Questions tagged [mapping-class-groups]

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

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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 ...
mme's user avatar
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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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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 $\...
Thomas's user avatar
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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 ...
Paul's user avatar
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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}^+...
Harry Reed's user avatar
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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 ...
Jun Lu's user avatar
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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}, \...
Paul's user avatar
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7 votes
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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 ...
Mauro's user avatar
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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, ...
Brian Lawrence's user avatar
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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"...
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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 ...
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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 ...
Montmorency's user avatar
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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) ...
Lukas Woike's user avatar
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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 ...
Nati's user avatar
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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 ...
Christoph Wockel's user avatar
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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 ...
Harry Reed's user avatar
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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 ...
user124543's user avatar
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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 ...
Rylee Lyman's user avatar
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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 $...
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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,...
Cusp's user avatar
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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 ...
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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 ...
Nati's user avatar
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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 ...
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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 ...
ThiKu's user avatar
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417 views

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 ...
Dan Petersen's user avatar
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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.) ...
Autumn Kent's user avatar
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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/...
Tommaso Rossi's user avatar
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94 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 ...
Filippo Bianchi's user avatar
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211 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 ...
Giacomo Bascapè's user avatar
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205 views

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 ...
qkqh's user avatar
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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 ...
Paul's user avatar
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176 views

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 ...
Mojtaba Shokrian's user avatar
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186 views

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\...
Overflowian's user avatar
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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 ...
Paul's user avatar
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4 votes
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319 views

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):\...
Nati's user avatar
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4 votes
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233 views

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 ...
Cusp's user avatar
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4 votes
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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+...
dvitek's user avatar
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4 votes
0 answers
178 views

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: ...
Federico Vigolo's user avatar
4 votes
0 answers
352 views

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 ...
John Pardon's user avatar
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4 votes
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467 views

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})...
skupers's user avatar
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3 votes
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217 views

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 ...
Chase's user avatar
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3 votes
0 answers
190 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 ...
Shijie Gu's user avatar
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3 votes
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75 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(...
JHM's user avatar
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3 votes
0 answers
110 views

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 ...
chronondecay's user avatar
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134 views

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?
Luca Iliesiu's user avatar
3 votes
0 answers
108 views

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}...
Renaud Detcherry's user avatar
3 votes
0 answers
65 views

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 $\leq i$. ...
qqqqqqw's user avatar
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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 ...
Marty Lee's user avatar
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3 votes
0 answers
217 views

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 ...
Daniele Zuddas's user avatar