# Celebratio Mathematica

## Joan S. Birman

### The Birman–Hilden theory

#### 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 $S^3$; and

• they 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: 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.

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