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

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3
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In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In Meyer, Werner Die Signatur von ...
3
votes
2answers
138 views

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 ...
4
votes
0answers
126 views

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 ...
3
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0answers
110 views

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 ...
0
votes
1answer
109 views

from Dehn twists to surgery diagram [closed]

Assume the relation $(ab)^6=1$, for $a$ and $b$ Dehn twists about the meridian and the longitude of a torus. Now if we glue the two ends of $T\times I$ together by either the diffeomorphism $(ab)^6$ ...
3
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1answer
56 views

(Un)distorted subgroups in the mapping class group: reference required.

Let $S$ be an orientable surface with negative Euler characteristic. Can somebody provide a reference for the following well-known results: the cyclic subgroup generated by a pseudo-Anosov element ...
8
votes
1answer
128 views

Mapping class group vs automorphism group in cobordism category

Let $3Cob$ be the category whose objects are closed surfaces and whose morphisms are diffeomorphism classes of cobordisms. By sending a diffeomorphism $\phi$ of a surface $X$ to its associated ...
2
votes
2answers
277 views

Origin of the name “Torelli group”

The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface. The first paper I am aware of that uses the ...
6
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1answer
104 views

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 $$ ...
6
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2answers
142 views

Are there some list of the finite subgroups of the mapping class groups of low genus surfaces?

We already know the bound of the order of the finite subgroups of the $Mod(S_g)$. If we take a further step, to find all the finite subgroups, then what is the result for low genus cases? For example, ...
1
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1answer
153 views

Explicit computation of the action of a Dehn twist on the fundamental group of a surface

Let $S$ be a compact orientable surface of genus $g$. Now let $p\in S$ and $\gamma$ a closed simple curve on $S$ disjoint from $p$. It is not very difficult to compute the action of a Dehn twist along ...
2
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2answers
186 views

Action of Mapping Class Group on Arc complex

Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of Mod(S)(mapping class group) on the set of all vertices has finitely many ...
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1answer
128 views

Homeomorphism of the punctured sphere which fixes an essential Jordan curve

$\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan ...
8
votes
1answer
139 views

Mapping class groups of small Seifert-fibred 3-manifolds

Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location? For most Seifert manifolds working out the mapping class group is easy-enough ...
1
vote
1answer
73 views

Centralizer of a pseudo-Anosov element

What is the centralizer of a pseudo-Anosov element in the mapping class group of an orientable punctured surface? Is it cyclic? If so, where can I find a proof?
10
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1answer
228 views

Mapping class group and CAT(0) spaces

I hope the questions are not too vague. Is the mapping class group of an orientable punctured surface $CAT(0)$ ? Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...
3
votes
0answers
119 views

Mapping One Curve to another using Dehn Twists

Let $M$ be an orientable surface with genus $g>1$. Let $\alpha$ and $\beta$ represent two different isotopy classes of essential curves on the surface. Is anyone aware of a technique or algorithm ...
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0answers
72 views

Isomorphism type of mapping class group

Let $MCG(S_{g,b}^s)$ be the mapping class group of a surface $S_{g,b}^s$. Assume that it is not trivial. Is it true that $MCG(S_{g,b}^s)$ is isomorphic to $MCG(S_{g',b'}^{s'})$ if and only if $ ...
1
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1answer
101 views

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 ...
1
vote
1answer
120 views

Quasi-isometric embeddings of the mapping class group into the Teichmuller space

Does there exist a quasi-isometric embedding $$MCG(S) \to (\mathrm{Teich}(S), d)$$ for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?
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0answers
83 views

Mapping class group of ciliated surfaces

Let $(S_{g,b}^n, \mathscr C)$ be a ciliated surface, that is, a surface with with $b$ boundary components, a finite set $\mathscr C$ of distinguished points ($n$ in the interior and $c$ on the ...
10
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4answers
336 views

How to detect a simple closed curve from the element in the fundamental group?

(1) Given a fundamental group representation of a hyperbolic surface, i.e. $<a_j,b_j|\prod[a_j,b_j]=1>$, and given an element in this group, can we determine whether this element can be ...
-1
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1answer
173 views

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 ...
0
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2answers
125 views

Convexity of a minimum function

I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following ...
2
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1answer
193 views

hyperelliptic involution on a surface

What is the Dehn twist factorization of the hyperelliptic involution on an oriented surface of genus g (with one boundary component)?
3
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1answer
235 views

Iterated Lefschetz numbers

Given a pseudo Anosov mapping class $f:S_{g,n}\rightarrow S_{g,n}$ is the Lefschetz number for $f^m$ negative for some $m$ depending only on $(g,n)$? The Lefschetz number of a mapping class $f$ can ...
4
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0answers
202 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 ...
2
votes
1answer
181 views

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 ...
7
votes
1answer
254 views

Reference request: Spin structures on surfaces and the spin mapping class group

I am looking for references on the following: Spin structures on surfaces, and particularly the spin mapping class group. What is known about generating the spin mapping class group? Has anybody ...
7
votes
1answer
213 views

Cohomology of the genus 2 mapping class group

Is the cohomology of the genus 2 mapping class group (that is, the cohomology of the moduli stack $M_2$ of genus 2 curves) known? I'd be interested in references. The rational cohomology is known to ...
4
votes
2answers
281 views

Mapping class group of once-punctured torus

Let $T$ be the 2-dimensional torus and let $S$ be $T$ minus one point. Then Birman exact sequence of mapping class groups becomes an isomorphism $$ \beta: Map(S)\to Map(T)=GL(2, {\mathbb Z}). $$ It ...
8
votes
1answer
175 views

Isotopy classes on the disk and mapping tori

Is the following true? "The conjugacy classes of two homeomorphisms of the n-times punctured disk have isotopic representatives iff the associated mapping tori are homeomorphic." By conjugacy class ...
6
votes
2answers
223 views

The action of torsion of $MCG(S)$ on curve complex

Hi everyone. Let $S$ be a closed surface with genus at least 3, $\alpha, \beta$ be the two vertices of curve complex of $S$ such that $d_{\mathcal {C}(S)}(\alpha, \beta)\geq 3$. My question is ...
0
votes
2answers
256 views

The action of periodic map on the complex of curves

Hi, everyone. Assume $S$ is a genus at least 2 orientable closed surface. And there is a simplical complex defined on $S$ called Curve complex. It is well known that any automorphism of surface ...
4
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0answers
208 views

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.) ...
20
votes
2answers
563 views

The image of the point-pushing group in the hyperelliptic representation of the braid group

Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation $\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$ called the "hyperelliptic representation," which ...
6
votes
2answers
361 views

Do the following set of Dehn twists generate the mapping class group?

If $S$ is the surface illustrated below, do the Dehn twists about the red curves generate the mapping class group $\operatorname{MCG}(S,\partial S)$?
1
vote
1answer
206 views

Requiring references

Assume $V$ be a genus larger than 1 handlebody, $S=\partial_{+} V$. Denote $N$ be the normal closure of $MCG(V)$ in $MCG(S)$. Is there any material related to the quotient group $MCG(S)/N$ ? Thanks! ...
2
votes
2answers
314 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
11
votes
1answer
352 views

Lower bounds on dimensions of faithful representations of braid groups

Let $B_n$ be the braid group on $n$ strands. It's a theorem of Daan Krammer and Stephen Bigelow that there is a a faithful representation $$B_n \to GL_{n \choose 2} \mathbb Z[t^{\pm}, q^{\pm}] $$ ...
15
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3answers
543 views

Nielsen-Thurston classification via the curve complex?

I am curious to see if anyone knows a proof of the Nielsen-Thurston classification of mapping classes that does not depend on results in Teichmuller theory. From a naive point of view, translation ...
3
votes
0answers
168 views

Extension of homeomorphism of boundaries to a homeomorphism of a cobordism

Suppos we have a cobordism $(M, \partial_{-}M, \partial_{+}M)$, where $M$ is a oriented compact (topological) 3-manifold. Assume we have orientation preserving homeomorphism $f_{\pm}: \Sigma \to ...
2
votes
1answer
300 views

Automorphism of first homology and mapping class group

It is known that for a torus $\Sigma$, every automorphism of $H_1(\Sigma; \mathbb{Z})$ is induce by an orientation preserving self-homeomorphism of $\Sigma$ unique up to isotopy. In onther words, ...
1
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1answer
234 views

Mapping class group and cylindrical structure

Let us fix a torus $\Sigma=S^1 \times S^1$. We consider a cylinder $\Sigma \times I$ and a data $(\Sigma\times I, \Sigma\times 0, \Sigma\times 1, f_{0},f_{1})$. Here $f_{i}$, called parametrization, ...
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270 views

Isomorphism of cobordisms

Let $(M, \partial_{-}M, \partial_{+}M)$ be a decorated 3-cobordism, where $M$ be a (topological) 3-manifold and $\partial M=-\partial_{-}M \cup \partial_{+}M$. (decorated in a sense of Turaev, Quantum ...
9
votes
3answers
387 views

Flips of triangulations on non-orientable surfaces

Let $N_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures. I'm looking at ideal triangulations of the surface, namely ...
7
votes
3answers
560 views

Automorphisms of Riemann surface and mapping class

For a higher genus Riemann surface $\Sigma$, is it true that every nontrivial (holomorphic) automorphism is of nontrivial mapping class, i.e., not isotopic to the identity?
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2answers
216 views

Elements of finite order in mapping class groups of high dimensional manifolds

Let $M$ be a manifold with boundary. Consider the following groups: (1) $\pi_0(\operatorname{Diff}(M,\partial M))$. (2) $\pi_0(\operatorname{Homeo}(M,\partial M))$. (3) ...
8
votes
1answer
580 views

Surface groups and non separating loops

QUESTION: Let $g \geq 4$, $S(g)$ be the fundamental group of the genus $g$ surface, and $G$ be finitely generated (the number of generators $\leq 3$) group with abelianization of rank less than equal ...
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190 views

Do the real polarization and Kahler polarization of the character varieties of closed surfaces give rise to equivalent representations of the Mappping 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 ...