Questions tagged [mapping-class-groups]
Topology of groups of automorphisms of surfaces, and high dimensional analogues.
29
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Homeomorphic but Non-Conjugate Mapping Tori
Suppose we fix a genus $g$ closed surface $S$. Let $f, g \in Map(S)$ be conjugate, for $Map(S)$ the mapping class group of $S$. Then I know that $M_f$ (the mapping torus of $M$ with monodromy $f$) is ...
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3
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Which mapping class group representations come from algebraic geometry?
Let $\Gamma_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping ...
- 39.3k
12
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1
answer
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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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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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0
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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 ...
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1
answer
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What is the order of the isotopy group of the Brieskorn homology 3-sphere?
Let $\Sigma(p,q,r)$ be the Brieskorn homology 3-sphere with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ (so not the 3-sphere or the Poincare sphere). The fundamental group is given by $$ \pi_1(\Sigma(p,...
9
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Mapping Class Groups and torus (JSJ) decomposition of closed 3-manifolds
I am wondering if some intuitive relation exists between Mapping Class Group (MCG) of a 3-manifold (assume "simple" enough manifolds: closed,compact,irreducible, orientable, non-hyperbolic) and its ...
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1
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Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
Let $\Sigma$ be an oriented compact surface with non-empty boundary that is not a disk or a cylinder (i.e. negative Euler characteristic). Let $\phi, \psi: \Sigma \to \Sigma$ be two orientation ...
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6
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0
answers
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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 ...
- 1,040
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1
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Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures
Let $S=S_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x_1,\ldots , x_n\}$ be a set of $n$ distinguished points in the ...
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6
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1
answer
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Mapping Class Group (MCG) of connected sum of 3-torus and $S^2\times S^1$
I know that the MCG (isotopy-classes of orientation preserving homeomorphisms) of 3-torus $(S^1\times S^1 \times S^1)$ is $SL(3,Z)$, since it is an Eilenberg–MacLane space, giving $ MCG(T^3)=Out(\pi_1(...
- 231
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votes
0
answers
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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 ...
- 1,971
5
votes
2
answers
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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$....
user127837
5
votes
1
answer
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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/...
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Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface
If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups:
$$\text{Mod}(\mathcal{R}')\longrightarrow \...
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votes
0
answers
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 ...
- 39.3k
4
votes
2
answers
211
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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 ...
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1
answer
376
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Divergence of geodesics in mapping class groups
I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the ...
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4
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2
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Nielsen-Thurston classification of homeomorphisms for open surfaces?
In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
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Mapping class groups of a punctured surface vs. surface with boundary
Let $S_{g,b}$ an orientable surface with genus $g$ and $b$ boundary components and $S_g^b$ be an orientable surface with $b$ punctures.
Denote by $PMCG(S_g^b)$ and $PMCG(S_{g,b}) $ the pure mapping ...
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3
votes
1
answer
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Isomorphism between a mapping class group and the fundamental group of a moduli space
For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
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1
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How do Dehn functions of special linear and mapping class groups behave?
Hi,
I apologize for the basic questions. I am looking for good references on the following problems:
1) What is known about the Dehn function of $SL_n(\mathbb{Z})$?
2) What is known about the Dehn ...
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3
votes
3
answers
746
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Reducible 3d torus bundles
Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,
could anyone give me a hint to classify them?
In contrast, do you agree ...
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votes
0
answers
406
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Geometric intersection number for product of elements of the fundamental group
Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...
- 1,703
3
votes
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Lie Group Isomorphisms
I'm not sure of the difficulty of the question I'm about to ask. If it does not fit the criteria for this site then I apologize in advance, I'm rather new here.
So here it goes: Let $G$ be a Lie ...
3
votes
1
answer
525
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Is the Action of the mapping class group transitive on embedded arcs?
Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The ...
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2
votes
0
answers
182
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Stabilizing an open book with Anosov piece
It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...
- 1,379
1
vote
1
answer
305
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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 ...
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0
votes
1
answer
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mapping class group of a surface
I want to know what techniques are known to present a diffeomorphism on a surface with boundary (the diffeomorphism is not necessarily the identity restricted to the boundary) as product of Dehn ...
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