#### by Dan Margalit and Rebecca R. Winarski

#### 1. Introduction

In the early 1970s Joan Birman and
Hugh Hilden wrote a series of
now-classic papers on the interplay between mapping class groups and
covering spaces. The initial goal was to determine a presentation for
the mapping class group of __\( S_2 \)__, the closed surface of genus two (it
was not until the late 1970s that Hatcher and Thurston
[e10]
developed an approach for finding explicit presentations for mapping
class groups).

The key innovation by Birman and Hilden is to relate the mapping class
group __\( \operatorname{Mod}(S_2) \)__ to the mapping class group of
__\( S_{0,6} \)__, a sphere with six marked points. Presentations for
__\( \operatorname{Mod}(S_{0,6}) \)__ were already known since that group is
closely related to a braid group.

The two surfaces __\( S_2 \)__ and __\( S_{0,6} \)__ are related by a two-fold
branched covering map __\( S_2 \to S_{0,6} \)__,
as in the figure below.

The six marked points in the base are branch points. The deck
transformation is called the *hyperelliptic involution*
of __\( S_2 \)__, and we denote it by __\( \iota \)__. Every element of
__\( \operatorname{Mod}(S_2) \)__ has a representative that commutes with
__\( \iota \)__, and so it follows that there is a map
__\[
\Theta : \operatorname{Mod}(S_2) \to \operatorname{Mod}(S_{0,6}).
\]__
The kernel of __\( \Theta \)__ is the cyclic group of order two generated by (the homotopy class of) the involution __\( \iota \)__. One can check that each generator for __\( \operatorname{Mod}(S_{0,6}) \)__ lifts to __\( \operatorname{Mod}(S_2) \)__ and so __\( \Theta \)__ is surjective. From this we have a short exact sequence
__\[ 1 \to \langle \iota \rangle \to \operatorname{Mod}(S_2)
\xrightarrow{\Theta} \operatorname{Mod}(S_{0,6}) \to 1,\]__
and hence
a presentation for __\( \operatorname{Mod}(S_{0,6}) \)__ can be lifted to a
presentation for __\( \operatorname{Mod}(S_2) \)__.

But wait — the map __\( \Theta \)__ is not a priori well defined! The problem
is that elements of __\( \operatorname{Mod}(S_2) \)__ are only defined up to isotopy, and
these isotopies are not required to respect the hyperelliptic
involution. The first paper by Birman and Hilden proves that in fact
all isotopies can be chosen to respect the involution. Birman and
Hilden quickly realized that the theory initiated in that first paper
can be generalized in various ways, and they wrote a series of papers
on the subject, culminating in the paper
*On isotopies of homeomorphisms of Riemann surfaces* [8], published in
*Annals of Mathematics* in 1973.

In the remainder of this article, we will discuss the history of the Birman–Hilden theory, including generalizations by Maclachlan–Harvey and the second author of this article, we will give several applications, explain three proofs, and discuss various open questions and new directions in the theory. As we will see, the Birman–Hilden theory has had influence on many areas of mathematics, from low-dimensional topology, to geometric group theory, to representation theory, to algebraic geometry and more, and it continues to produce interesting open problems and research directions.

##### The other article by Birman and Hilden

Before getting
on with our main business, we would be remiss not to mention the other
paper by Birman and Hilden [9], the 1975 paper
*Heegaard splittings of branched coverings of *__\( S^3 \)__, published in
*Transactions of the American Mathematical Society* (there is
also the corresponding research announcement,
*The homeomorphism problem for *__\( S^3 \)__, published two years earlier [7]). In
this paper, Birman and Hilden discuss the relationship between
branched covers and Heegaard splittings of 3-manifolds. Their results
cover a lot of territory. For instance:

they prove that every closed, orientable 3-manifold of Heegaard genus 2 is a two-fold branched covering space of

__\( S^3 \)__branched over a 3-bridge knot or link;they give an algorithm for determining if a Heegaard splitting of genus two represents

__\( S^3 \)__;they prove that any simply connected two-fold cover of

__\( S^3 \)__branched over the closure of a braid on three strands is itself__\( S^3 \)__; andthey disprove a conjecture of Haken that among all simply connected 3-manifolds, and among all group presentations for their fundamental groups arising from their Heegaard splittings, the presentations for

__\( \pi_1(S^3) \)__have a certain nice property.

While this paper has also been influential and well-cited, and in fact relies on their work on surfaces, we will restrict our focus in this article to the work of Birman and Hilden on mapping class groups.

#### 2. Statements of the main theorem

Let __\( p: S \to X \)__ be a covering map of surfaces, possibly branched,
possibly with boundary. We say that __\( f : S \to S \)__ is
*fiber preserving* if for each __\( x \in X \)__ there is a __\( y \in X \)__ so that
__\[ f(p^{-1}(x)) = p^{-1}(y) ;\]__
in other words, as the terminology
suggests, __\( f \)__ takes fibers to fibers.

Given two homotopic fiber-preserving homeomorphisms of __\( S \)__, we can ask
if they are homotopic through fiber-preserving homeomorphisms. If the
answer is yes for all such pairs of homeomorphisms, we say that the
covering map __\( p \)__ has the *Birman–Hilden property*. An equivalent
formulation of the Birman–Hilden property is: whenever a
fiber-preserving homeomorphism is homotopic to the identity, it is
homotopic to the identity through fiber-preserving homeomorphisms.

We are now ready to state the main theorems of the Birman–Hilden
theory. There are several versions, proved over the years by various
authors, each generalizing the previous. The first version is the one
that appears in the aforementioned 1973 *Annals of Mathematics*
paper by Birman and Hilden and also in the accompanying research
announcement *Isotopies of Homeomorphisms of Riemann surfaces*
[5]. Throughout, we will say that a surface is
*hyperbolic* if its Euler characteristic is negative.

Theorem 2.1 [Birman–Hilden]:
Let __\( p : S \to X \)__ be a finite-sheeted regular branched covering map
where __\( S \)__ is a hyperbolic surface. Assume that __\( p \)__ is either
unbranched or is solvable. Then __\( p \)__ has the Birman–Hilden property.

If we apply Theorem 2.1 to the branched covering map __\( S_2 \to
S_{0,6} \)__ described earlier, then it exactly says that the map
__\[ \Theta: \operatorname{Mod}(S_2) \to \operatorname{Mod}(S_{0,6}) \]__
is well defined.

It is worthwhile to compare our Theorem 2.1 to what is actually
stated by Birman and Hilden. In their paper, they state two theorems,
each of which is a special case of Theorem 2.1. Their Theorem 1
treats the case of regular covers where each deck transformation fixes
each preimage of each branch point in __\( X \)__. This clearly takes care of
the case of unbranched covers, and also the case of certain solvable
branched covers (on one hand a finite group of homeomorphisms of a
surface that fixes a point must be a subgroup of a dihedral group, and
on the other hand there are solvable — even cyclic — branched covers
that do not satisfy the condition of Theorem 1). Birman and Hilden’s
Theorem 2 deals with the general case of solvable covers, which
includes some unbranched covers.

In early 1973
Maclachlan and
Harvey
[e5]
published a paper called
*On Mapping Class Groups and Covering Spaces*, in which they give
the following generalization of Theorem 2.1.

Theorem 2.2 [Maclachlan–Harvey]:
Let __\( p : S \to X \)__ be a finite-sheeted regular branched covering map
where __\( S \)__ is a hyperbolic surface. Then __\( p \)__ has the Birman–Hilden
property.

Maclachlan and Harvey’s work was contemporaneous with the work of Birman and Hilden cited in Theorem 2.1, and was subsequent to the original paper by Birman and Hilden on the hyperelliptic case. Their approach is completely different, and is framed in terms of Teichmüller theory.

The 2014 Ph.D. thesis of the second author of this article is a
further generalization
[e38].
For the statement, a preimage
of a branch point is *unramified* if some small neighborhood is
mapped injectively under the covering map, and a cover is
*fully ramified* if no branch point has an unramified preimage.

Theorem 2.3 [Winarski]:
Let __\( p: S \to X \)__ be a finite-sheeted branched covering map where __\( S \)__
is a hyperbolic surface, and suppose that __\( p \)__ is fully ramified.
Then __\( p \)__ has the Birman–Hilden property.

Note that all regular covers are fully ramified and also that all
unbranched covers are fully ramified. Thus Theorem 2.3 indeed
implies Theorems 2.1 and 2.2. In Section 2.3 of her paper,
Winarski gives a general construction of irregular branched covers
that are fully ramified. Thus there are many examples of covering
spaces that satisfy the hypotheses of Theorem 2.3 but not those
of Theorem 2.2.
We will briefly remark on the assumption that __\( S \)__ is hyperbolic. It is
not hard to construct counterexamples in the other cases. For instance
suppose __\( S \)__ is the torus __\( T^2 \)__ and __\( p : S \to X \)__ is the branched cover
corresponding to the hyperelliptic involution of __\( T^2 \)__. In this case
__\( X \)__ is the sphere with four marked points. Rotation of __\( T^2 \)__ by __\( \pi \)__
in one __\( S^1 \)__-factor is a fiber-preserving homeomorphism homotopic to
the identity, but the induced homeomorphism of __\( X \)__ acts nontrivially
on the marked points and hence is not homotopic to the identity. Thus
this cover fails the Birman–Hilden property. One can construct a
similar example when __\( S \)__ is the sphere __\( S^2 \)__ and __\( p : S^2 \to X \)__ is
the branched cover induced by a finite order rotation.

#### 3. Restatement of the main theorem

We will now give an interpretation of the Birman–Hilden
property — hence all three theorems above — in terms of mapping class
groups. Here, the *mapping class group* of a surface is the group
of homotopy classes of orientation-preserving homeomorphisms that fix
the boundary pointwise and preserve the set of marked points
(homotopies must also fix the boundary and preserve the set of marked
points).

Let __\( p: S \to X \)__ be a covering map of surfaces, possibly branched. We
treat each branch point in __\( X \)__ as a marked point, and so
homeomorphisms of __\( X \)__ are assumed to preserve the set of branch
points. Let __\( \operatorname{LMod}(X) \)__ denote the subgroup of the mapping class group
__\( \operatorname{Mod}(X) \)__ consisting of all elements that have representatives that
lift to homeomorphisms of __\( S \)__. This group is called the
*liftable mapping class group* of __\( X \)__.

Let __\( \operatorname{SMod}(S) \)__ denote the subgroup of __\( \operatorname{Mod}(S) \)__ consisting of the
homotopy classes of all fiber-preserving homeomorphisms. Here we
emphasize that two homeomorphisms of __\( S \)__ are identified in __\( \operatorname{SMod}(S) \)__
if they differ by an isotopy that is not necessarily fiber preserving
(so that we have a subgroup of __\( \operatorname{Mod}(S) \)__). We also emphasize that
preimages of branch points are not marked. Fiber-preserving
homeomorphisms are also called *symmetric homeomorphisms*; these
are exactly the lifts of liftable homeomorphisms of __\( X \)__. The group
__\( \operatorname{SMod}(S) \)__ is called the *symmetric mapping class group* of __\( S \)__.

Let __\( D \)__ denote the subgroup of __\( \operatorname{SMod}(S) \)__ consisting of the homotopy
classes of the deck transformations (it is a fact that nontrivial deck
transformations represent nontrivial mapping classes).

Proposition 3.1:
Let __\( p : S \to X \)__ be a finite-sheeted branched covering map where __\( S \)__ is a hyperbolic surface without boundary. Then the following are equivalent:

__\( p \)__has the Birman–Hilden property,the natural map

__\( \operatorname{LMod}(X) \to \operatorname{SMod}(S)/D \)__is injective,the natural map

__\( \operatorname{SMod}(S) \to \operatorname{LMod}(X) \)__is well defined, and__\( \operatorname{SMod}(S)/D \cong \operatorname{LMod}(X) \)__.

The proposition is straightforward to prove. The main content is the equivalence of the first two statements. The other statements, while useful in practice, are equivalent by rudimentary abstract algebra. Using the proposition, one obtains several restatements of Theorems 2.1, 2.2, and 2.3 in terms of mapping class groups.

Birman and Hilden also proved that for a regular cover
__\( \operatorname{SMod}(S) \)__ is the normalizer in __\( \operatorname{Mod}(S) \)__
of the deck group __\( D \)__ (regarded as a subgroup of
__\( \operatorname{Mod}(S) \)__), and so we can also write the last statement
in Proposition 3.1 as
__\[
N_{\operatorname{Mod}(S)}(D) / D \cong \operatorname{LMod}(X).
\]__
Birman and Hilden only stated the result about normalizers in the case
where the deck group is cyclic. However, by combining their argument
with Kerckhoff’s resolution of the Nielsen realization problem
[e9]
one obtains the more general version.

There is also a version of Proposition 3.1 for surfaces with
boundary. Since the mapping class group of a surface with boundary is
torsion free, the deck transformations do not represent elements of
__\( \operatorname{Mod}(S_g) \)__. And so in this case we can simply replace __\( D \)__ with the
trivial group. For example, in the presence of boundary the
Birman–Hilden property is equivalent to the statement that
__\[ \operatorname{SMod}(S) \cong \operatorname{LMod}(X) .\]__
This will become especially important in the
discussion of braid groups below.

#### 4. Application to presentations of ~~mapping class groups~~

The original work on the Birman–Hilden theory concerns the case of
the hyperelliptic involution and is reported in the 1971 paper
*On the mapping class groups of closed surfaces as covering spaces*
[2]. We will explain how Theorem 2.1 specializes in this case and helps to give presentations for the
associated symmetric mapping class group and the full mapping class
group in genus two.

Consider the covering space
__\[ S_g \to S_{0,2g+2} \]__
induced by a
hyperelliptic involution of __\( S_g \)__. In general a
*hyperelliptic involution* of __\( S_g \)__ is a homeomorphism of order two that acts by
__\( -I \)__ on __\( H_1(S_g;\mathbb{Z}) \)__; we remark that the hyperelliptic
involution is unique up to homotopy for __\( S_1 \)__ and __\( S_2 \)__ but there are
infinitely many distinct hyperelliptic involutions of __\( S_g \)__ when __\( g
\geq 3 \)__.

Theorem 2.1 and Proposition 3.1 give an isomorphism
__\[
\operatorname{SMod}(S_g) / \langle \iota \rangle \cong \operatorname{LMod}(S_{0,2g+2}).
\]__
In the special case of the hyperelliptic involution we have
__\[ \operatorname{LMod}(S_{0,2g+2}) = \operatorname{Mod}(S_{0,2g+2}) .\]__
Indeed, we can check directly
that each half-twist generator for __\( \operatorname{Mod}(S_{0,2g+2}) \)__ lifts to a Dehn
twist in __\( S_g \)__.

In the case __\( g=2 \)__ we further have
__\[
\operatorname{SMod}(S_2) = \operatorname{Mod}(S_2).
\]__
In other words, every mapping class of __\( S_2 \)__ is symmetric with respect to the hyperelliptic involution. The easiest way to see this is to note that each of the Humphries generators for __\( \operatorname{Mod}(S_2) \)__ is a Dehn twist about a curve that is preserved by the hyperelliptic involution. We thus have the following isomorphism:
__\[ \operatorname{Mod}(S_2) / \langle \iota \rangle \cong \operatorname{Mod}(S_{0,6}). \]__
Simple presentations for __\( \operatorname{Mod}(S_{0,n}) \)__ were found by
Magnus, and so from his presentation for __\( \operatorname{Mod}(S_{0,6}) \)__
Birman and Hilden use the above isomorphism to derive the following
presentation for __\( \operatorname{Mod}(S_2) \)__. The generators are the
Humphries generators for __\( \operatorname{Mod}(S_2) \)__, and we denote them
by __\( T_1,\dots \)__, __\( T_5 \)__. The relations are:
__\begin{align*}
[T_i,T_j]=1\phantom{,}& \quad\text{ for } |i-j| > 2, \\
T_iT_{i+1}T_i=T_{i+1}T_i T_{i+1}\phantom{,}& \quad \text{ for } 1 \leq i \leq 4, \\
(T_1T_2T_3T_4T_5)^6=1,& \\
(T_1T_2T_3T_4T_5T_5T_4T_3T_2T_1)^2=1,& \\
[T_1T_2T_3T_4T_5T_5T_4T_3T_2T_1,T_1]=1.&
\end{align*}__
The first two relations are the standard braid relations from __\( B_6 \)__, the next relation describes the kernel of the map
__\[ B_6 \to \operatorname{Mod}(S_{0,6}) ,\]__
and the last two relations come from the two-fold cover: the mapping class
__\[
T_1T_2T_3T_4T_5T_5T_4T_3T_2T_1
\]__
is the hyperelliptic involution. This presentation is the culmination
of a program begun by
Bergau and
Mennicke
[e1],
who approached
the problem by studying the surjective homomorphism
__\[ B_6 \to\operatorname{Mod}(S_2) \]__
that factors through the map
__\[\operatorname{Mod}(S_{0,6}) \to \operatorname{Mod}(S_2)\]__
used here.

Birman used the above presentation to give a normal form for elements
of __\( \operatorname{Mod}(S_2) \)__ and hence a method for enumerating 3-manifolds of
Heegaard genus two [3].

As explained by Birman and Hilden, the given presentation for
__\( \operatorname{Mod}(S_2) \)__ generalizes to a presentation for __\( \operatorname{SMod}(S_g) \)__. The latter
presentation has many applications to the study of __\( \operatorname{SMod}(S_g) \)__. It
was used by
Meyer
[e3]
to show that if a surface bundle over
a surface has monodromy in __\( \operatorname{SMod}(S_g) \)__ then the signature of the
resulting 4-manifold is zero; see also the related work of
Endo
[e20].
Endo and
Kotschick
used the Birman–Hilden presentation
to show that the second bounded cohomology of __\( \operatorname{SMod}(S_g) \)__ is
nontrivial
[e25].
Also,
Kawazumi
[e18]
used it to
understand the low-dimensional cohomology of __\( \operatorname{SMod}(S_g) \)__.

In 1972 Birman and
Chillingworth
published the paper
*On the homeotopy group of a nonorientable surface* [4]. There, they
determine a generating set for the mapping class group (= homeotopy
group) of an arbitrary closed nonorientable surface using similar
ideas, namely, they exploit the associated orientation double cover
and pass information through the Birman–Hilden theorem from the
orientable case. They also find an explicit finite presentation for
the mapping class group of a closed nonorientable surface of genus
three, which admits a degree two cover by __\( S_2 \)__.

One other observation from the 1971 paper is that __\( \operatorname{Mod}(S_2) \)__ is both
a quotient of and a subgroup of __\( \operatorname{Mod}(S_{2,6}) \)__. To realize
__\( \operatorname{Mod}(S_2) \)__ as a quotient, we consider the map
__\[ \operatorname{Mod}(S_{2,6}) \to \operatorname{Mod}(S_2) \]__
obtained by forgetting the marked points/punctures; this is
a special case of the Birman exact sequence studied by Birman in her
thesis [1]. And to realize __\( \operatorname{Mod}(S_2) \)__ as a subgroup,
we use the Birman–Hilden theorem: since every element of __\( \operatorname{Mod}(S_2) \)__
has a symmetric representative that preserves the set of preimages of
the branch points in __\( S_{0,6} \)__ and since isotopies between symmetric
homeomorphisms can also be chosen to preserve this set of six points,
we obtain the desired inclusion. Birman and Hilden state that “the
former property is easily understood but the latter much more
subtle.” As mentioned by
Mess
[e13],
the inclusion
__\[\operatorname{Mod}(S_2)\to \operatorname{Mod}(S_{2,6})\]__
can be rephrased as describing a multisection of
the universal bundle over moduli space in genus two.

#### 5. More applications to the genus-two ~~mapping class group~~

In the previous section we saw how the Birman–Hilden theory allows us to transport knowledge about the mapping class group of a punctured sphere to the mapping class group of a surface of genus two. As the former are closely related to braid groups, we can often push results about braid groups to the mapping class group. Almost every result about mapping class groups that is special to genus two is proved in this way.

A prime example of this is the result of
Bigelow and
Budney
[e26]
and
Korkmaz
[e22]
which states that __\( \operatorname{Mod}(S_2) \)__ is linear, that is,
__\( \operatorname{Mod}(S_2) \)__ admits a faithful representation into __\( \operatorname{GL}_N(\mathbb{C}) \)__
for some __\( N \)__. Bigelow and
Krammer independently proved that braid
groups were linear, and so the main work is to derive from this the
linearity of __\( \operatorname{Mod}(S_{0,n}) \)__. They then use the isomorphism
__\[ \operatorname{Mod}(S_2)/ \langle \iota \rangle \cong \operatorname{Mod}(S_{0,6}) \]__
to push the linearity up
to __\( \operatorname{Mod}(S_2) \)__.

A second example is from the thesis of
Whittlesey, published in 2000.
She showed that __\( \operatorname{Mod}(S_2) \)__ contains a normal subgroup where every
nontrivial element is pseudo-Anosov
[e21].
The starting
point is to consider the Brunnian subgroup of __\( \operatorname{Mod}(S_{0,6}) \)__. This is
the intersection of the kernels of the six forgetful maps
__\[ \operatorname{Mod}(S_{0,6}) \to \operatorname{Mod}(S_{0,5}) ,\]__
so it is obviously normal in
__\( \operatorname{Mod}(S_{0,6}) \)__. She shows that all nontrivial elements of this group
are pseudo-Anosov and proves that the preimage in __\( \operatorname{Mod}(S_2) \)__ has a
finite-index subgroup with the desired properties.

We give one more example. In the 1980s, before the work of Bigelow and
Krammer,
Vaughan Jones
discovered a representation of the braid group
defined in terms of Hecke algebras
[e12].
As in the work of
Bigelow–Budney and Korkmaz, one can then derive a representation of
__\( \operatorname{Mod}(S_{0,2g+2}) \)__ and then — using the Birman–Hilden theory — of
__\( \operatorname{SMod}(S_g) \)__. When __\( g=2 \)__ we thus obtain a representation of
__\( \operatorname{Mod}(S_2) \)__ to __\( \operatorname{GL}_5(\mathbb{Z}[t,t^{-1}]) \)__. This representation was
used by
Humphries
[e14]
to show that the normal closure in
__\( \operatorname{Mod}(S_2) \)__ of the __\( k \)__-th power of a Dehn twist about a nonseparating
curve has finite index if and only if __\( |k| \leq 3 \)__.

There are many other examples, such as the computation of the
asymptotic dimension of __\( \operatorname{Mod}(S_2) \)__ by
Bell and
Fujiwara
[e28]
and
the determination of the minimal dilatation in
__\( \operatorname{Mod}(S_2) \)__ by
Cho and
Ham
[e29];
the list goes on, but so must we.

#### 6. Application to representations of the braid group

The Birman–Hilden theorem also gives a way to embed braid groups into mapping class groups. This is probably the most oft-used application of their results.

Let __\( S_g^1 \)__ the orientable surface of genus __\( g \)__ with one boundary
component and let __\( D_{2g+1} \)__ denote the closed disk with __\( 2g+1 \)__ marked
points in the interior. Consider the covering space __\( S_g^1 \to
D_{2g+1} \)__ induced by a hyperelliptic involution of __\( S_g^1 \)__. It is well
known that __\( \operatorname{Mod}(D_{2g+1}) \)__ is isomorphic to the braid group
__\( B_{2g+1} \)__. As in the closed case, it is not hard to see that
__\[ \operatorname{LMod}(D_{2g+1}) = \operatorname{Mod}(D_{2g+1}) \]__
(again, each of the standard
generators for __\( B_{2g+1} \)__ lifts to a Dehn twist).

One is thus tempted to conclude that
__\[ \operatorname{SMod}(S_g^1) / \langle \iota \rangle \cong B_{2g+1} .\]__
But this is not the right statement, since __\( \iota \)__ does not represent an element of __\( \operatorname{Mod}(S_g^1) \)__. Indeed, for surfaces with boundary we insist that homeomorphisms and homotopies fix the boundary pointwise. Therefore, the correct isomorphism is:
__\[ \operatorname{SMod}(S_g^1) \cong B_{2g+1}. \]__
The most salient aspect of this isomorphism is that there is an injective homomorphism
__\[ B_{2g+1} \to \operatorname{Mod}(S_g^1). \]__
The injectivity here is sometimes
attributed to Perron–Vannier
[e16].
It is possible that they were
the first to observe this consequence of the Birman–Hilden theorem
but the only nontrivial step is the Birman–Hilden theorem.

In the case of __\( g=1 \)__ the representation of __\( B_3 \)__ is onto __\( \operatorname{Mod}(S_1^1) \)__, and so
__\[ \operatorname{Mod}(S_1^1) \cong B_3. \]__
Similarly we have
__\[ \operatorname{Mod}(S_1^2) \cong B_4 \times \mathbb{Z}. \]__
The point here is that
__\( B_4 \)__ surjects onto __\( \operatorname{SMod}(S_1^2) \)__ and the latter is almost isomorphic
to __\( \operatorname{Mod}(S_1^2) \)__; the extra __\( \mathbb{Z} \)__ comes from the Dehn twist
about a single boundary component.

One reason that the embedding of __\( B_{2g+1} \)__ in __\( \operatorname{Mod}(S_g^1) \)__ is so important is that if we compose with the standard symplectic representation
__\[ \operatorname{Mod}(S_g^1) \to \operatorname{Sp}_{2g}(\mathbb{Z}) \]__
then we obtain a representation of the braid group
__\[ B_{2g+1} \to \operatorname{Sp}_{2g}(\mathbb{Z}) .\]__
This representation is called
the standard symplectic representation of the braid group. It is also
called the *integral Burau representation* because it is the only
integral specialization of the Burau representation besides the
permutation representation. The symplectic representation is obtained
by specializing the Burau representation at __\( t=-1 \)__, while the
permutation representation is obtained by taking __\( t=1 \)__.

The image of the integral Burau representation has finite index in the
symplectic group: it is an extension of the level two symplectic group
by the symmetric group on __\( 2g+1 \)__ letters. The projection onto the
symmetric group factor is the standard permutation representation of
the braid group. See
A’Campo’s paper
[e8]
for details.

The kernel of the integral Burau representation is known as the hyperelliptic Torelli group. This group is well studied, as it describes the fundamental group of the branch locus of the period mapping from Teichmüller space to the Siegel upper half-space; see, for instance, the paper [e37] by Brendle, Putman, and the first author of this article and the references therein.

There are plenty of variations on the given representation. Most important is that if we take a surface with two boundary components __\( S_g^2 \)__ and choose a hyperelliptic involution, that is, an order two homeomorphism that acts by __\( -I \)__ on the first homology of the surface, then the quotient is __\( D_{2g+2} \)__ and so we obtain an isomorphism:
__\[ \operatorname{SMod}(S_g^2) \cong B_{2g+2}.\]__
Also, since the
inclusions __\( S_g^1 \to S_{g+1} \)__ and __\( S_g^2 \to S_{g+1} \)__ induce
injections
__\[\operatorname{SMod}(S_g^1) \to \operatorname{Mod}(S_{g+1}) \]__
and
__\[\operatorname{SMod}(S_g^2) \to\operatorname{Mod}(S_{g+1}),\]__
we obtain embeddings of braid groups into
mapping class groups of closed surfaces.

In the 1971 paper Birman and Hilden discuss the connection with
representations of the braid group. They point out the related fact
that __\( B_{2g+2} \)__ surjects onto __\( \operatorname{SMod}(S_g) \)__ (this follows immediately
from their presentation for the latter). In the special case __\( g=1 \)__
this becomes the classical fact that __\( B_4 \)__ surjects onto
__\[\operatorname{SMod}(S_1) =\operatorname{Mod}(S_1) \cong \operatorname{SL}_2(\mathbb{Z}) .\]__
We can also derive this fact from
our isomorphism
__\[\operatorname{Mod}(S_1^1) \cong B_3,\]__
the famous surjection __\( B_4\to B_3 \)__, and the surjection
__\[\operatorname{Mod}(S_1^1) \to \operatorname{Mod}(S_1)\]__
obtained by
capping the boundary.

One useful application of the embeddings of braid groups in mapping class groups is that we can often transport relations from the former to the latter. In fact, almost all of the widely used relations in the mapping class group have interpretations in terms of braids. This is especially true in the theory of Lefschetz fibrations; see for instance the work of Korkmaz [e24] and Hamada [e43] and of Baykur and Van Horn-Morris [e41].

#### 7. Application to a question of Magnus

The last application we will explain is beautiful and unexpected. It is the resolution of a seemingly unrelated question of Magnus about braid groups.

As mentioned in the previous section, the braid group __\( B_n \)__ is isomorphic
to the mapping class group of a disk __\( D_n \)__ with __\( n \)__ marked points. Let us
write __\( D_n^\circ \)__ for the surface obtained by removing from __\( D_n \)__ the marked
points. There is then a natural action of __\( B_n \)__ on __\( \pi_1(D_n^\circ) \)__
(with base point on the boundary). The latter is isomorphic to the free
group __\( F_n \)__ on __\( n \)__ letters. Basic algebraic topology tells us that this
action is faithful. In other words, we have an injective homomorphism:
__\[
B_n \to \operatorname{Aut}(F_n).
\]__
This is a fruitful way to view the braid group; for instance, since the
word problem in __\( \operatorname{Aut}(F_n) \)__ is easily solvable, this gives a solution to
the word problem for __\( B_n \)__.

Let __\( F_{n,k} \)__ denote the normal closure in __\( F_n \)__ of the elements
__\( x_1^k,\dots \)__, __\( x_n^k \)__. The quotient __\( F_n/F_{n,k} \)__ is isomorphic to
the __\( n \)__-fold free product
__\[\mathbb{Z}/k\mathbb{Z} \ast \cdots \ast\mathbb{Z}/k\mathbb{Z} .\]__
Since the elements of __\( B_n \)__ preserve the set of
conjugacy classes __\( \{[x_1],\dots \)__, __\( [x_n]\} \)__, there is an induced homomorphism
__\[
B_n \to \operatorname{Aut}(F_n/F_{n,k}).
\]__
Let __\( B_{n,k} \)__ denote the image of __\( B_n \)__ under this map. Magnus asked:
*Is* __\( B_n \)__ *isomorphic to* __\( B_{n,k} \)__*?* In other words, is the map
__\[B_n \to \operatorname{Aut}(F_n/F_{n,k})\]__
injective?

In their *Annals* paper, Birman and Hilden answer Magnus’ question in the
affirmative. Here is the idea. Let __\( H_{n,k} \)__ denote the kernel of the map
__\[ F_n \to \mathbb{Z}/k\mathbb{Z}, \]__
where each generator of __\( F_n \)__ maps to 1. The covering space of __\( D_n^\circ \)__
corresponding to __\( H_{n,k} \)__ is a __\( k \)__-fold cyclic cover __\( S^\circ \)__. If we
consider a small neighborhood of one of the punctures in __\( D_n^\circ \)__,
the induced covering map is equivalent to the connected __\( k \)__-fold covering
space of __\( \mathbb{C} \setminus \{0\} \)__ over itself (i.e., the one induced
by __\( z \mapsto z^k \)__). As such, we can “plug in” to __\( S^\circ \)__ a total of
__\( n \)__ points in order to obtain a surface __\( S \)__ and a cyclic branched cover __\( S
\to D_n \)__. The fundamental group of __\( S^\circ \)__ is __\( H_{n,k} \)__ by definition.
It follows from Van Kampen’s theorem that
__\[\pi_1(S) \cong H_{n,k}/F_{n,k}.\]__
Indeed, a simple loop around a puncture in __\( S^\circ \)__ projects to a loop in
__\( D_n^\circ \)__ that circles the corresponding puncture __\( k \)__ times.

As in the case of the hyperelliptic involution, we can check directly that
each element of __\( B_n \)__ lifts to a fiber-preserving homeomorphism of __\( S \)__.
Therefore, to answer Magnus’ question in the affirmative it is enough to check
that the map
__\[B_n \to \operatorname{Aut} \pi_1(S)\]__
is injective. Suppose __\( b \in B_n \)__ lies
in the kernel. Then the corresponding fiber-preserving homeomorphism of __\( S \)__
is homotopic, hence isotopic, to the identity. By the Birman–Hilden theorem
(the version for surfaces with boundary), __\( b \)__ is trivial, and we are done.

Bacardit and
Dicks
[e30]
give a purely algebraic treatment of Magnus’
question; they credit the argument to
Crisp and
Paris
[e27].
Another
algebraic argument for the case of even __\( k \)__ was given by
D. L. Johnson
[e11].
Yet another combinatorial proof was given by Krüger
[e15].

#### 8. A famous (but false) proof of the ~~Birman–Hilden theorem~~

When confronted with the Birman–Hilden theorem, one might be tempted to
quickly offer the following easy proof: given the branched cover __\( p : S \to
X \)__, the fiber-preserving homeomorphism __\( f : S \to S \)__, the corresponding
homeomorphism __\( \bar f : X \to X \)__, and an isotopy __\( H : S \times I \to S \)__
from __\( f \)__ to the identity map, we can consider the composition __\( p \circ H \)__,
which gives a homotopy from __\( \bar f \)__ to the identity. Then, since homotopic
homeomorphisms of a surface are isotopic, there is an isotopy from __\( \bar f \)__
to the identity, and this isotopy lifts to a fiber-preserving isotopy from
__\( f \)__ to the identity. Quod erat demonstrandum.

This probably sounds convincing, but there are two problems. First of all,
the composition __\( p \circ H \)__ is really a homotopy between __\( p \circ f \)__ and __\( p \)__
which are maps from __\( S \)__ to __\( X \)__; since __\( H \)__ is not fiber preserving, there is
no way to convert this to a well-defined homotopy between maps __\( X \to X \)__.
The second problem is that __\( H \)__ might send points that are not preimages of
branch points to preimages of branch points; so even if we could project
the isotopy, we would not obtain a homotopy of __\( X \)__ that respects the
marked points.

In the next two sections will outline proofs of the Birman–Hilden theorem in various cases. The reader should keep in mind the subtleties uncovered by this false proof.

#### 9. The unbranched (= easy) case

Before getting to the proof of the Birman–Hilden theorem, we will warm up with the case of unbranched covers. This case is much simpler, as all of the subtlety of the Birman–Hilden theorem lies in the branch points. Still the proof is nontrivial, and later we will prove the more general case by reducing to the unbranched case.

In 1972 Birman and Hilden published the paper
*Lifting and projecting homeomorphisms* [6], which gives a quick proof of Theorem 2.1 in the case of regular unbranched covers. Following along the same lines,
Aramayona, Leininger, and
Souto generalized their proof to the case of
arbitrary (possibly irregular) unbranched covers
[e31].
We will now
explain their proof.

Let __\( p : S \to X \)__ be an unbranched covering space of surfaces, and let
__\( f : S \to S \)__ be a fiber-preserving homeomorphism that is isotopic to
the identity. Without loss of generality, we may assume that __\( f \)__ has a
fixed point. Indeed, if __\( f \)__ does not fix some point __\( x \)__, then we can push
__\( p(x) \)__ in __\( X \)__ by an ambient isotopy, and lift this isotopy to __\( S \)__ until __\( x \)__
is fixed. As a consequence, __\( f \)__ induces a well-defined action __\( f_\star \)__
on __\( \pi_1(S) \)__. Since __\( f \)__ is isotopic to the identity, __\( f_\star \)__ is the
identity. If __\( \bar f \)__ is the corresponding homeomorphism of __\( X \)__, then it
follows that __\( \bar f_\star \)__ is the identity on the finite-index subgroup
__\( p_\star(\pi_1(S)) \)__ of __\( \pi_1(X) \)__. From this, plus the fact that roots
are unique in __\( \pi_1(X) \)__, we conclude that __\( \bar f_\star \)__ is the identity.
By basic algebraic topology, __\( \bar f \)__ is homotopic to the identity, and
hence it is isotopic to the identity, which implies that __\( f \)__ is isotopic
to the identity through fiber-preserving homeomorphisms, as desired.

#### 10. Three (correct) proofs of the ~~Birman–Hilden theorem~~

In this section we present sketches of the proofs of all three versions of the Birman–Hilden theorem given in Section 2. We begin with the original proof by Birman and Hilden, which is a direct attack using algebraic and geometric topology. Then we explain the proof from Maclachlan and Harvey’s Teichmüller theoretic approach, and finally the combinatorial topology approach of the second author, which gives a further generalization.

##### The Birman–Hilden proof: ~~Algebraic and geometric topology~~

As in the
statement of Theorem 2.1, let __\( p : S \to X \)__ be a regular branched
covering space where __\( S \)__ is a hyperbolic surface. As in Theorem 1 of
the *Annals* paper by Birman and Hilden, we make the additional
assumption here that each deck transformation for this cover fixes each
preimage of each branch point in __\( X \)__. Theorem 2.1 will follow easily
from this special case. Let __\( f \)__ be a fiber-preserving homeomorphism of __\( S \)__
and assume that __\( f \)__ is isotopic to the identity.

Let __\( x \)__ be the preimage in __\( S \)__ of some branch point in __\( X \)__. The first
key claim is that __\( f(x)=x \)__
([8], Lemma 1.3).
Thus if we take the
isotopy __\( H \)__ from __\( f \)__ to the identity and restrict it to __\( x \)__, we obtain an
element __\( \alpha \)__ of __\( \pi_1(S,x) \)__. Birman and Hilden argue that __\( \alpha \)__
must be the trivial element. The idea is to argue that __\( \alpha \)__ is fixed by
each deck transformation (this makes sense since the deck transformations
fix __\( x \)__), and then to argue that the only element of __\( \pi_1(S) \)__ fixed by
a nontrivial deck transformation is the trivial one (to see this, regard
__\( \alpha \)__ as an isometry of the universal cover __\( \mathbb{H}^2 \)__ and regard
a deck transformation as a rotation of __\( \mathbb{H}^2 \)__).

Since __\( \alpha \)__ is trivial, we can deform it to the trivial loop, and by
extension we can deform the isotopy __\( H \)__ to another isotopy that fixes
__\( x \)__ throughout. Proceeding inductively, Birman and Hilden argue that __\( H \)__
can be deformed so that it fixes all preimages of branch points throughout
the isotopy. At this point, by deleting branch points in __\( X \)__ and their
preimages in __\( S \)__, we reduce to the unbranched case.

Finally, to prove their Theorem 2, which treats the case of solvable covers, Birman and Hilden reduce it to Theorem 1 by factoring any solvable cover into a sequence of cyclic covers of prime order. Such a cover must satisfy the hypotheses of their Theorem 1.

It would be interesting to use the Birman–Hilden approach to prove the more general theorem of Winarski. There is a paper by Zieschang from 1973 that uses similar reasoning to Birman and Hilden and recovers the result of Maclachlan and Harvey [e4].

##### Maclachlan and Harvey’s proof: __Teichmüller theory__

__
__

Let __\( p : S \to X \)__
be a regular branched covering space where __\( S \)__ is a hyperbolic surface.
We will give Maclachlan and Harvey’s argument for Theorem 2.2
and at the same time explain why the argument gives the more general result
of Theorem 2.3.
The mapping class group __\( \operatorname{Mod}(S) \)__ acts on the Teichmüller space __\( \operatorname{Teich}(S) \)__,
the space of isotopy classes of complex structures on __\( S \)__ (or conformal
structures on __\( S \)__, or hyperbolic structures on __\( S \)__, or algebraic structures
on __\( S \)__). Let __\( X^\circ \)__ denote the complement in __\( X \)__ of the set of branch
points. There is a map
__\[ \Xi : \operatorname{Teich}(X^\circ) \to \operatorname{Teich}(S) \]__
defined by
lifting complex structures through the covering map __\( p \)__ (one must apply
the removable singularity theorem to extend over the preimages of the
branch points).

The key point in the proof is that __\( \Xi \)__ is injective. One way to see
this is to observe that Teichmüller geodesics in __\( \operatorname{Teich}(X^\circ) \)__ map
to Teichmüller geodesics in __\( \operatorname{Teich}(S) \)__ of the same length. Indeed,
the only way this could fail would be if we had a Teichmüller geodesic
in __\( \operatorname{Teich}(X^\circ) \)__ where the corresponding quadratic differential had a
simple pole (= 1-pronged singularity) at a branch point and some preimage of
that branch point was unramified (1-pronged singularities are only allowed
at marked points, and preimages of branch points are not marked). This is
why the most natural setting for this argument is that of Theorem 2.3, namely, where __\( p \)__ is fully ramified.

Let __\( Y \)__ denote the image of __\( \Xi \)__. The symmetric mapping class group
__\( \operatorname{SMod}(S) \)__ acts on __\( Y \)__ and the kernel of this action is nothing other
than __\( D \)__. The liftable mapping class group __\( \operatorname{LMod}(X) \)__ acts faithfully on
__\( \operatorname{Teich}(X^\circ) \)__ and hence — as __\( \Xi \)__ is injective — it also acts faithfully
on __\( Y \)__. It follows immediately from the definitions that the images of
__\( \operatorname{SMod}(S) \)__ and __\( \operatorname{LMod}(X) \)__ in the group of automorphisms of __\( Y \)__ are equal.
It follows that __\( \operatorname{SMod}(S)/D \)__ is isomorphic to __\( \operatorname{LMod}(X) \)__, as desired.

##### Winarski’s proof:

##### Winarski’s proof: ~~Combinatorial topology~~

Let __\( p : S \to X \)__ be a
fully ramified branched covering space where __\( S \)__ is a hyperbolic surface.
To prove Theorem 2.3 we will show that
__\[ \Phi : \operatorname{LMod}(X) \to \operatorname{SMod}(S)/D \]__
is injective.

Suppose __\( f \in \operatorname{LMod}(X) \)__ lies in the kernel of __\( \Phi \)__. Let __\( \varphi \)__ be
a representative of __\( f \)__. Since __\( \Phi(f) \)__ is trivial we can choose a lift
__\( \tilde \varphi : S \to S \)__ that is isotopic to the identity; thus __\( \tilde
\varphi \)__ fixes the isotopy class of every simple closed curve in __\( S \)__.
The main claim is that __\( \varphi \)__ fixes the isotopy class of every simple
closed curve in __\( X \)__. From this, it follows that __\( f \)__ has finite order in
__\( \operatorname{LMod}(X) \)__. Since __\( \ker(\Phi) \)__ is torsion free
([e38], Prop 4.2),
the theorem will follow.

So let us set about the claim. Let __\( c \)__ be a simple closed curve in __\( X \)__, and
let __\( \tilde c \)__ be its preimage in __\( S \)__. By assumption __\( \tilde \varphi(\tilde
c) \)__ is isotopic to __\( \tilde c \)__ and we would like to leverage this to show
__\( \varphi(c) \)__ is isotopic to __\( c \)__. There are two stages to the argument:
first dealing with the case where __\( \varphi(c) \)__ and __\( c \)__ are disjoint, and
then in the case where they are not disjoint we reduce to the disjoint case.

If __\( \varphi(c) \)__ and __\( c \)__ are disjoint, then __\( \tilde \varphi( \tilde
c) \)__ and __\( \tilde c \)__ are disjoint. Since the latter are isotopic, they
cobound a collection of annuli __\( A_1,\dots \)__, __\( A_n \)__. Then, since orbifold
Euler characteristic is multiplicative under covers, we can conclude that
__\( p(\bigcup A_i) \)__ is an annulus with no branch points (branch points decrease
the orbifold Euler characteristic), and so __\( c \)__ and __\( \varphi(c) \)__ are isotopic.

We now deal with the second stage, where __\( \varphi(c) \)__ and __\( c \)__ are not
disjoint. In this case __\( \tilde \varphi(\tilde c) \)__ and __\( \tilde c \)__ are
not disjoint either, but by our assumptions they are isotopic in __\( S \)__.
Therefore, __\( \tilde \varphi(\tilde c) \)__ and __\( \tilde c \)__ bound at least one bigon.

Consider an innermost such bigon __\( B \)__. Since __\( B \)__ is innermost, __\( p(B) \)__ is an
innermost bigon bounded by __\( \varphi(c) \)__ and __\( c \)__ in __\( X \)__ (the fact that __\( B \)__
is innermost implies that __\( p|B \)__ is injective). If there were a branch
point in __\( p(B) \)__ then since __\( p \)__ is fully ramified, this would imply that
__\( B \)__ was a __\( 2k \)__-gon with __\( k > 1 \)__, a contradiction. Thus, we can apply an
isotopy to remove the bigon __\( p(B) \)__ and by induction we reduce to the case
where __\( \varphi(c) \)__ and __\( c \)__ are disjoint.

For an exposition of Winarski’s proof in the case of the hyperelliptic involution, see the book by Farb and the first author of this article [e33].

#### 11. Open questions and new directions

One of the most striking aspects of the Birman–Hilden story is the breadth of open problems related to the theory and the constant discovery of related directions. We mentioned a number of questions already. Perhaps the most obvious open problem is the following.

Question 11.1: Which branched covers of surfaces have the Birman–Hilden property?

Based on the discussion in Section 2 above, one might hope
that all branched coverings — at least where the cover is a hyperbolic
surface — have the Birman–Hilden property. However, this is not true.
Consider, for instance, the simple threefold cover __\( p : S_g \to X \)__,
where __\( X \)__ is the sphere with __\( 2g+4 \)__ branch points (this cover is unique).
As shown in
the figure below
we can find an essential curve __\( a \)__ in
__\( X \)__ whose preimage in __\( S_g \)__ is a union of three homotopically trivial
simple closed curves. It follows that the Dehn twist __\( T_a \)__ lies in the
kernel of the map
__\[ \operatorname{LMod}(X) \to \operatorname{SMod}(S_g) \]__
and so __\( p \)__ does not have the
Birman–Hilden property. See Fuller’s paper for further discussion of this
example and the relationship to Lefschetz fibrations
[e23].

Berstein and Edmonds generalized this example by showing that no simple cover of degree at least three over the sphere has the Birman–Hilden property [e7], and Winarski further generalized this by proving that no simple cover of degree at least three over any surface has the Birman–Hilden property [e38].

Having accepted the fact that not all covers have the Birman–Hilden
property, one’s second hope might be that a cover has the Birman–Hilden
property if and only if it is fully ramified. However, this is also false.
Chris Leininger
[e44]
has explained to us how to construct a
counterexample using the following steps. First, let __\( S \)__ be a surface
and let __\( z \)__ be a marked point in __\( S \)__. Let __\( p : \tilde S \to S \)__ be a
characteristic cover of __\( S \)__ and let __\( \tilde z \)__ be one point of the full
preimage __\( p^{-1}(z) \)__.
Ivanov and
McCarthy
[e19]
observed that there
is an injective homomorphism
__\[ \operatorname{Mod}(S,z) \to \operatorname{Mod}(\tilde S,p^{-1}(z)) \]__
where for each element of __\( \operatorname{Mod}(S,z) \)__, we choose the lift to __\( \tilde S \)__
that fixes __\( \tilde z \)__. Aramayona–Leininger–Souto
[e31]
proved that
the composition of the Ivanov–McCarthy homomorphism with the forgetful
map
__\[ \operatorname{Mod}(\tilde S,p^{-1}(z)) \to \operatorname{Mod}(\tilde S, \tilde z) \]__
is injective.
If we then take a regular branched cover __\( S^{\prime} \to \tilde S \)__ with branch
locus __\( \tilde z \)__, the resulting cover __\( S^{\prime} \to S \)__ is not fully ramified but
it has the Birman–Hilden property.

Here is another basic question.

Question 11.2:
For which cyclic branched covers of __\( S_g \)__ over the sphere is __\( \operatorname{SMod}(S_g) \)__
equal to a proper subgroup of __\( \operatorname{Mod}(S_g) \)__? When is it finite index?

Theorem 5 in the *Annals* paper by Birman and Hilden states for a
cyclic branched cover __\( S \to X \)__ over the sphere we have
__\[ \operatorname{LMod}(X) =\operatorname{Mod}(X) .\]__
Counterexamples to this theorem were recently discovered by
Ghaswala and
the second author (see the erratum [10]), who wrote a paper
[e42]
classifying exactly which branched covers over the sphere have
__\[ \operatorname{LMod}(X)=\operatorname{Mod}(X). \]__
Theorem 6 in the paper by Birman and Hilden states that
for a cyclic branched cover of __\( S_g \)__ over the sphere with __\( g \geq 3 \)__ the
group __\( \operatorname{SMod}(S_g) \)__ is a proper subgroup of __\( \operatorname{Mod}(S_g) \)__. The proof uses their
Theorem 5, so Question 11.2 should be considered an open question.
Of course this question can be generalized to other base surfaces besides
the sphere and other types of covers. For simple branched covers over the
sphere (which, as above, do not have the Birman–Hilden property) Berstein
and Edmonds
[e7]
proved that __\( \operatorname{SMod}(S_g) \)__ is equal to __\( \operatorname{Mod}(S_g) \)__.

We can also ask about the Birman–Hilden theory for orbifolds and 3-manifolds.

Question 11.3: Which covering spaces of two-dimensional orbifolds have the Birman–Hilden property?

Earle proved some Birman–Hilden-type results for orbifolds in his recent
paper
[e32],
which he describes as a sequel to his 1971 paper
*On the moduli of closed Riemann surfaces with symmetries*
[e2].

Question 11.4: Which covering spaces of 3-manifolds enjoy the Birman–Hilden property?

Vogt proved that certain regular unbranched covers of certain Seifert-fibered 3-manifolds have the Birman–Hilden property [e6]. He also explains the connection to understanding foliations in codimension two, specifically for foliations of closed 5-manifolds by Seifert 3-manifolds.

A specific 3-manifold worth investigating is the connect sum of __\( n \)__ copies of
__\( S^2 \times S^1 \)__; call it __\( M_n \)__. The outer automorphism group of the free
group __\( F_n \)__ is a quotient of the mapping class group of __\( M_n \)__ by a finite
group. Therefore, one might obtain a version of the Birman–Hilden theory
for the outer automorphism group of __\( F_n \)__ by developing a Birman–Hilden
theory for __\( M_n \)__.

Question 11.5:
Does __\( M_n \)__ enjoy the Birman–Hilden property? If so, does this give a
Birman–Hilden theory for free groups?

For example, consider the hyperelliptic involution __\( \sigma \)__ of __\( F_n \)__,
the outer automorphism that (has a representative that) inverts each
generator of __\( F_n \)__. This automorphism is realized by the homeomorphism of
__\( M_n \)__ that reverses each __\( S^1 \)__-factor. The resulting quotient of __\( M_n \)__
is the 3-sphere with branch locus the __\( (n+1) \)__-component unlink. This is
in consonance with the fact that the centralizer of __\( \sigma \)__ in the outer
automorphism group of __\( F_n \)__ is the palindromic subgroup and that the latter
is closely related to the configuration space of unlinks in __\( S^3 \)__; see the
paper by
Collins
[e17].

Next, there are many questions about the hyperelliptic Torelli group and its
generalizations. As discussed in Section 6, the hyperelliptic Torelli
group is the kernel of the integral Burau representation of the braid group.
With Brendle and Putman, the first author of this article proved
[e37]
that this group is generated by the squares of Dehn twists about curves
that surround an odd number of marked points in the disk __\( D_n \)__.

Question 11.6: Is the hyperelliptic Torelli group finitely generated? Is it finitely presented? Does it have finitely generated abelianization?

There are many variants of this question. By changing the branched cover __\( S
\to D_n \)__, we obtain many other representations of (the liftable subgroups of)
the braid group. Each representation gives rise to its own Torelli group.
Except for the hyperelliptic involution case, very little is known. One set
of covers to consider is the set of superelliptic covers studied by Ghaswala
and the second author of this article
[e45].

Another aspect of this question is to determine the images of the braid
groups in __\( \operatorname{Sp}_{2N}(\mathbb{Z}) \)__ under the various representations of (finite
index subgroups of) the braid group arising from various covers __\( S \to D_n \)__.
By work of
McMullen
[e35]
and
Venkatarmana
[e36],
it is known that when the degree of the cover is at least three and __\( n \)__ is
more than twice the degree, the image has finite index in the centralizer
of the image of the deck group.

Question 11.7:
For which covers __\( S \to D_n \)__ does the associated representation of the braid
group have finite index in the centralizer of the image of the deck group?

There are still many aspects to the Birman–Hilden theory that we have not
touched upon.
Ellenberg
and
McReynolds
[e34]
used the theory to prove
that every algebraic curve over __\( \mathbb{\bar Q} \)__ is birationally equivalent
over __\( \mathbb{C} \)__ to a Teichmüller curve.
Nikolaev
[e40]
uses the embedding of the braid group into the mapping class group to give
cluster algebraic representations of braid groups.
Kordek applies the
aforementioned result of Ghaswala and the second author of this article
to deduce information about the Picard groups of various moduli spaces of
Riemann surfaces
[e39].
A Google search for “Birman–Hilden”
yields a seemingly endless supply of applications and connections (the
*Annals* paper has 139 citations on Google Scholar at the time of
this writing). We hope that the reader is inspired to learn more about
these connections and pursue their own developments of the theory.

#### Acknowledgments

The authors would like to thank John Etnyre, Tyrone Ghaswala, Allen Hatcher, and Chris Leininger for helpful conversations.