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There are many ways to embed the braid group into the mapping class group of a surface. To describe one of them, let ${C}_{2g+2}(\mathbb{D}^2)$ be the configuration of unordered $2g+2$ points in the unit disk and let $\mathcal{M}_{g,2}$ be the moduli space of connected Riemann surfaces of genus $g$ with two ordered and parametrized boundary components. There is a map

$$ \psi: {C}_{2g+2}(\mathbb{D}^2)\to \mathcal{M}_{g,2} $$ which sends $2g+2$ points ${\bf z}= \{z_1,z_2,\dots, z_{2g+2}\}$ to the Riemann surface $\Sigma_{{\bf z}}$ given by $$ f_{{\bf z}}(z)^2 =\prod_{i=1}^{2g+2} (z-z_i) $$ which gives a branch cover over the unit disk with ${\bf z}$ as branched points. We consider the map that is induced on the fundamental groups $$ \phi: \mathrm{Br}_{2g+2}\to \mathrm{Mod}(\Sigma_{g,2}). $$ Tillmann and Song proved that $\phi$ induces the trivial map in the stable homology.

Motivated by the Lie theoretic version of Margulis Superrigidity which asserts that homomorphisms between lattices (virtually) extend to homomorphisms of the ambient groups, Aramayona and Suoto in asked if the same is true for group homomorphisms between mapping class groups. In our case, the question becomes

Question: Is it known whether the group homomorphism $\phi$ is induced by a homomorphism $\Phi$ between diffeomorphism groups, in other words does there exist a homomorphim $\Phi$ that makes the following diagram commutative

$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}$ $$ \begin{array}{c} \mathrm{Diff}^{\delta}(\mathbb{D}^2- (2g+2) \text{ points},\partial\mathbb{D}^2) & \ra{\Phi} & \mathrm{Diff}^{\delta}(\Sigma_{g,2},\partial\Sigma_{g,2}) \\\da{} & & \da{} \\ \mathrm{Br}_{2g+2} & \ra{\phi} & \mathrm{Mod}_{g,2} \end{array}, $$ where $\mathrm{Diff}^{\delta}(\mathbb{D}^2- (2g+2) \text{ points},\partial\mathbb{D}^2)$ denotes the discrete group of diffeomorphisms of the punctured disk which are identity near $\partial\mathbb{D}^2$ and $\mathrm{Diff}^{\delta}(\Sigma_{g,2},\partial\Sigma_{g,2}) $ denotes the discrete group of diffeomorphisms of the surface $\Sigma_{g,2}$ of genus $g$ and $2$ boundary components?

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The real question is: does every (virtual) homomorphism of mapping class groups have a “geometric meaning”? Your example is defined by geometry, so the answer is yes. Aramayona and Souto make “geometric meaning” precise by defining it as (virtually) extending to the diffeomorphism group. So they are implying that your example should easily extend. And it does.

Such ramified space constructions work just as well for the diffeomorphism group as for the mapping class group. They all give virtual homomorphisms, which is all that Aramayona and Souto ask. It takes some work to figure out in which cases they give actual homomorphisms, but the answer is the same for the mapping class group and the diffeomorphism group. Your construction does not seem to include that extra work, which may be fooling you into thinking that the mapping class group case is easier than the diffeomorphism case.

Your construction is just taking the double cover ramified in the specified points. This is a coordinate-free construction, so it ought to work just as well in the smooth category as in the holomorphic category. That is, if it gave you a map of holomorphic moduli stacks, it should give you a map of stacks of $C^\infty$ surfaces. But that is just a functor between groupoids, which is pretty much the desired homomorphism. However, maps between moduli stacks are not quite that easy to write down. You cannot associate to a surface “the” double cover because the cover has automorphisms. In your example, there is a real homomorphism of mapping class groups, but in other examples, it is only a virtual homomorphism. For example, the hyperelliptic mapping class group is not a genus 0 mapping class group, but a nontrivial central extension thereof: $M_{0,2g+2}\leftarrow H_g\to M_g$.

It is difficult to construct maps between moduli spaces, but fairly easy to construct such zig-zags or correspondences. For any finite map of surfaces $\Sigma’\to\Sigma$, there is a moduli of such maps $M_{\Sigma’\to\Sigma}$, a diffeomorphism group $Diff(\Sigma’\to\Sigma)$, and a mapping class group. The forgetful map from the relative object to the downstairs object is almost an isomorphism. For the diffeomorphism group and the mapping class group, they are virtual isomorphisms, while in the case of the moduli space, it is a finite covering space. Thus they give virtual homomorphisms from the groups associated to $\Sigma$ to the groups associated to $\Sigma’$.

More precisely, let $\Sigma’\to\Sigma$ be a finite map of surfaces, in some category: algebraic, holomorphic, or just smooth. They could have decorations, like a tangent vector or the boundary you ask, but I will suppress that in the notation. Let $D\subset \Sigma$ be the branch locus. We will produce zig-zags $M_{\Sigma,D}\leftarrow M_{\Sigma’\to\Sigma}\to M_{\Sigma’}$ and $Diff(\Sigma,D)\leftarrow Diff(\Sigma’\to\Sigma)\to Diff(\Sigma’)$. Here $M_{\Sigma,D}$ is the moduli of curves of the same genus as $\Sigma$ marked by $D$ and anything else suppressed in the notation (such as the boundary in your case). If $\Sigma’$ is unmarked and $n$ is the size of $D$, then $M_{\Sigma,D}=M_{g,n}$. The upstairs object $M_{\Sigma’}$ could be the moduli of curves of the genus of $\Sigma’$ with or without a marking of size the preimage of $D$. The most structure is to include a marking, but people usually prefer to compose with the forgetful map.

The key ingredient is $M_{\Sigma’\to\Sigma}$. This is just what is has to be to fit in the middle: it is the moduli of pairs of surfaces with a map and appropriate branching structure. The maps are simply forgetting $\Sigma$ or $\Sigma’$. Similarly, $Diff(\Sigma’\to\Sigma)$ is the group of pairs of a diffeomorphism of $\Sigma’$ covering a diffeomorphism of $\Sigma$. Thus the diffeomorphism of $\Sigma$ must preserve $D$ and the diffeomorphism of $\Sigma’$ must preserve the inverse image of $D$.

The key theorem is that $Diff(\Sigma’\to\Sigma)\to Diff(\Sigma,D)$ is a virtual isomorphism. That is, its kernel and index are both finite. This divides into three parts: that the kernel is finite; that the map on the connected component is surjective; and that the map on mapping class groups is a virtually surjective. The kernel consists of diffeomorphisms of $\Sigma’$ that cover the identity of $\Sigma$. They are nontrivial because they act on the fibers of the map. The generic fibers are all “the same” so the kernel acts faithfully by permutations on a single fiber, which is a finite set, so the kernel is itself finite. The connected component of the diffeomorphism group is a Lie group with Lie algebra consisting of vector fields that vanish at the marked points. That is, it is generated by short flows along such vector fields. Vector fields on $\Sigma$ that vanish at $D$ lift to vector fields on $\Sigma’$, showing surjectivity (thence isomorphism).

This reduces to understanding the corresponding map on mapping class groups. After deleting the branch locus $D$ and its preimage $D’$, the map $\Sigma’-D’\to\Sigma-D$ becomes a finite covering space. The basic idea is that this corresponds to a subgroup of $\pi_1(\Sigma-D)$ of finite index $i$. Since the fundamental group is finitely generated, there are only finitely many subgroups of index $i$. A mapping class of $(\Sigma,D)$ cannot lift to a relative mapping class if it does not preserve the subgroup of the covering space, but this is the only obstruction. This is not quite right because we need a basepoint to talk about fundamental groups. I included the step about surjectivity of the connected component diffeomorphism group just to make it easier to talk about basepoints; the alternative is to work with groupoids and multiple basepoints. Now, more rigorously: take a mapping class of $(\Sigma,D)$. Compose with a diffeomorphism in the connected component so that it preserves the basepoint. It now induces a self-map on the fundamental group $\pi(\Sigma,D)$. If that does not preserve the conjugacy class of the subgroup corresponding to the cover $\Sigma’-D’$, then it cannot lift. If it does, find a path of diffeomorphism of $\Sigma-D$ that moves the basepoint around the conjugating path. The path exhibits the final diffeomorphism as being in the connected component of $Diff(\Sigma-D)$ (but not $Diff(\Sigma-D,*)$). Compose with the final diffeomorphism to get a diffeomorphism of $\Sigma-D$ that is in the same mapping class as the original, but now preserves the subgroup corresponding to the cover (and not just its conjugacy class). By the standard theorems about covering spaces, the two covering spaces are isomorphic, exhibiting the covering diffeomorphism of $\Sigma’-D’$, thence of $\Sigma’$.

Since $Diff(\Sigma’\to\Sigma)\to Diff(\Sigma,D)$ is a virtual isomorphism, $Diff(\Sigma,D)\leftarrow Diff(\Sigma’\to\Sigma)\to Diff(\Sigma’)$ is a virtual homomorphism corresponding to the virtual homomorphism on mapping class groups. That is the most that is true in general.

In your case, this can be promoted to an actual homomorphism, both for diffeomorphism group and thence mapping class group. First, $Diff(\Sigma’\to\Sigma)\to Diff(\Sigma,D)$ is surjective. As in the discussion of surjectivity above, this must be because there is a unique subgroup of $\pi_1(\Sigma-D)$ of index 2 that does not contain the loop about any puncture. To put it another way, there is a unique double cover of the plane with specified ramification. This is not true in higher genus, because the fundamental group is not generated by the punctures. Specifically, the set of covers is a torsor over $H^1(\Sigma;\mathbb Z/2)$. So in the genus 0 case $Diff(\Sigma’\to\Sigma)\to Diff(\Sigma,D)$ is surjective, but it is not an isomorphism; its kernel is $\mathbb Z/2$, the diffeomorphism that swaps the two sheets. In some examples, such as the hyperelliptic group, the extension is nontrivial, but in this example it is split by the homomorphism $Diff(\Sigma’\to\Sigma)\to \mathbb Z/2$ that detects whether the diffeomorphism permutes the two boundary components of $\Sigma’$. Thus the kernel of this homomorphism is isomorphic to $Diff(\Sigma,D)$, but it is a subgroup of $Diff(\Sigma’\to\Sigma)$ and thus of $Diff(\Sigma’)$, so it gives a homomorphism $Diff(\Sigma,D)\to Diff(\Sigma’)$.

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  • $\begingroup$ I am not sure if I understand the idea. It is clear that $\phi$ can be extended to a group homomorphism from $\mathrm{Diff}^{\delta}(\mathbb{D}^2- (2g+2) \text{ points},\partial\mathbb{D}^2)$ to $\mathrm{Homeo}^{\delta}(\Sigma_{g,2},\partial\Sigma_{g,2})$ but it is not clear that it can be extended to even $C^1$-diffeomorphisms. By covering space theory from every element of $\mathrm{Diff}^{\delta}(\mathbb{D}^2- (2g+2) \text{ points},\partial\mathbb{D}^2)$, we obtain a diffeomorphism of $\Sigma_{g,2}-\text{ramification locus}$ $\endgroup$ – Sam Nariman Jan 15 '16 at 10:12
  • $\begingroup$ but I am almost sure that it ought not be possible to extend the diffeomorphism to the ramification locus. I appreciate if you explain how you overcome this issue. $\endgroup$ – Sam Nariman Jan 15 '16 at 10:13
  • $\begingroup$ @SamNariman Let me try that again. The homeomorphism is canonical, so it will preserve be a diffeomorphism if the smooth structure is canonical. I claim that a ramified cover of a surface inherits a smooth structure from the smooth structure downstairs. Let me specialize to a double cover. A function upstairs is the sum of a function that descends to downstairs and an “odd” function that takes negated values on the two sheets. Define the smooth structure by saying that an odd function is smooth if its square (which descends) is smooth. $\endgroup$ – Ben Wieland Jan 21 '16 at 23:18

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