22
votes

### Fixed-point free diffeomorphisms of surfaces fixing no homology classes

Here's an example with $g=2$. Let $T$ be the torus $\mathbb C/L$, where $L$ is the lattice spanned by $1$ and $\zeta=e^{2\pi i/6}$. Let $f:T\to T$ be induced by multiplication by $\zeta$. This is a ...

18
votes

Accepted

### On trivial mapping class group of 3-manifolds

Dave Gabai proved that the mapping class group of a closed hyperbolic 3-manifold is isomorphic to its isometry group. For a hyperbolic knot $K$ without any symmetries, for large enough $n$, $S^3_{1/n}(...

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 ...

16
votes

Accepted

### 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 ...

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 ...

15
votes

Accepted

### 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{...

15
votes

Accepted

### 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 ...

15
votes

Accepted

### Fixed-point free diffeomorphisms of surfaces fixing no homology classes

Goodwillie's construction (in genus two) generalises to all higher genus as follows.
Let $P_n$ be the regular $n$-gon in the plane with vertices at roots of unity. When $n$ is even, we can glue ...

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 $...

13
votes

Accepted

### 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 ...

13
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, ...

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 ...

12
votes

Accepted

### 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 ...

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 ...

11
votes

Accepted

### 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” ...

10
votes

Accepted

### 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 $A_5$ ...

10
votes

Accepted

### 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 ...

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 ...

9
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 ...

9
votes

Accepted

### 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 ...

9
votes

Accepted

### 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 ...

9
votes

### Mapping Out(F_n) to the mapping class group

As pointed out by YCor, $\mathrm{Out}(F_g)$ is not linear (for $g \geq 3$). Also, the linearity of $\mathrm{Mod}(S_g)$ is unknown. So the existence of such an embedding would solve a long-standing ...

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 $n$-...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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\{ ...

8
votes

Accepted

### Why is the mapping class group of a surface with nonempty boundary torsion-free?

I think the reason that Dehn twists enter is that we can take the differential of a (orientation-preserving) diffeomorphism $f$ of $S_g$ that fixes a chosen basepoint $\ast \in S_g$ at this point, ...

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 ...

8
votes

Accepted

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

Your answer is correct if appropriately understood, but it's a little subtle. Here I should note that I'm interpreting your question as a purely homotopy theoretic one (in particular ignoring smooth ...

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