17 votes
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Which mapping class group representations come from algebraic geometry?

Dan, Although I'm no longer very active on MO, I thought I'd make a few comments, since your question is an interesting one (and you're not anonymous). The paper of Looijenga referenced in Igor's ...
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16 votes

Mapping Class Group (MCG) of connected sum of 3-torus and $S^2\times S^1$

The mapping class groups of all compact orientable 3-manifolds are essentially known. A fairly detailed summary of the results, focusing on the nonprime case and with references to proofs in the ...
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16 votes
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Conjugacy classes of the mapping class group

An exponential-time solution to the conjugacy problem in the mapping class group was given by Jing Tao, in: Tao, Jing(1-OK) Linearly bounded conjugator property for mapping class groups. (English ...
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15 votes
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Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$

See Keith Conrad's notes http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,Z).pdf, particularly Example 2.5. Let us write (as Conrad does) $S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{...
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  • 4,715
15 votes

Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$

To expand my comment, and combine it with some points from Zach Teitler's answer, but more in a generators and relations framework: ${\rm SL}(2,\mathbb{Z})$ is well-known to be isomorphic to the group ...
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15 votes
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Pseudo-Anosov maps with same dilatation.

Yes. If two pseudo-Anosov mapping classes are conjugate then they must have the same dilatation. So take any pseudo-Anosov $f$ and any mapping class $h$ not in the centraliser of $f$ and let $g = h f ...
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  • 3,055
13 votes

Compact manifolds with big mapping class group

Take $M^d$ to be a connected sum of $n$ copies of $S^1\times S^{d-1}$, where $d\ge 3$. Then $M$ is a closed, orientable manifold of dimension $d$ with $\pi_1(M)=F_n$, the free group of rank $n$. If $...
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  • 2,143
13 votes
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Mapping class group of certain 3-manifolds

Since you write ${\rm Diff}_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientation-preserving. Every diffeomorphism of $M$ can be isotoped to take fibers to ...
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12 votes
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Ivanov's metaconjecture on surface homeomorphisms

EDIT: Brendle-Margalit have released their paper. See here. One should observe that these are not all examples of Ivanov's metaconjecture (for instance, the automorphism group of the disk complex is ...
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  • 39.8k
12 votes

Finite subgroups of mapping class groups

You can also use Serre's theorem which says that kernel of the natural homomorphism from the mapping class group of $\Sigma$ to $\text{Sp}(2g;\mathbb{Z}/3\mathbb{Z})$ is torsion free, and therefore ...
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  • 14.6k
12 votes

Compact manifolds with big mapping class group

In Infinitesimal computations in topology Sullivan shows in Theorem 13.3 that if $M$ is a simply-connected manifold of dimension $>5$, then $\pi_0(\mathrm{Diff\,} M)$ is commensurable to an ...
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12 votes
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Mapping class group of torus, why is $(ST)^3=S^2$?

Flip the direction of rotation for $S$, or choose the other meridian for $T$. We can see this at the level of matrices. Define $$S_1 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad ...
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  • 1,661
12 votes

Mapping class group and pure mapping class group

Just to give an explicit description of the difference: if one takes a loop "around a boundary component," the Dehn twist around this loop is not isotopic to the identity. On the other hand, ...
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  • 20.5k
11 votes

Homeomorphic but Non-Conjugate Mapping Tori

Counterexamples are easily constructed using the Thurston norm. In fact, any example of a fibered, oriented, closed 3-manifold $M$, with a fiber of genus $\ge 2$ and with pseudo-Anosov monodromy, and ...
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  • 14.6k
10 votes
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A query about Hatcher flow on arc complex

The flow consists of a sequence of surgeries using one fixed oriented arc $\alpha$ to cut (and isotope) all other arcs $\beta$ to remove one point of $\alpha\cap\beta$ at a time. Each surgery cuts ...
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10 votes
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Mapping class group and representation of fundamental group of Riemann surfaces

There are counterexamples as soon as $g > 1$. Let $n$ be the number of surjective homomorphisms $\pi_1(S) \to A_5$, up to $S_5$-conjugacy. (We can see that $n \geq 1$ using the fact that $A_5$ ...
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  • 114k
10 votes
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All non-compact simply connected $2$-manifolds with boundary

Here is one proof, using the Uniformization Theorem. This proof will be easier in the setting of the "Primer" since the authors are considering universal covering spaces of complete ...
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  • 7,328
10 votes
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Mapping class group and pure mapping class group

I do not recall the conventions adopted in the Primer, but there is a wide difference between boundary components (which will be embedded circles or lines) and punctures (which are “missing points” ...
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  • 18.5k
9 votes

Homeomorphic but Non-Conjugate Mapping Tori

McMullen and Taubes 4-manifolds with inequivalent symplectic forms and 3-manifolds with inequivalent fibrations constructs 3-manifolds $N$ with different fibrations, whose Euler classes do not lie in ...
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9 votes
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Abelianization of mapping class groups $\Gamma_{g,n}$

The following statement can be found in Section 5 of Low-dimensional homology groups of mapping class groups: a survey: Theorem: Let $g \geq 1$. Then $$H_1(\Gamma_{g,r}^n,\mathbb{Z}) \simeq \left\{ ...
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  • 6,053
9 votes
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The largest group acting on a non-orientable surface of genus 5

The group $F$ is isomorphic to the symmetric group $S_5$. In fact, since $N_5$ is non-orientable of genus $5$, both $F$ and the extended group $F^*$ (of order twice the order of $F$) act on its ...
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8 votes

Realizing braid group by homeomorphisms

Hot off the presses: a new preprint by Nick Salter and Bena Tshishiku proves that the braid groups cannot be realized by diffeomorphisms for $n \geq 5$.
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  • 39.8k
8 votes
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mapping class group relations

I can answer your last question: there exist natural ways of embedding the fundamental group of the unit tangent bundle of a surface into the mapping class group, and the lantern relation is the image ...
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  • 39.8k
8 votes

Compact manifolds with big mapping class group

There are situations in which surfaces are the "unique" examples with big mapping class groups. One such is closed manifolds of negative sectional curvature. Theorem (Paulin): If $M$ is a closed $n$-...
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  • 22.3k
8 votes
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Categorical mapping class group action

[This is an elaboration of parts of Mark Penney's answer] A natural source of categorical actions of the mapping class group is the category assigned by any 4d TFT to a surface. Such categories are ...
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8 votes
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Well definedness of square roots of separating Dehn Twists

They are different. In fact, they act differently on $H_1(\Sigma_2;\mathbb{Z})$. Let $V \subset H_1(\Sigma_2;\mathbb{Z})$ be the span of the homology classes of $c_1$ and $c_2$, and let $W \subset ...
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8 votes

Is there a Morita cocycle for the mapping class group Mod(g,n) when n > 1?

The answer is "yes" -- in fact one can do better and get a class in $$H^1(\text{Aut}(F_m), \text{Hom}(H, \wedge^2 H)),$$ where $F_m$ is the free group on $m$ generators and $H$ is the ...
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  • 20.5k
8 votes
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Minimal number of (Dehn twists) generators of the mapping class group of a marked sphere

The minimum number of Dehn twist generators (and in fact the minimum number of generators of any kind) for $\Gamma_{0,n}$ is ${n-1 \choose 2} - 1$. Here's why. A presentation for $\Gamma_{0,n}$ is ...
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8 votes

$ \mathbb{R}P^n $ bundles over the circle

No. Every smooth bundle over $S^1$ with fiber $M$ is the mapping torus of some diffeomorphism $f:M\to M$. Isomorphism classes of bundles correspond to conjugacy classes in the group of isotopy classes ...
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