagged [mapping-class-groups]
Topology of groups of automorphisms of surfaces, and high dimensional analogues.
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questions with no upvoted or accepted answers
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Is the mapping class group of $\Bbb{CP}^n$ known?
In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an ...
18
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843
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Are these local systems on $\mathscr{M}_{g,1}$ motivic?
Let $(\Sigma_g, x)$ be a pointed topological surface of genus $g$, and let $MCG(g,1)$ be the mapping class group of this pointed surface. Then $MCG(g,1)$ has a natural action on $\pi_1(\Sigma_g, x)$ $$...
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675
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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 $\...
12
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869
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On a corollary of a paper by Colin and Honda
The question is about the last sentence of the last corollary of Stabilizing the monodromy of an open book decomposition by Vicent Colin and Ko Honda. This question is also related to this other ...
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569
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What is known about mapping class groups of 4-manifolds?
I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+...
11
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559
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How to get a Dehn-twist presentation of a periodic map of a Riemann surface?
Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ).
A classical result says such $f$ is ...
10
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Some questions about geodesic lamination
I'm learning geodesic laminations on surfaces. Here are some questions I thought a lot but could not understand well.
We consider a complete finite area hyperbolic surface $S$ w/o geodesic boundary. ...
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Mapping classes as Lefschetz fibrations over surfaces with positive genus
Let $\Sigma_{g,r}$ be the surface of genus $g$ and $r$ boundary components. It is known that, from a positive factorization of a mapping class $\phi$ in the mapping class group $MCG(\Sigma_{g,r}, \...
7
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Purely analytic proof of the Nielsen-Thurston classification theorem
I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory
and Applications to ...
7
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459
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Reference request: Mapping class group action on homology of surface with boundary
This is a request for a reference to a proof of a result. The result is not very hard, but I'd rather cite than reprove.
I'm looking for a generalization of the following result (Farb and Margalit, ...
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351
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Making diffeomorphism of submanifolds boring
This is probably very well known in surgery theory... I'm looking for a modern reference on the following questions (the only one I know is Browder's "Diffeomorphism of 1-connected manifolds"...
6
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Borel conjecture and arbitrary surface
Before starting my question I want to write something that I already know.
Borel Conjecture: Any homotopy equivalence between two closed
aspherical manifolds is homotopic to a homeomorphism.
Now, my ...
6
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J. F. Adams Proof of Cellular Approximation Theorem
In Ronald Brown's discussion of the proof of The Cellular Approximation Theorem in Topology and Groupoids Sec. 7.6 he writes that, "the elegant formulation of the proof is due to J. F. Adams." Does ...
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Mapping class group orbits of principal bundles
Suppose $M$ is a manifold (I would be happy with low-dimensional examples like surfaces, but let me ask more generally).
Then for any discrete group $G$ (again, I would be happy with a finite group) ...
6
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What's the meaning of the Johnson filtration in terms of configuration spaces?
This question is inspired of course by the remarkable paper of Tetsuhiro Moriyama from 2008.
Let $\Sigma$ be a genus $g \geq 3$ closed surface.
Let $\phi : \Sigma \to \Sigma$ be an orientation ...
6
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Third cohomology of mapping class group
I would like to know the third cohomology with coefficients in $U(1)$ or $\mathbb{C}^\ast$ of the mapping class group of a surface of genus at least one. I found many results on the rational ...
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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 ...
5
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Center of symplectic derivation Lie algebra
Morita–Sakasai–Suzuki studied the graded Lie algebra $\mathfrak{h}_{g,1}$ of symplectic derivations, as well as variations $\mathfrak{h}_{g,\ast}$ and $\mathfrak{h}_g$. This is the Lie algebra of ...
5
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Is there an equivariant simplicial deformation retract of Teichmüller space?
Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...
5
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Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
5
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148
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Relation between point pushing pseudo-Anosov map and the minimum length
Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space.
By Birman exact sequence we get the point pushing map $Push:\pi_1(S,...
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What is the fundamental group of Kontsevich's space of stable maps?
... at least in the case where the target is a rationally connected variety.
This question is a follow-up to question
Constructing embedded families of curves with general moduli
and Jason Starr's ...
5
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297
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Does the Torelli space appear "in nature"?
What I mean by the (slightly facetious) title is:
The classical theory of algebraic curves from the 19th century was split in two in the 20th century (much like the theory of groups): the theory of ...
5
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100
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Reference Request: Mapping Class Group of Seifert-Fibered spaces
It seems to be a well understood and old topic, but even after a few days of searching, I am having trouble finding a good/more pedagogical introduction to Mapping Class Group of Seifert-Fibered ...
5
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158
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Second homology of mapping class group via universal coefficient theorem
Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$.
It is known ...
5
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417
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Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves
Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
5
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Eilenberg-Mac Lane spaces for surface group extensions.
(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which case $G$ is finite.)
...
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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/...
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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 ...
4
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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 ...
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Image of the mapping class group of surfaces into automorphism group?
Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
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Are these two arguments incompatible?
I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong.
First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...
4
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Symmetries of MCG in terms of Humphries generators?
The Riemann-Hurwitz formula gives $84(g-1)$ as the upper bound to the order of a finite group acting faithfully on a closed genus g surface. Famously the bound is realized when $g=3$ by a simple ...
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Mapping class group of $\mathbb{S}^3$
If I recall correctly from a lecture I attended the last year we have that
$MCG(\mathbb{S^2})\simeq\frac{\mathbb{Z}}{2\mathbb{Z}}$ by Smale in the 60' and $MCG(\mathbb{S^3})\simeq\frac{\mathbb{Z}}{2\...
4
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Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy
Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...
4
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319
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Higher genus Cohen-Jones-Segal's conjecture?
Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism
$$(ev,\Phi):\...
4
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Kra's theorem of Pseudo-Anosov maps
Let $S$ be a surface of negative Euler characteristic. Consider the Birman exact sequence:
$$1\xrightarrow{ }\pi_1(S,p)\xrightarrow{P} Mod(S,p)\xrightarrow{ }Mod(S)\xrightarrow{ }1$$
In his paper ...
4
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How do we see the rank of the braid group?
The only presentation of the braid group that most people ever see is the standard Artin presentation
$$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i σ_{i+...
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Centralizers and intersections in the Gromov-boundary of the mapping class group
The mapping class group of a punctured surface $\Sigma$ is weakly relatively hyperbolic (see below), hence it is well defined the Gromov-boundary with respect to the relative metric.
First question: ...
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Do real polarization and Kahler polarization of character varieties of closed surfaces give equivalent representations of the Mapping Class Group?
This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.
These ...
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Does a Dehn twist in the mapping class group of an cobordism give a BV-operator in string topology?
In her article Higher string topology operations, Godin in particular construct for each surface with $n$ incoming and $m \geq 1$ outgoing boundary circles an operation $H_\ast(BMod(S);det^{\otimes d})...
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217
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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 ...
3
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190
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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 ...
3
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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(...
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Can a path-connected domain be completely surrounded by 4 translates?
Question: Does there exist a compact path-connected set $A\subseteq\mathbb C$ such that:
$A\cap(A+1)=A\cap(A+i)=\emptyset$,
$A\cap(A+1+i)\neq\emptyset$, and
$A\cap(A+1-i)\neq\emptyset$?
Remarks: If ...
3
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134
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Mapping class groups of $T^2 \times [0, 1]$ and $T^2 \times S^1$
Are the mapping class groups of $T^2 \times [0, 1$] and $T^2 \times S^1$ explicitly known?
3
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108
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Order of separating Dehn twists in the image of Johnson homomorphisms
Let $S$ be a closed surface, $\Gamma=\pi_1 S$ and $\Gamma_i$ be the lower central series defined by $\Gamma_0=\Gamma$ and $\Gamma_i=[\Gamma,\Gamma_{i-1}].$
The Johnson filtration $\lbrace \mathcal{K}...
3
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65
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Stability for mapping class groups, spaces of sections, and polynomial coefficient systems
Let $X$ be a simply connected space. Cohen and Madsen https://arxiv.org/abs/math/0601750 proved that the functor sending a surface with boundary M to $H_i(Map(M,X))$ has polynomial degree $\leq i$. ...
3
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115
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Preimages of certain subgroups of $\bigwedge^{3} H $ under Johnson's homomorphism
We have Johnson's homomorphism $\tau_{2}\colon \mathcal{I}_{g,1} \to \bigwedge^{3}H$, where $\mathcal{I}_{g,1}$ - is a Torelli group, i.e. it consists of all elements of a mapping class group that ...
3
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217
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Lutz twist and open book decompositions
Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...