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

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Let $\Sigma_g$ be the closed orientable surface of genus $g\ge 2$. Suppose $\phi_1$ and $\phi_2$ are periodic mapping classes on $\Sigma_g$ such that the mapping tori $M_{\phi_1}(\Sigma_g)$ and $M_{\...
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I am interested in the following natural question about the realization of automorphisms of the cohomology ring of a manifold, by diffeomorphisms. Question: Let $M$ be a closed and connected smooth ...
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In Casson and Bleiler's Automorphisms of Surfaces after Nielsen and Thurston, they include a diagram of an infinite non-closed simple geodesic on a closed hyperbolic surface, which limits on either ...
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Let $\overline{B}_{2n+1}$ be the mapping class group of a $(2n+1)$-punctured sphere fixing one distinguished puncture. Let $f\in \overline{B}_{2n+1}$ be a periodic element. Since there is a concrete ...
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I'd like to know the $G$-equivariant mapping class groups of the torus --- by which I mean the groups of connected components of the groups of $G$-equivariant diffeomorphisms, $$ \pi_0\big( \mathrm{...
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Fix a Heegaard splitting of the 3-sphere, and let $\operatorname{Mod}(A)$ and $\operatorname{Mod}(B)$ be the mapping class groups of the two genus $g$ handlebodies $A$ and $B$ in this splitting. We ...
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I am learning Chapter 8 of the book A Primer on Mapping Class Groups by Farb and Margalit, and I have a question regarding the paragraph "once-punctured versus closed" on page 235. Let me ...
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An irreducible orientable 3-manifold $M$ can be decomposed along a collection of incompressible tori $T = (T_i)$ in pieces $(M_j)$ which are either atoroidal or Seifert fibered, according to the JSJ ...
berto's user avatar
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I am studying about the center of the mapping class group of a genus $g$ surface. I am having some difficulties in understanding the proof for the triviality of the group $Z(Mod(S_{g}))$. Here is the ...
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Let $B_{n}$ be the braid group on $n$-strands with $n\ge 3$. Then $B_{n}$ can be realised as the mapping class group of the $n$-times punctured disk. The centre of $B_{n}$ is infinite cyclic and ...
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I am looking for an explicit presentation of the mapping class group of the annulus $\mathbb{A}^2$, after equipping it with $n$ interior punctures/marked points $\{x_1, \cdots, x_n\} \hookrightarrow \...
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Let $\operatorname{Mod}_g^1(k)$ be the Johson filtration of the mapping class group of a surface with one boundary component. $k=1$ gives all of $\operatorname{Mod}_g^1$, $k=2$ is the Torelli group, ...
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What I aim to understand is under what circumstances the Dehn twist along the boundary 2-sphere of a punctured 3-manifold $M' = M \setminus \text{int}(B^3)$ results in a non-trivial element of the ...
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I do not work in topology, but for some reason we need to know the mapping class group of certain non-orientable 3 manifolds. We found answers online for a lot of orientable manifolds. But we still ...
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This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them. Recall that a subgroup $K$ of a ...
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I have a curious question about a natural sequence, which I haven't seen answered in the literature. Let $\Sigma$ be an oriented surface of genus $g$ without boundary with a set $\mathcal P_n$ of $n$ ...
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Let $\mathcal{T}_A$ be a Teichmüller space of the sphere $S^2$ with a finite set $A$ of marked points, and suppose that $f \colon \mathcal{T}_A \to \mathcal{T}_A$ is a holomorphic map that has a ...
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Let $\Sigma$ be a finite-type orientable surface with negative Euler characteristic, and $\mathrm{Mod}(\Sigma)$ denote the mapping class group. What are the finite normal subgroups in $\mathrm{Mod}(\...
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There are well-known symplectic forms on the Teichmuller space $\mathcal{T}(\Sigma)$ of a closed surface $\Sigma$ (Wolpert gave a formula in Fenchel-Nielsen coordinates) and the space of measured ...
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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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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 $...
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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 ...
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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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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 $\...
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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 ...
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$\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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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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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 ...
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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 \...
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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 ...
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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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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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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 $\...
Don's user avatar
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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 ...
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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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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 ...
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$\DeclareMathOperator\Mod{Mod}$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 $p_1,\...
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$\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 ...
Mike's user avatar
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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 $...
Andrey Ryabichev's user avatar
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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 $(X,d)$. ...
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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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$\DeclareMathOperator\MCG{MCG}$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 $...
Renaud Detcherry's user avatar
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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 ...
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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 ...
Anubhav Mukherjee's user avatar
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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 $H^...
Fred's user avatar
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Why do we define the pants complex? I learned for the first time in A Presentation for the mapping class group of a closed orientable surface (by A. Hatcher and W. Thurston) that we have definition of ...
Usa's user avatar
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I'm considering some complex 1-dimensional/real 2-dimensional orbifolds that I expect to be hyperbolic. However, some of them seem to be Euclidean or spherical. Any thoughts what's going on here? Here ...
Ethan Dlugie's user avatar
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$\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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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-...
Robert's user avatar
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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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