Surface branched covers and Hurwitz numbers

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In Supremae Dignitatis · Universit`

a di Pisa

Corso di Laurea Magistrale in Matematica

Surface branched covers

and Hurwitz numbers

Relatore:

Chiar.mo Prof.

Carlo Petronio

Presentata da:

Filippo Sarti

Sessione di Luglio

Anno Accademico 2018/19

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Introduction

This thesis deals with the realizability problem of branched coverings between surfaces. A branched covering f : eΣ −→ Σ is a map modeled on maps of the form

(C, 0) z7→zk (C, 0)

with k ≥ 1. If k > 1, the points in the target corresponding to 0 are called branching points and the set of local degrees of f over each branching point is a partition of the total degree d. To a surface branched covering one can associate a set of data called branching datum where the two surfaces, the total degree, the number of branching points and the local degrees are specified. On the other hand, for a set of data to be associated with an existing branched covering some easy necessary conditions are required, and these are given by the Riemann-Hurwitz formula (see [EKS84]); if a branching datum satisfies these conditions it is called compatible. Moreover, it is said to be realizable if it comes from some branched covering, exceptional if not. A classical problem, first proposed by A. Hurwitz in 1891 (see [Hur91]), asks whether a compatible datum is realizable, namely whether the necessary conditions above are also sufficient for existence. During the last 60 years quite some progress has been made to understanding the exceptional candidates, in particular it has been shown that exceptions can only occur if Σ is either S or the projective plane. Therefore, since the case χ(Σ) ≤ 0 has already been solved, the efforts have been focused on the case Σ = S (the case of the projective plane follows immediately from this one). More precisely, the following conjecture proposed in [EKS84] appears to be still open:

a compatible branching datum with prime degree is realizable.

In order to investigate the branched coverings of the sphere, some different techniques has been developed over the time. In particular in this thesis we make use of:

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INTRODUCTION ii

• dessins d’enfant, which are a combinatorial technique first introduced by A. Grothendieck (see [Gro97]) and generalized by E. Pervova and C. Petro-nio in [PP06] and [PP08] to give some results about the Hurwitz existence problem;

• constellations, which are an algebraic technique first introduced by A. Hurwitz in [Hur91] and developed by A. Edmonds, S. Kulkarni and E. Stong in [EKS84].

A parallel approach to the surface branched covering existence problem has been developed by A. Mednykh in [Med84]. Given two branched coverings f, f0 : eΣ −→ Σ, Mednykh says that f and f0 are equivalent if there exists a homeomorphism eh : eΣ −→ eΣ such that the following diagram commutes

e Σ Σe Σ. e h f f0

In his article, Mednykh provides a formula for the number of inequivalent cov-erings realizing a fixed branching datum (called Hurwitz numbers). Even if such a formula in fact gives an answer to the existence problem (a datum will be exceptional if it gives 0 and realizable otherwise), it is very hard to apply except for some very simple case. Starting from a particular case and making use of the platform Mathematica, we managed to apply the formula and com-pare the results with some theoretical results obtained using dessins d’enfant (see Chapter 5). During this process we found some discrepancy between the two approaches due to a rough interpretation of the notion of equivalence; this gave us the motivation to formalize the notion of equivalence between surface branched coverings.

The aim of this thesis is to formalize the notion of equivalence between surface branched coverings and to understand the behavior of dessins d’enfant and constellations in relation to covering equivalence. More precisely we trans-late, with some restrictions where necessary, the different notions of equivalence between coverings in terms of the two objects mentioned above.

In particular, in the first chapter we recall the definitions of surface branched covering and branching datum. Then we define four sets

C(D), C+(D), C∗(D), C∗,+(D)

of branched covering realizing a datum D where we take or less into account the order of the branching points and the positivity of the coverings. Starting

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INTRODUCTION iii

from these sets we define some quotient spaces and some maps between them. Finally we prove some easy relations between these quotients.

In the second chapter we introduce the dessins d’enfants. We give the def-inition of such an object and we describe a correspondence between dessins d’enfant and branched coverings. Then we introduce some new combinatorial moves between dessins d’enfant, that describe the equivalence relations in the quotient spaces in terms of dessins. Finally we show an example where we compute the cardinality of the quotients using the dessins and the moves just described.

In the third chapter we describe the constellations, that are collections of permutations with specific properties associated to a branched covering. We re-alize a correspondence between constellations and branched coverings, passing through the representations of the fundamental group of the covered surface. Then, as in the second chapter, we provide some moves between constellations that describe the equivalence relations in the quotient spaces in terms of con-stellations. Finally we repeat the computations in the example of Chapter 2 using the constellations and the moves just described.

In the fourth chapter, following the proof of Mednykh, we describe a formula for the cardinality of one of the quotient spaces defined in the first chapter.

In the last chapter we provide some explicit computations, describing an application of the Mednykh’s formula.

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Branched coverings and

notions of equivalence

1.1 Branched coverings

In this section we recall the definition of branched covering between surfaces, which will be the fundamental object studied in this thesis.

A surface is a topological 2-dimensional manifold; we will consider only compact and connected surfaces without boundary.

In a genuine surface covering f : eΣ −→ Σ, each p ∈ Σ has an evenly covered open neighborhood. The notion of branched covering generalizes that of genuine coverings, allowing some “special points”. More precisely, we start with the following:

Definition 1.1.1. A branched covering is a map f : eΣ −→ Σ where eΣ, Σ are surfaces and f is locally modeled on maps of the form

(C, 0) z7→zk (C, 0)

with k ≥ 1. The integer k is called the local degree of f at the point of eΣ corresponding to 0 in the source C. If k > 1 the point of Σ corresponding to 0 in the target is called branching point of f . The branching points are isolated, whence finite in number, and denoted by x1, . . . , xn. The surfaces eΣ

and Σ are respectively the covering surface and the covered surface; the set B = {x1, . . . , xn} is the branching set and, if f−1(xi) = {yi1, . . . , yimi}, we

denote by dij the local degree of f at yij. The degree d of the covering is the

cardinality of f−1(p), where p is any point of Σ \ B. 1

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1.1 Branched coverings 2

Definition 1.1.2. A partition of an integer d is an unordered set of integers with repetitions allowed and summing up to d. If d1, . . . , dn form a partition µ

of d and d1≥ · · · ≥ dn, we will write

µ = [d1, . . . , dn].

Remark 1.1.1. Given a branched covering with local degrees (dij)i=1,...,n j=1,...,mi

at the yij, the integers di1, . . . , dimi satisfy

mi

X

j=1

dij = d

(namely, they form a partition of the integer d).

Therefore we can give the following:

Definition 1.1.3. Let f : eΣ −→ Σ a branched covering. The 5-tuple

D = (eΣ, Σ, d, n, λ)

where λ = (λ1, . . . , λn) and λi= [di1, . . . , dimi] for all i = 1, . . . , n is the

branch-ing datum associated to the coverbranch-ing f ; we say that the coverbranch-ing f matches (or realizes) the datum D.

During the rest of the dissertation we will adopt the following:

Notation 1.1.1. • eΣ is the covering surface and Σ is the covered surface; we will assume that both eΣ and Σ are oriented;

• _eg is the genus of eΣ and g is the genus of Σ;

• f is the projection map;

• d is the degree of the covering;

• n is the number of branching points;

• B = {x1, . . . , xn} is the branching set and the xi’s are the branching

points;

• for all xi ∈ B, we denote by mi the cardinality of the fiber f−1(xi) and

by yij the preimages of xi for j = 1, . . . , mi;

• for all i = 1, . . . , n and j = 1, . . . , mi the local degree of f at yij is dij;

• for all i = 1, . . . , n we denote by λithe partition of d given by [di1, . . . , dimi];

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1.2 Notions of equivalence between coverings 3

1.2 Notions of equivalence between coverings

In this section we define some quotients starting from four different sets of branched coverings matching a datum D.

Let

D = (eΣ, Σ, d, n, λ)

be a branching datum with a fixed orientation for eΣ and Σ and consider n fixed and ordered points p1, . . . , pn in Σ.

Definition 1.2.1. For the sake of brevity, we call positive a local homeomor-phism between oriented surfaces that respects the orientation. For a branched covering f : eΣ −→ Σ with branching set B we will say that f is positive if

f |

e

Σ\f−1_(B): eΣ \ f−1(B) −→ Σ \ B

is.

We define the sets

C(D) =nf : eΣ −→ Σ realizing Do, C+(D) = n f : eΣ −→ Σ positive realizing Do, C∗(D) = n

f : eΣ −→ Σ realizing D with λj over pj

o ,

C∗,+(D) =

n

f : eΣ −→ Σ positive realizing D with λj over pj

o .

Starting from these sets, we define some different quotient spaces

Cb,r

m,s(D) = Cm,s(D)/∼

where m is ∗ or omitted, s is + or omitted, b is an over-bar or omitted, r is + or omitted, b must be omitted if m is, and where f ∼ f0 if there exist homeomorphisms eh, h such that the diagram

e Σ Σe Σ Σ e h f f0 h commutes; moreover

• if r = + then eh, h are positive;

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1.3 Basic results 4

Hence we get the following quotients:

C∗(D) C∗+(D) C∗,+(D) C∗,++ (D)

C(D) C+_(D) _C

+(D) C++(D)

C∗(D) C∗+(D) C∗,+(D) C∗,++ (D).

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From the definitions it immediately follows that the following maps exist:

• the quotient maps

q_m,sb : C_m,sb,+_{(D) C}b_m,s(D); • the quotient maps

crs: C r

∗,s(D) C∗,sr (D);

• the inclusions

j_mb,r: C_m,+b,r (D) ,→ C_mb,r(D); • the maps corresponding to “forgetting the marking”

ors: C∗,sr (D) → Csr(D).

Therefore we get the following commutative diagram:

C+ ∗(D) C∗+(D) C+(D) C_∗,++ (D) C_∗,++ (D) C++(D) C∗(D) C∗(D) C(D) C∗,+(D) C∗,+(D) C+(D). c+ q∗ j∗+ q∗ j∗+ o+ q j+ c+₊ q∗,+ q∗,+ o+₊ q+ c j∗ j∗ o j c+ o+

1.3 Basic results

In this section we investigate the relations between the quotients in (1.1) and we give some basic results. We start with the following:

Remark 1.3.1. In C∗,+(D) we have that f ∼ f0 if there exists eh such that

e Σ Σe Σ e h f f0

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1.3 Basic results 5

commutes; since f, f0 are positive eh then automatically is, therefore the map q∗,+is a bijection.

We now show that more of the above maps are always bijections.

Proposition 1.3.1. The maps j, j∗ and j∗ are bijections.

Proof. Let r : eΣ −→ eΣ be a negative homeomorphism such that r ◦ r = id

e Σ.

Consider first the inclusion

j : C+(D) ,→ C(D).

Then we can define a map

φ : C(D) −→ C+(D) as φ([f ]) =    [f ] if f is positive [f ◦ r] if f is negative.

Such a map is well-defined:

• if f ∈ C(D) then:

– if f is positive then [f ] ∈ C+(D);

– if f is negative then [f ◦ r] ∈ C+(D);

• take f, f0_{∈ C(D) such that [f}0_{] = [f ] in C(D); then there exist e}_{h, h such}

that e Σ Σe Σ Σ e h f f0 h

commutes. We have four cases:

– if f, f0 are positive, then [f0] = [f ] in C+(D);

– if f is positive and f0is negative, then [f ] = [f0◦ r] in C+(D) because

the following diagram commutes:

e Σ Σe Σ Σ; r ◦ eh f f0◦ r h

– if f is negative and f0is positive, then [f ◦ r] = [f0] in C+(D) because

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1.3 Basic results 6 e Σ Σe Σ Σ; e h ◦ r f ◦ r f0 h

– if f, f0 are negative, then [f ◦ r] = [f0◦ r] in C+(D) because the

fol-lowing diagram commutes:

e Σ Σe Σ Σ. r ◦ eh ◦ r f ◦ r f0◦ r h

We show that φ ◦ j = idC+(D) and that j ◦ φ = idC(D).

• If f ∈ C+(D) then

(φ ◦ j)([f ]) = φ([f ]) = [f ];

• if f ∈ C(D), we have two cases:

– if f is positive, then

(j ◦ φ)([f ]) = j([f ]) = [f ];

– if f is negative, then

(j ◦ φ)([f ]) = j([f ◦ r]) = [f ◦ r]

but [f ] = [f ◦ r] in C(D) because the following diagram commutes:

e Σ Σe Σ Σ. r f ◦ r f id

Consider now the inclusion

j∗: C∗,+(D) ,→ C∗(D).

φ∗: C∗(D) −→ C∗,+(D) as φ∗([f ]) =    [f ] if f is positive [f ◦ r] if f is negative.

Such a map is well defined:

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1.3 Basic results 7

– if f is positive then [f ] ∈ C∗,+(D);

– if f is negative then [f ◦ r] ∈ C∗,+(D);

• take f, f0 _{∈ C}

∗(D) such that [f0] = [f ] in C∗(D); then there exists eh such

that e Σ Σe Σ e h f f0

– if both f and f0 are positive, then eh is positive too and [f0] = [f ] in C∗,+(D);

– if f is positive and f0 is negative, then eh is negative and [f ] = [f0◦ r] in C∗,+(D) because the following diagram commutes:

e Σ Σe Σ; r ◦ eh f f0◦ r

– if f is negative and f0 is positive, then eh is negative and [f ◦ r] = [f0] in C∗,+(D) because the following diagram commutes:

e Σ Σe Σ; e h ◦ r f ◦ r f0

– if both f and f0 are negative, then eh is positive and [f ◦ r] = [f0◦ r] in C∗,+(D) because the following diagram commutes:

e Σ Σe Σ. r ◦ eh ◦ r f ◦ r f0◦ r

We show that φ∗◦ j∗= idC∗,+(D) and that j∗◦ φ∗= idC∗(D).

• If f ∈ C∗,+(D) then

(φ∗◦ j∗)([f ]) = φ∗([f ]) = [f ];

• if f ∈ C∗(D), we have two cases:

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1.3 Basic results 8

(j∗◦ φ∗)([f ]) = j∗([f ◦ r]) = [f ◦ r]

but [f ] = [f ◦ r] in C∗(D) because the following diagram commutes:

e Σ Σe Σ. r f f ◦ r

Finally, consider the inclusion

j∗: C∗,+(D) ,→ C∗(D).

φ∗: C∗(D) −→ C∗,+(D) as follows: φ∗([f ]) =    [f ] if f is positive [f ◦ r] if f is negative.

Such a map is well defined:

• if f ∈ C∗(D) then:

– if f is positive then [f ] ∈ C∗,+(D);

– if f is negative then [f ◦ r] ∈ C_∗,++ (D); • take f, f0_{∈ C}

∗(D) such that [f0] = [f ] in C∗(D); then there exist eh, h such

that e Σ Σe Σ Σ e h f f0 h

– if both f and f0 are positive, then [f0] = [f ] in C∗,+(D);

– if f is positive and f0is negative, then [f ] = [f0◦ r] in C∗,+(D) because

e Σ Σe Σ Σ; r ◦ eh f f0◦ r h

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1.3 Basic results 9

– if f is negative and f0is positive, then [f ◦ r] = [f0] in C∗,+(D) because

e Σ Σe Σ Σ; e h ◦ r f ◦ r f0 h

– if both f and f0are negative, then [f ◦ r] = [f0◦ r] in C∗,+(D) because

e Σ Σe Σ Σ. r ◦ eh ◦ r f ◦ r f0◦ r h

We show that φ∗◦ j∗= id_C_∗,+(D) and that j∗◦ φ∗= idC∗(D).

• If f ∈ C∗,+(D) is positive then

(φ∗◦ j∗)([f ]) = φ∗([f ]) = [f ];

• if f ∈ C∗(D), we have two cases:

(j∗◦ φ∗)([f ]) = j∗([f ]) = [f ];

(j∗◦ φ∗)([f ]) = j∗([f ◦ r]) = [f ◦ r]

but [f ] = [f ◦ r] in C∗(D) because the following diagram commutes:

e Σ Σe Σ Σ. r f ◦ r f id

The following result shows that the maps or

s are also bijections.

Proposition 1.3.2. The maps o, o+, o+, o++ are bijections.

Proof. Let us consider first the map

o : C∗(D) −→ C(D)

where [f ] 7→ [f ] forgetting the marking. Then we can define a map

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1.3 Basic results 10

as

[f ] 7−→ [g ◦ f ]

where g : Σ −→ Σ is an automorphism such that g−1(pj) is the branching point

xj of f with local degrees λj over it. Such a g exists because for each pair

of points p and q in Σ one can choose a path γ joining them (and eventually avoiding some other points in Σ), a tubular neighborhood U of γ realizing a homeomorphism which is the identity out of U and on ∂U and which sends p to q.

The map φ is well-defined:

• since

(g ◦ f )−1(pj) = f−1(g−1(pj)) = f−1(xj),

the map g ◦ f has local degrees λj over pj;

• let g0 _{: Σ −→ Σ be another map with the same properties of g. Then}

[g ◦ f ] = [g0◦ f ] in C∗(D) because the following diagram commutes:

e Σ Σe Σ Σ. id g ◦ f g0◦ f g0_{◦ g}−1

• take f, f0_{∈ C(D) such that [f ] = [f}0_{] in C(D), then there exist e}_{h, h such}

that the following diagram commutes:

e Σ Σe Σ Σ. e h f f0 h Then [g ◦ f ] = [g0◦ f0_{] in C}

∗(D) because the following diagram commutes:

e Σ Σe Σ Σ. e h g ◦ f g0◦ f0 g0_{◦ h ◦ g}−1

We show that o ◦ φ = idC(D) and φ ◦ o = idC∗(D).

• if f ∈ C(D) then

(o ◦ φ)([f ]) = o([g ◦ f ]) = [g ◦ f ]

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1.3 Basic results 11 e Σ Σe Σ Σ. id f g ◦ f g • if f ∈ C∗(D) then (φ ◦ o)([f ]) = φ([f ]) = [g ◦ f ]. Taking g = id we get [g ◦ f ] = [f ] in C∗(D).

Before investigating the map o+_{, we note that the map g can be chosen to}

be positive (we can take it isotopic to the identity). Consider now the map

o+: C∗+(D) −→ C+(D)

φ+ : C+(D) −→ C∗+(D)

as

[f ] 7−→ [g ◦ f ]

where g : Σ −→ Σ is a positive automorphism such that g−1(pj) is the branching

point xi of f with local degrees λj over it. From what observed and repeating

the above proof it can be proved that φ+_{is well-defined, that o}+_{◦ φ}+_{= id} C+_(D)

and that φ+_{◦ o}+_{= id} C+

∗(D)

.

Consider now the map

o+:C∗,+(D) −→ C+(D)

φ+ : C+(D) −→ C∗,+(D)

as

[f ] 7−→ [g ◦ f ]

where g is as usual and positive. Again from what noted above it follows that φ+ is well-defined and that o+ and φ+ are the inverse of each other.

Finally, for the map

o++: C + ∗,+(D) −→ C + +(D) we define φ+₊: C₊+(D) −→ C_∗,++ (D)

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as

[f ] 7−→ [g ◦ f ]

where g is as usual. The same arguments shows that o+₊ is a bijection with inverse φ+₊.

Proposition 1.3.2 allow to consider the following easier commutative dia-gram: C+ ∗(D) C+(D) C_∗,++ (D) C++(D) C∗(D) C(D) C∗,+(D) C+(D) c+ q∗ j+ ∗ q j+ c+₊ q+ c j∗ j q∗,+ c+

where j and j∗ are bijections from Proposition 1.3.1 and, with a slight abuse of

notation, we write cr

s instead of ors◦ crs.

The following result explains the maps j+ and j_∗+:

Proposition 1.3.3. The maps j+, j_∗+are never surjective. Moreover, the space C+

∗(D) (respectively C+(D)) can be split into two subspace each of which is in

bijection with C_∗,++ (D) (respectively C₊+(D)). Proof. Let us consider first the quotient C+

∗(D) where f ∼ f0 if there exists a

positive eh such that the following diagram commutes:

e Σ Σe Σ. e h f f0

Then we can split that space into two complementary subspace:

C+ ∗(D) = C + ∗,+(D) t C + ∗,−(D) where C+ ∗,−(D) = {[f ] ∈ C∗+(D) : f negative }

We show that the map

j_∗−: C_∗,−+ (D) −→ C+_∗,+(D)

defined as j−_∗([f ]) = [f ◦ r] is a bijection. Such a map is clearly well-defined and the inverse is

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defined as φ([f ]) = [f ◦ r]. Setting

C₋+(D) = {[f ] ∈ C+(D) : f negative }, with the same argument it can be proved that

C+_{(D) = C}+ +(D) t C

+ −(D),

and that there exists a bijection between C₊+(D) and C₋+(D). Then we get the diagram

C+ ∗(D) C+(D) C+ ∗,+(D) C + +(D) C∗(D) C(D) C∗,+(D) C+(D) c+ q∗ j+ ∗ q j+ c+₊ q+ c j∗ j q∗,+ c+

where the red arrows are the maps which are left to investigate (since c+ _{is “two}

copies” of c+₊ from Proposition 1.3.3 and c = j ◦ c+◦ j−1∗ ). Similarly, if we are

interested only in the cardinality of the quotients defined in (1.1), we can focus only on the spaces

C+

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Chapter 2

Dessins d’enfant

In this chapter we investigate the notions of equivalence between branched cov-erings via certain objects called dessins d’enfant. This combinatorial technique was introduced for the first time by A. Grothendieck in [Gro97] and has been generalized to give some results about the Hurwitz existence problem by E. Pervova and C. Petronio in [PP06] and [PP08]. Here we give the definition of such an object and we describe a correspondence between dessins d’enfant and marked branched coverings. Then we translate in terms of dessins d’enfant the equivalence relations defined in Chapter 1, thereby providing a combinatorial method to describe the sets C_∗,++ (D), C₊+(D) and C+(D) to which we have shown

that one can restrict his attention.

2.1 From positive marked coverings

to dessins and back

In this section we give the definition of dessin d’enfant and a method for con-structing one starting from a branched covering (see also [PP06]). More pre-cisely, we realize a bijection between C+_∗,+(D) and a quotient of the set of dessins realizing a datum D with n ≤ 4.

We start with the following:

Definition 2.1.1. Let eΣ be a surface. A dessin d’enfant on eΣ is an embedded graph Γ ⊂ eΣ such that:

• for some n ≥ 3 the set of vertices of Γ is split as V1t . . . t Vn−1and the

set of edges of Γ is split as E1t . . . t En−2;

• for i = 1, . . . , n − 2 each edge in Ei joins a vertex of Vi to one of Vi+1 ;

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2.1 From positive marked coverings

to dessins and back 15

• for i = 2, . . . , n − 2 any vertex of Vi has even valence and going around it

we alternatively encounter edges from Ei−1and edges from Ei;

• eΣ \ Γ consists of open discs.

The length of one of the discs in eΣ \ Γ is the number of edges of Γ along which the boundary of the disc passes (with multiplicity).

If D is a branching datum, we say that a dessin d’enfant realizes D if:

• for i = 1 and i = n − 1 the vertices in Vi have valences (dij)j=1,...,mi;

• for i = 2, . . . , n − 2 the vertices in Vi have valences (2dij)j=1,...,mi;

• the discs in eΣ \ Γ have lengths (2(n − 2)dnj)j=1,...,mn.

In this thesis we will always consider a dessin d’enfant together with the marking of its vertices.

Consider now the datum

D = (e_{Σ, S, d, n, λ)} with n ≤ 4 and the set

G(D) = {Γ dessins realizing D} .

Then we have the following:

Proposition 2.1.1. For a datum D with 3 ≤ n ≤ 4, the set C_∗,++ (D) corresponds to the quotient of G(D) where we identify two dessins Γ and Γ0 if there exists a positive homeomorphism eh : eΣ −→ eΣ such that eh(Γ) = Γ0respecting the marking (namely, such that eh(Vi) = Vi0 for each i).

Proof. Let f ∈ C∗,+(D) and let Γ be a dessin constructed as follows:

• choose a simple arc α in S joining each pi with pi+1 for i = 1, . . . , n − 2

and avoiding pn; let αibe the segment of α from pi to pi+1;

• set Vi= f−1(pi) for i = 1, . . . , n − 1 and Ei= f−1(αi) for i = 1, . . . , n − 2;

• set Γ = f−1_(α).

Then we have the the following facts:

• Γ ∈ G(D); to show that Γ is in fact a dessin, the only non-obvious fact concerns the components of e_{Σ\Γ. Since S\α is an open disk, the restriction} of f to any component of eΣ \ Γ is a covering onto a disk with a single branching point and such a covering is modeled on the covering z 7−→ zk

of the open unit disk onto itself, so the components of eΣ \ Γ are discs. Moreover, by construction, Γ realizes D.

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• Let α0 _{be another arc with the same properties of α and set Γ}0_{= f}−1_(α0_);

if n = 3 there exists an isotopy between the arcs α1 and α01. If n = 4

we can find an isotopy between α1 and α01 relative to p1, p2, p3 and such

an isotopy can be chosen to be the identity around α2. Repeating for

the arcs α,₂α02and composing the two isotopies, we get an isotopy H of S

relative to p1, . . . , pn−1such that H0= id and H1 sends α to α0; such an

isotopy defines an isotopy eH of eΣ between Γ and Γ0 which gives a positive homeomorphism eh : eΣ −→ eΣ such that eh(Γ) = Γ0_{respecting the markings.}

• If f0 _{is equivalent to f , then there exists a positive homeomorphism e}_{h :}

e

Σ −→ eΣ such that f = f0◦ eh; hence, using the same α in the construction of Γ and Γ0_{, we have that e}_{h(Γ) = Γ}0 _{respecting the markings, because}

Vi= f−1(pi) and Vi0= (f0)−1(pi).

We show the inverse construction: let Γ be a dessin d’enfant realizing D and choose a simple arc α in S as above. Then we define a function f from Γ to α mapping the vertices in Vi to pi and each edge of Ei homeomorphically onto αi

for i = 1, . . . , n − 1. For each region R of eΣ \ Γ, assuming R has length 2(n − 2)k, we fix y in R and extend f to R so that:

• f maps y to pn locally as

(C, 0) z7→zk (C, 0); • f is continuous on the closure of R;

• f is a genuine covering of degree k from R \ {y} to S \ (α ∪ {pn});

• f is positive.

The following facts prove that f is well-defined up to equivalence in C_∗,++ (D) and that equivalent dessins in G(D) give equivalent f ’s:

• Since the local degrees over each piare given by the valences of the vertices

in Vi, the covering f realizes D .

• Let f0 _{be the coverings defined choosing different points in the regions of}

e

Σ \ Γ; then for each region R, if the fixed point is y0, we can define a positive homeomorphism eh : R \ {y} −→ R \ {y0} so that f0◦ eh = f on R \ {y}. Setting eh(y) = y0 and repeating the construction for each region, we get a positive automorphism eh such that f = f0◦ eh.

• If f and f0differ for the definition on the edges of Γ, then we can define a homeomorphism eh between Γ and Γ0 such that f = f0◦ eh on Γ and then extending on the regions, getting an equivalence between f and f0.

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• Let α0 _{be another simple arc as above and let f}0 _{be the covering}

con-structed using α0. Then, as above, an isotopy H relative to p1, . . . , pn−1

between α and α0 translates into an isotopy of eΣ which gives a positive homeomorphism eh : eΣ −→ eΣ such that f = f0◦ eh.

• Let f, f0 _{be two coverings constructed as above starting from equivalent}

dessins Γ and Γ0 and choosing the same α; then there exists eh : eΣ −→ eΣ which sends Γ to Γ0and we can define a homeomorphism g : eΣ −→ eΣ such that f0_{◦ g ◦ e}_{h = f . Then the homeomorphism e}_{h ◦ g realizes an equivalence}

between f and f0 in C_∗,++ (D).

Finally, we show that the two constructions are one the inverse of the other.

• Let Γ ⊂ eΣ be a dessin d’enfant realizing D, choose a simple arc α and con-struct a branched covering f ; if we choose the same α in the concon-struction of the dessin associated to f , we get the again Γ.

• Let f ∈ C∗,+(D), choose a simple arc α and let Γ be the dessin constructed

as above: if we choose again α in the realization of the branched covering associated to Γ, then we can extend to the regions of eΣ \ Γ using f , getting again f .

Remark 2.1.1. When n > 4, the construction of a dessin starting from a branched covering is not unique: if α and α0 are the two arcs in Figure 2.1, then there exists no isotopy relative to p1, . . . , pn−1 sending α to α0, whence,

in general, there exists no positive homeomorphism eh : eΣ −→ eΣ sending Γ = f−1(α) to Γ0= f−1(α0). α1 α2 p1 p2 p3 p4 α3 α1 α2 p1 p2 p3 p4 α3 etc

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2.2 Unmarked coverings 18

2.2 Unmarked coverings

In this section we define some combinatorial moves which translates the equiv-alence relations in the quotients C₊+(D) and C+(D) in terms of dessins d’enfants

realizing D.

We define certain moves σ1, σ2, σ3 on the dessins realizing a datum D with

n = 4 as follows:

• the move σ1 acts on a dessin as follows:

– it switches the labels V1 and V2;

– it erases the edges in E2;

– for each region R of Γ, assuming its length is 4k, note that around ∂R (before switching labels) in a positive cyclic order we see the sequence of vertices marked as

V1, V2, V3, V2 (2.1)

repeated k times; then σ1 acts repeating k times the following

con-struction: it draws k edges labeled E2joining by a small arc parallel

to ∂R each appearance in (2.1) of V1 to each appearance in (2.1) of

V3;

– it switches the labels V2 and V3;

– for each region R of length 4k it repeats k times the following con-struction: it draws k edges labeled E1joining by a small arc parallel

to ∂R each appearance in (2.1) of V1to appearance in (2.1) of V3;

– it erases the vertices in V3 and ads a new vertex inside each region

R labeled as V3;

– for each region R of length 4k it repeats k times the following con-struction: it draws k edges labeled E2joining by a small arc parallel

to ∂R each first appearance in (2.1) of V2 to the new vertex.

Similarly, for data with n = 3 we define the moves σ1, σ2 as follows:

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– it erases vertices in V2 and ads a new vertex inside eah region R

labeled as V2;

– it erases all edges;

– for each region R of length 2k we see, in a positive cyclic order, the sequences of vertices marked as

V1, V2

repeated k times; then σ2 acts repeating k times the following

con-struction: it draws k edges labeled E1joining by a small arc parallel

to ∂R each appearance in (2.1) of V1to the new vertex.

We give the following:

Example 2.2.1. The dessin d’enfant Γ in Figure 2.2 realizes the datum

D = (S, S, 6, 4, ([3111], [2211], [321], [321])). 2 2 2 2 1 1 1 1 3 3 3 1 1 1 1 1 2 2 2 2 2 2 1

Figure 2.2: A dessin d’enfant realizing D.

In Figures 2.3 to 2.5 we show the action of σ1, σ2and σ3on Γ (the red arcs

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2.2 Unmarked coverings 20 2 2 2 2 1 1 1 1 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1

Figure 2.3: The dessin σ1(Γ).

2 2 2 2 1 1 1 1 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1

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2.2 Unmarked coverings 21 2 2 2 2 1 1 1 1 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1

Figure 2.5: The dessin σ3(Γ).

Definition 2.2.1. We define ∼ as the equivalence relation between dessins realizing D generated by:

• Γ ∼ Γ0 _{if there exists a homeomorphism e}_{h : e}_{Σ −→ e}_{Σ such that e}_{h(Γ) = Γ}0

respecting the marking;

• Γ ∼ Γ0 _{if Γ}0_{= σ}

j(Γ) for some i = 1, . . . n − 1.

If we require that the homeomorphism eh in the first instance is positive, Γ and Γ0 are said to be positively equivalent (we write Γ ∼+Γ0).

Then, for data with n ≤ 4, we have the following:

Proposition 2.2.1. The set C₊+(D) corresponds to the quotient of G(D) where we identify two dessins Γ and Γ0 if Γ ∼+ Γ0.

Proof. Suppose that Γ ∼+Γ0 and let f, f0 be two branched covering associated

to Γ and Γ0 constructed as in Proposition 2.1.1. It is enough to prove that f and f0 are equivalent in C₊+(D) for the two instances of Definition 2.2.1:

• if there exists eh positive such that eh(Γ) = Γ0then, from Proposition 2.1.1, the coverings f and f0 _{are equivalent in C}+

∗,+(D) and hence also in C + +(D);

• suppose that n = 4 and Γ0_{= σ}

1(Γ); we can find a positive homeomorphism

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2.2 Unmarked coverings 22 – switches p1 and p2; – fixes p3, p4 and α1; – h(α2) joins p1 to p3 as in Figure 2.6. α1 α2 p1 p2 p3 α1 h(α2) h(p2) h(p1) p3 α1 h(α2) h(p2) h(p1) p3 p4

Figure 2.6: The possible action of h on the path α when n = 4 and Γ0= σ1(Γ).

The covering h ◦ f is associated to Γ0 so it is equivalent to f0 in C_∗,++ (D) and we get an equivalence between f and f0 _{in C}+

+(D). h(α1) α2 h(p2) p1 h(p3) h(α1) α2 h(p2) p1 h(p3) α1 α2 p1 p2 p3 p4

Figure 2.7: The possible action of h on the arc α when n = 4 and Γ0 = σ2(Γ).

The proof is the similar if Γ0= σ2(Γ) using the homomorphisms in Figure

2.7. If Γ0 = σ3(Γ), we choose an automorphism h of S that:

– switches p3 and p4;

– fixes p1, p2 and α1;

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2.2 Unmarked coverings 23 α1 α2 p1 p2 p3 α1 h(α2) p2 p1 h(p3) h(α2) p4 h(p4) p2 p1 h(p4) h(p3)

As above, h ◦ f is associated to Γ0 so it is equivalent to f0 in C_∗,++ (D) and we get an equivalence between f and f0 _{in C}+

+(D).

If n = 3 and Γ0= σ1(Γ), we repeat the proof above using an automorphism

h that:

– switches p1 and p2;

– fixes p3 and α.

If Γ0 = σ2(Γ) we use an automorphism h that:

– switches p2 and p3; – fixes p1; – h(α) joins p1to p3 as in Figure 2.9. α1 p1 p2 p3 h(p2) p1 h(p3) h(p2) p1 h(p3) h(α1) h(α1)

For the converse let f, f0 be equivalent coverings in C₊+(D) and let Γ and Γ0 be two associated dessins constructed using the same α. Then there exist two positive homeomorphisms eh, h such that h ◦ f = f0◦ eh. Using the σi’s on Γ and

Γ0we can reduce to the case where the vertices Viand Vi0have valences given by

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repetitions we may still have that h permutes the pi’s non-trivially, but up to

replacing f0 _{by g ◦ f where g : S −→ S is a positive homeomorphism, we can} suppose that h fixes p1, . . . , pn−1 and α. Then we have eh(Γ) = Γ0 respecting

the marking.

Similarly we have the following:

Proposition 2.2.2. The set C+(D) corresponds to the quotient of G(D) where

we identify two dessins Γ and Γ0 if Γ ∼ Γ0.

Proof. The proof is the same as that of Proposition 2.2.1 where we allow also negative homeomorphisms.

Example 2.2.2. If n = 3, we note that the action of σ1, σ2 on the dessins in

G(D) can lead to

|C++(D)| < |C + ∗,+(D)|

only if λ contains repetitions. In this case, two dessins are equivalent in C++(D)

if:

• they are equivalent in C+ ∗,+(D);

• λ1= λ2 and they are related by σ1and by planar isotopies;

• λ2= λ3 and they are related by σ2and by planar isotopies;

• λ1= λ2= λ3and they are related by σ2σ1 and by planar isotopies;

• λ1= λ2= λ3and they are related by σ1σ2 and by planar isotopies;

• λ1= λ2= λ3and they are related by σ1σ2σ1and by planar isotopies,

since σi can be applied only if λi= λi+1 and the others combinations of σ1, σ2

can be reduced to these ones. In other words, the moves σ1, σ2 act on G(D) as

S3.

Let us consider the following data:

D1= (S, S, 7, 3, ([4111], [3211], [7])),

D2= (S, S, 7, 3, ([61], [211111], [7])),

D3= (S, S, 7, 3, ([3211], [3211], [7])).

Since λ36= λ1, λ2, two dessins representing Di are equivalent in C++(Di) only if:

• they are equivalent in C_∗,++ (Di);

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Here we consider all the dessins where the black vertices have valences given by λ1, the white vertices have valences given by λ2and the only region has length

2(n − 2)d31 = 2(3 − 2)7 = 14. We note also that two inequivalent dessins in

C+

+(D) are equivalent in C+(D) if they are related by a reflection.

A set of dessin representing C_∗,++ (D1) is shown in Figure 2.10.

Γ1

Γ2

Γ3

Figure 2.10: Dessins d’enfant representing the equivalence classes in C_∗,++ (D1).

Then we have

|C_∗,++ (D1)| = 3.

Since λ does not contain repetitions, we have

|C+

+(D1)| = |C∗,++ (D1)| = 3.

However, Γ1and Γ2are related by a reflection, whence Γ1∼ Γ2. We get

|C+(D1)| = 2.

The only dessin representing C_∗,++ (D2) is shown in Figure 2.11.

Then we have

|C+

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A set of dessins representing C_∗,++ (D3) is shown in Figure 2.12.

Γ1

Γ2

Γ3

Γ4

Γ9

Γ5

Γ6

Γ7

Γ8

Then we have

|C_∗,++ (D3)| = 9.

Since λ1= λ2, the action of σ1can be non-trivial. The dessins Γ2and Γ3are

related by a switch of colors, namely the vertices in V1 and in V2 are switched.

Hence they are related by the move σ1, so the related coverings are equivalent

in C++(D3). Similarly, also the coverings related to Γ6and Γ9and the coverings

related to Γ7 and Γ8 are equivalent in C++(D3) and we get

|C+

+(D3)| = 6.

Reflecting Γ4we obtain Γ5, so the related coverings are equivalent in C+(D3).

Similarly, reflecting Γ6 we obtain Γ7, so the related coverings are equivalent in

C+(D3). Finally we get

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The following table summarizes the cardinality of the different quotients in the previous examples:

λ |C+

∗,+(D)| |C++(D)| |C+(D)|

[7], [4111], [3211] 3 3 2 [7], [61], [211111] 1 1 1 [7], [3211], [3211] 9 6 4

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Chapter 3

Constellations

In this chapter we describe an algebraic approach to the problem of existence of branched coverings which makes use of permutations. Such a point of view was introduced for the first time by A. Hurwitz in [Hur91] and then exploited by Edmonds, Kulkarni and Stong in [EKS84] (see also [Eze78]). Here we show the correspondence between a realization of a branching datum and the existence of a collection of permutations with specific properties, called constellation. Then, as in Chapter 2, we (partially) translate in terms of constellations the notions of equivalence defined in Chapter 1.

3.1 From positive marked coverings

to constellations and back

In this section we give a correspondence between branched coverings and certain objects called constellations.

Let

D = (eΣ, Σ, d, n, λ)

be a branching datum and fix a n-tuple of points p1, . . . , pnin Σ. Let x0∈ Σ\B,

choose disjoint discs ∆1, . . . , ∆n with pi ∈ int(∆i) and let γi = ∂∆i. Then we

define the quotient set

Rx0(D) = {ρ : π1(Σ \ B, x0) −→ Sdhomomorphism :

Im(ρ) transitive on {1, . . . , d}, ρ(ci) has cyclic structure λi ∀i}/∼

where ci is the class represented by the loop iγi−1i with i a path from x0 to

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to constellations and back 29

γi and ρ ∼ ρ0 if ρ0 is conjugate with ρ, namely if there exists σ ∈ Sd such that

ρ0(α) = σρ(α)σ−1 for all α ∈ π1(Σ \ B, x0).

Lemma 3.1.1. The definition of Rx0(D) does not depend on the choice of the

i’s and of the γi.

Proof. Let 0_i be another path from x0 to the same point of γi. Then

ρ(iγi−1i ) = ρ(i(0i)−10iγi(0i)−10i −1

i ) = ρ(i(0i)−1)ρ(0iγi(0i)−1)ρ(i(0i)−1)−1

so ρ(ci) and ρ(c0i) have the same cyclic structure since they are conjugate in Sd.

If we take a path from x0 to a different point of γi, the proof is similar but we

must take into account the path between the two ending points.

If γ0_i = ∂∆0_i where ∆i is another disk around pi, we can find an isotopy

between γi and γi0 and so ci= c0i.

Lemma 3.1.2. If x0, x00 ∈ Σ \ B, there exists a canonical bijection between

Rx0(D) and Rx00(D).

Proof. Let δ be a path from x0 to x00. Then we have an isomorphism

δ : π1(Σ \ B, x0) −→ π1(Σ \ B, x00)

defined as δ([ω]) = [δ−1ωδ]. Moreover, if δ0 is another path from x0 to x00 and

α = [(δ0)−1δ] ∈ π1(eΣ \ B, x0), we have

δ0_{([ω]) = [(δ}0₎−1_ωδ0_{] = [(δ}0₎−1_δδ−1_ωδδ−1_δ0_{] = αδ([ω])α}−1 _(3.1)

We now claim that the map

Rx0

0(D) −→ Rx0(D)

[ρ] 7−→ [ρ ◦ δ]

is a well-defined bijection. In fact the well-definition follows from (3.1) and the inverse can be constructed taking the path δ−1 and the isomorphism

δ−1_{: π}

1(Σ \ B, x00) −→ π1(Σ \ B, x0).

By Lemma 3.1.2 we can consider the set R(D) avoiding to specify the base point. We now show that representations are closely related to branched cover-ings. More precisely, we have the following:

Proposition 3.1.3. There exists a canonical bijection between C_∗,++ (D) and R(D).

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Proof. We define a map

φ : C_∗,++ (D) −→ R(D) where φ([f ]) = [ρ] and ρ is constructed as follows:

• label f−1_(x

0) as {xi}di=1;

• given [γ] ∈ π1(Σ \ B, x0) define ρ([γ])(i) = j if the lifting of γ−1 starting

at xi ends at xj.

The map φ is well-defined:

• since eΣ is connected, for each pair (xi, xj) there exists a lifting of a loop

in π1(Σ \ B, x0) joining xi and xj, hence Im(ρ) is transitive on {1, . . . , d}.

• The image ηi∈ Sd of ci has cyclic structure λi since a lifts of ci can join

xi and xj only if i and j are in the same cycle of ρ(ci).

• A relabeling of the preimages f−1_(x

0) induces another representation ρ0

which is conjugate to ρ.

• Let f0 be equivalent to f and let eh : eΣ −→ eΣ so that f = f0◦ eh. We can assume that (f0)−1(x0) is labeled so that eh respects the labels. Since a

lift with respect to f0of γ is the image under eh of a lift with respect to f , with this choice of labels we obtain two equivalent representations.

The inverse map

ψ : R(D) −→ C_∗,++ (D)

sends a class [ρ] into R(D) to a class [f ] in C+_∗,+(D) where f is constructed as follows (see also [Eze78]):

• choose a graph Γ in Σ \ (˚∆1∪ . . . ∪ ˚∆n) cutting along which we get the

polygon P in Figure 3.1; so the surface Σ \ (˚∆1∪ . . . ∪ ˚∆n) is obtained by

edge-pairing as follows:

– α+_i with α−_i for all i = 1, . . . , g; – β+_i with β−_i for all i = 1, . . . , g; – +_i with −_i for all i = 1, . . . , n.

The loops α+_i , β_i+ and δi = +i γi(−i )−1 correspond to a set of generators

{ai, bi, ci} of π1(Σ \ B, x0); moreover, * a1, b1, . . . , ag, bg, c1, . . . , cn g Y i=1 [ai, bi] n Y i=1 ci + is a presentation for π1(Σ \ B, x0).

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α1 β1 α1 β1 _αg βg αg βg γ1 γ2 γ n ε1 ε2 ε2 ε1 εn εn

Figure 3.1: The polygon P .

• Let ΣΓ be the closed and connected surface obtained as follows:

– let P(1), . . . , P(d) be d copies of P ;

– if (either an αi, βi or δi) is an edge in ∂P(k) (and denoting by e

the corresponding ai, bi or ci in π1(Σ \ B, x0)) for which ρ(e)(k) = j,

attach P(k)to P(j) by identifying ((k))+ to ((j))−; – cap boundary loops with discs.

• Define a branched cover fΓ : ΣΓ −→ Σ using the identity on each P(k)

and extending on each disk as follows: if the boundary consists of k edges labeled as γi extend using the map z 7→ zk.

• Since ΣΓ ∼= eΣ, choose a positive homeomorphism g : eΣ −→ ΣΓ and set

f = fΓ◦ g.

The map ψ is well defined:

• by construction the only possible branching points are the {pi} and x0. If

ρ(ci) has a cycle (q1· · · qt), then δi lifts to arcs in sheets q1, . . . , qt. Hence

pihas a preimage through fΓof multiplicity t and the set of its preimages

have multiplicities given by the cyclic structure of ρ(ci), namely λi. Since

ρ is a homomorphism, we must have

ρ g Y i=1 [ai, bi] ! ρ n Y i=1 ci ! = id

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The fact that ΣΓis homeomorphic to eΣ follows from the Riemann-Hurwitz

condition (see [Hur91] or [EKS84]);

• if g0 _{is another homeomorphism between Σ}

Γ and eΣ, then the branched

covering f0 = fΓ0◦ g0 is equivalent to f : setting eh = (g0)−1◦ g, we get

f = f0◦ eh;

• if Γ0 _{is another graph with covering f}0_{, we have a polygon P}0 _{with edges}

α0±_i , β_i0± and δ0_i. Then we can define a homeomorphism between ΣΓ and

ΣΓ0 sending each (P )(k) to (P0)(k) with a positive homeomorphism

ob-tained by pairing the edges e and e0 and extending on the discs; hence we get an equivalence between f and f0;

• let ρ0 _{be conjugate to ρ}0 _{via σ ∈ S}

d; up to replacing each (P0)(k) by

P(σ−1(k)), the two coverings constructed are the same. We show that ψ ◦ φ = id_C+

∗,+(D) and that φ ◦ ψ = idR(D).

Let [f ] ∈ C_∗,++ (D); in the construction of φ([f ]) a label of f−1_(x

0) is

neces-sary; letρ ∈ φ([f ]) be the representation realized with such a label. Consider_e now a graph Γ as above and cut along it getting the polygon P ; note that, if eΓ is the graph f−1(Γ), cutting eΣ along eΓ and removing on open neighborhood of each yij, we obtain d copies of P . If we label such copies as P(1), . . . , P(d)where

P(j)_{is the sheet with the point labeled as x}

j, the branched covering realized by

edge pairing as above results to be f .

Let f be the covering constructed starting from ρ and label f−1(x0) such

that xj ∈ P(j) for each j. Taking e ∈ {ai, bi, ci} one of the generators of

π1(Σ \ B, x0), the lift ei of e−1 starting at some xi is the common edge of P(i)

and P(ei(1))_{. Hence, applying φ, we get a representation}

e

ρ such that

e

ρ(e)(i) = ei(1) = ρ(e)(i)

for each i.

We now give the following:

Definition 3.1.1. Let

S(D) = S(D)/∼

where:

• S(D) is the set of (2g + n)-tuples

(σ1, τ1, . . . , σg, τg, η1, . . . , ηn) ∈ S2g+n_d

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i)

ηi has cyclic structure λi ∀i = 1, . . . , n; (3.2)

ii) g Y i=1 [σi, τi] n Y i=1 ηi= id; (3.3) iii) hσ1, τ1, . . . , σg, τg, η1, . . . , ηni is transitive on {1, . . . , d}; (3.4) • (σ1, τ1, . . . , σg, τg, η1, . . . , ηn) ∼ (σ01, τ10, . . . , σg0, τg0, η10, . . . , η0n) if there

ex-ists σ ∈ Sd such that x = σx0σ−1 for each x ∈ {σi, τi, ηi}.

A tuple of permutations which satisfies the properties (3.2), (3.3) and (3.4) is called constellation. Two representatives of the same class in S(D) are said to be conjugate.

We realize a correspondence between R(D) and S(D).

Proposition 3.1.4. There exists a non-canonical bijection between R(D) and S(D). Proof. Let * a1, b1, . . . , ag, bg, c1, . . . , cn g Y i=1 [ai, bi] n Y i=1 ci + (3.5) be a presentation of π1(Σ \ B, x0).

Let [ρ] be an equivalence class in R(D). Then a representative of a class in S(D) is the constellation obtained taking a representative ρ and setting:

σi:= ρ(ai) ∀i = 1, . . . , g;

τi:= ρ(bi) ∀i = 1, . . . , g;

ηi:= ρ(ci) ∀i = 1, . . . , n.

The properties (3.2), (3.3) and (3.4) are clearly satisfied; another choice of representativeρ gives, by definition, an equivalent constellation._e

Conversely, given a constellation (σ1, τ1, . . . , σg, τg, η1, . . . , ηn) ∈ S 2g+n d , we

can define a representation ρ : π1(Σ \ B, x0) −→ Sd as follows:

ρ(ai) := σi ∀i = 1, . . . , g;

ρ(bi) := τi ∀i = 1, . . . , g;

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3.2 Notions of equivalence and constellations 34

and extending. Such a representation is well-defined from (3.2) and (3.3) and has transitive image from (3.4). Moreover, two equivalent constellations gives by definition two equivalent representations.

The two constructions are clearly one inverse of the other, hence realize the desired bijection.

3.2 Notions of equivalence and constellations

In this section, making use of the bijections shown in the previous section, we translate in terms of constellations the notions of equivalence giving rise to C_∗,++ (D), C₊+(D) and C+(D). We consider only branched coverings of the sphere,

namely data of the form

D = (eΣ, S, d, n, λ).

3.2.1 Marked coverings

Let us first consider the coverings with marking, where the local degrees over pj are given by λj and precisely the set C∗,++ (D).

From the propositions 3.1.3 and 3.1.4 it follows that the quotient C_∗,++ (D) corresponds to the quotient of S(D) where we identify two constellations if they are conjugate. Such a correspondence is non-canonical because it depends on the presentation (3.5) of π1(S \ B, x0) which, since g = 0, becomes

* c1, . . . , cn n Y i=1 ci + .

3.2.2 Unmarked positive coverings

Forgetting the marking (or equivalently allowing homeomorphisms at the lower level) allows us to exchange the positions of the points p1, . . . , pn. If a

homeo-morphism h : S −→ S switches pi and pi+1, then the image under h of the loops

δi and δi+1 around pi and pi+1 is shown in Figure 3.2-(b). Then, if we express

the loops δ0_j in figure 3.2-(c) in terms of the h(δj)’s, we get

δ0_j= h(δj) = δj if j 6= i, i + 1;

δ0_i= h(δi+1);

δ0_i+1= h(δ−1_i+1)h(δi)h(δi+1).

In Figure 3.3 we show the expression of δ0

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3.2 Notions of equivalence and constellations 35 x₀ h(δi) h(δi+1) h(pi) h(pi+1) pi pi+1 x₀ x₀ δi δi+1 (a) (b) (c)

Figure 3.2: a) Switching of pi and pi+1. b) The effect on loops of a switch of pi

and pi+1. c) Two generators of π1(S \ B, x0)

x₀

Figure 3.3: Expression of the new generator δ_i+10 in terms of the h(δj)’s.

Hence, since the representations are groups homomorphisms, on the related constellations we get the following transformation:

(η1, . . . , ηn) 7→ (η1, . . . , ηi+1, ηi+1−1ηiηi+1, . . . , ηn0).

Remark 3.2.1. If we denote by σi the transformation

(η1, . . . , ηn) 7→ (η1, . . . , ηi+1, η−1i+1ηiηi+1, . . . , ηn)

the following statements can be easily proved:

• if [ω] = [ω0_{] in S(D), then [σ}

i(ω)] = [σi(ω0)];

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• σiσi+1σi= σi+1σiσi+1.

Then the transformations σ1, . . . , σn−1generate an action of the braid group Bn

on S(D) (for further details see [LZ13]).

We give the following natural:

Definition 3.2.1. We define an equivalence relation between constellations in S(D) generated by:

• ω ∼+ω0 if they are conjugate;

• ω ∼+ω0 if ω0 = σi(ω) for some i = 1, . . . , n − 1.

The following theorem, due to H. Kneser, will be the crucial to translate the equivalence relation that defining C₊+(D) in terms of constellations.

Theorem 3.2.1. The space of automorphisms of S is path-connected. Proof. See [Kne26].

Then we have the following:

Theorem 3.2.2 (Hurwitz - Zdravkovska). The space C++(D) corresponds to the

quotient of S(D) where we identify two constellations ω and ω0 if ω ∼+ω0.

Proof. Let ω and ω0be two constellations in S(D) and let f, f0be two associated branched coverings. We first suppose that ω ∼+ ω0. If they are conjugate, then

from the result of Section 3.2.1 we have [f ] = [f0] in C_∗,++ (D) and hence also in C++(D). If ω0 = σi(ω) for some i = 1, . . . , n − 1 then we can choose a positive

homeomorphism h : S −→ S that permutes pi and pi+1 and leaves invariant the

other pk. Then h ◦ f and f0 are associated to the same ω0 in S(D); hence, again

from the result of Section 3.2.1, we have [h ◦ f ] = [f0] in C_∗,++ (D) so [f ] = [f0] in C+

+(D).

For the converse suppose that there exist positive homeomorphisms eh, h such that h ◦ f = f0_{◦ e}_{h. It is enough to prove that ω ∼}

+ ω0 when eh = id or when

h = id, since we can decompose the diagram

e Σ Σe Σ Σ e h f f0 h as follows: e Σ Σe Σe Σ Σ Σ. id f h ◦ f e h f0 h id

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If h = id, from the result of Section 3.2.1 we have ω = τ ω0τ−1 for a suitable τ ∈ Sd. If eh = id, according to Theorem 3.2.1, there exists a family Vt of

homeomorphisms of S depending on t ∈ [0, a] such that V0 = id and Va = h

and such that V is continuous on the space of homeomorphisms. Then, taking a point p ∈ S, its trajectory Vt(p) is a continuous function of t. Applying this

principle to the branching points p1, . . . , pn and recalling that h preserves this

set (as a set), we conclude that the collection of trajectories Vt(p1), . . . , Vt(pn)

is in fact a braid. But this concludes the proof, because the homeomorphism h acts on the constellations ω and ω0 as a braid, hence from Remark 3.2.1 they are related by the transformations σ1, . . . , σn−1(see also [LZ13]).

3.2.3 Unmarked coverings

During the rest of this section we will consider only branching data with three branching points and where the covering surface is S, namely of the form

D = (S, S, d, 3, λ)

From dessins to constellations and back. In Chapter 2 we have shown a correspondence between dessins d’enfant and branched coverings, while in the Section 3.1 we realized a correspondence between coverings and constellations. We explain such correspondences in the case of three branching point and when

e Σ = S.

Remark 3.2.2. If Γ is a dessin d’enfant realizing D, his the number of edges equals to 1 2 m1 X j=1 d1j+ m2 X j=1 d2j = 1 2(d + d) = 2.

If Γ is a dessin d’enfant realizing D, we can define a constellation (η1, η2, η3)

as follows:

• label the edges of Γ as {1, . . . , d};

• η1 is the permutation with one cycle for each preimage in f−1(p1) and

each cycle is obtained reading in a positive order the edge labels around each preimage;

• η2 is the permutation with one cycle for each preimage in f−1(p2) and

each cycle is obtained reading in a positive order the edge labels around each preimage;

• η3= η−12 η −1 1 .

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Conversely, to a constellation (η1, η2, η3) we associate a dessin d’enfant Γ

where:

• for each cycle (i1, . . . , ik) in the decomposition of η1the dessin has a vertex

in V1 of valence k labeled as (i1, . . . , ik);

• for each cycle (j1, . . . , ih) in the decomposition of η2the dessin has a vertex

in V2 of valence h labeled as (j1, . . . , jh);

• for all i = 1, . . . , d the dessin has an edge joining the vertices of V1and V2

labeled as the cycle containing i.

These two constructions, which are the compositions of the bijections described in Propositions 2.1.1, 3.1.3 and 3.1.4, are clearly the inverse of each other; then we can pass from constellations to dessins using such a correspondence.

Definition 3.2.2. We define a weaker equivalence relation ∼ between constel-lations in S(D) generated by ∼+and

(η1, η2, η3) ∼ (η3η2, η−12 , η −1 3 ).

Note that if ω ∼+ω0 then ω ∼ ω0. Then we have the following:

Proposition 3.2.3. The space C+(D) corresponds to the quotient of S(D) where

we identify two constellations ω and ω0 _{if ω ∼ ω}0_.

Proof. Let ω, ω0 be two constellations in S(D) and let f, f0 be two branched covering associated. We first suppose that [f ] = [f0] in C+(D). If [f ] = [f0] also

in C₊+(D), then ω ∼+ ω0 and then ω ∼ ω0. If [f ] 6= [f0] in C++(D), then there

exist eh, h satisfying h ◦ f = f0◦ eh; as in the proof of Theorem 3.2.1, it is enough to prove that ω ∼ ω0 in the following two cases:

• eh = id and h is negative;

• eh is negative and h = id.

Moreover, using the transformations σ1, σ2, we can suppose that h fixes p1, p2, p3.

If e_{h = id then the effect of a reflection h of S on the lifts of loops is the changing} of the base point x0; hence, from Lemma 3.1.2, we have [ω] = [ω0]. If h = id,

then we have a reflection eh and we can suppose that the points f−1(p1) and

f−1(p2) are on the reflection line; then, if Γ is a dessin d’enfant associated to f ,

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by construction, the constellation ω = (η1, η2, η3) becomes ω0= (η3η2, η2−1, η −1 3 ),

hence ω ∼ ω0.

Conversely, suppose that ω ∼ ω0. If ω ∼+ ω0 then, from Theorem 3.2.2,

[f ] ∼ [f0] in C₊+(D) and clearly also in C+(D). If

ω = (η1, η2, η3)

and

ω0= (η3η2, η2−1, η −1 3 )

then, taking e_{h a suitable reflection of S (which fixes f}−1(p1) and f−1(p2)),

the covering f0◦ eh is related to the constellation ω (since the effect of eh on a dessin related to f is that described above). Then [f ] = [f0] in C+(D) and this

concludes the proof.

Example 3.2.1. If n = 3, we note that the action of σ1, σ2 on constellations

can lead to

|C+

+(D)| < |C + ∗,+(D)|

only if λ contains repetitions. In this case, as we claimed for dessins in Chapter 2, two constellations are equivalent in C++(D) if:

• they are equivalent in C_∗,++ (D);

• λ1= λ2 and they are related by σ1and by conjugacy;

• λ2= λ3 and they are related by σ2and by conjugacy;

• λ1= λ2= λ3and they are related by σ2σ1 and by conjugacy;

• λ1= λ2= λ3and they are related by σ1σ2 and by conjugacy;

• λ1= λ2= λ3and they are related by σ1σ2σ1and by conjugacy,

since σican be applied only if λi = λi+1 and the others combinations of σ1and

σ2 can be reduced to these ones.

Let us consider the same data of Example 2.2.2:

D1= (S, S, 7, 3, ([4111], [3211], [7])),

D2= (S, S, 7, 3, ([61], [211111], [7])),

D3= (S, S, 7, 3, ([3211], [3211], [7])).

We enumerate, for each datum Di, all the inequivalent constellations (η1, η2, η3)

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• Up to conjugating by a permutation of S7, we can suppose that

η1= (1234567).

• If we want to leave η1 = (1234567) invariant, we can only conjugate the

constellations by powers of η1: if τ satisfies τ η1= η1τ and τ (1) = i, then

τ η1(1) = η1τ (1) ⇒ τ (2) = η1(i) = i + 1

τ (3) = i + 2 · · ·

• Since λ3 6= λ1, λ2, two constellations representing one of these data are

equivalent in C₊+(D) only if:

– they are equivalent in C_∗,++ (D);

– λ1= λ2 and they are related by σ1and by conjugacy;

We note also that two inequivalent constellations in C₊+(D) are equivalent in C+(D) if they are related by the move

(η1, η2, η3) ∼ (η3η2, η2−1, η −1 3 ).

Taking the dessins d’enfant in Figure 2.10, we can label the edges as in Figure 3.4. 7 6 1 2 3 4 5 7 4 1 2 3 5 6 4 2 1 3 6 7 5

Figure 3.4: A labeling of the edges of the dessins representing C_∗,++ (D1).

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ω1 ω2 ω3

η1= (7654321) η1= (7654321) η1= (4532671)

η2= (1456) η2= (1467) η2= (1234)

η3= (67)(123) η3= (45)(123) η3= (35)(176)

After conjugations, a set of constellations representing C_∗,++ (D1) is

ω1 ω2 ω3

η1= (1234567) η1= (1234567) η1= (1234567)

η2= (4327) η2= (1742) η2= (1743)

η3= (12)(657) η3= (34)(765) η3= (23)(657)

Hence we find again

|C_∗,++ (D1)| = 3.

Since λi6= λj we get

|C+

+(D1)| = |C∗,++ (D1)| = 3.

However, for ω2 we obtain

(η3η2, η−12 , η −1

3 ) = ((1657342), (2471), (34)(567))

which is conjugated to ((1234567), (7641), (56)(243)) and hence to ω1, so we find

again

|C+(D1)| = 2.

If we consider the dessin in Figure 2.11 we can label its edges as in Figure 3.5. 1 6 5 4 3 2 7

Figure 3.5: A labeling of the edges of the dessin representing C_∗,++ (D2).

From the above construction we get the following constellation:

ω η1= (1234567)

η2= (654321)

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Hence we find again

|C_∗,++ (D2)| = |C++(D2)| = |C+(D2)| = 1

Similarly we can consider the dessins in Figure 2.12 labeling the edges as in Figure 3.6.

Γ1

Γ2

Γ3

Γ4

Γ9

Γ5

Γ6

Γ7

Γ8

2 5 1 6 7 4 3 1 4 2 7 6 5 3 1 5 3 7 6 4 2 2 6 1 4 3 7 5 3 6 5 4 7 1 2 6 3 4 2 5 1 7 3 1 2 7 6 5 4 5 1 7 6 4 2 3 1 4 6 3 2 7 5

Figure 3.6: A labeling of the edges of the dessins representing C_∗,++ (D3).

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3.2 Notions of equivalence and constellations 43 ω1 ω2 ω3 η1= (1234567) η1= (1234567) η1= (1234567) η2= (15)(243) η2= (23)(154) η2= (34)(152) η3= (25)(176) η3= (24)(176) η3= (35)(176) ω4 ω5 ω6 η1= (1234567) η1= (1234567) η1= (1234567) η2= (12)(364) η2= (16)(243) η2= (34)(162) η3= (56)(173) η3= (17)(265) η3= (17)(365) ω7 ω8 ω9 η1= (1234567) η1= (1234567) η1= (1234567) η2= (46)(132) η2= (12)(365) η2= (23)(165) η3= (56)(174) η3= (45)(173) η3= (17)(154)

Hence we find again

|C+

∗,+(D3)| = 9.

The following pairs are related by σ1 and by conjugacy:

• (ω2, ω7); • (ω3, ω5); • (ω8, ω9); so we get |C+ +(D3)| = 6.

The following pairs are related by the relation (η1, η2, η3) ∼ (η3η2, η2−1, η −1 3 ) and

by conjugacy:

• (ω2, ω3);

• (ω4, ω6);

hence we find again

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Chapter 4

Mednykh’s formula

In this chapter we provide a formula for the cardinality of C_∗,++ (D) following the idea of A. Mednykh in [Med84]. In particular we make use of the Burnside Lemma (Lemma 4.1.1) and of the wreath product.

Fix a branching datum

D = (eΣ, Σ, d, n, λ).

From Propositions 3.1.3 and 3.1.4 we know that |C_∗,++ (D)| equals the number of orbits of the action of Sd on the set of constellations

S(D) = (

(σ1, τ1, . . . , σg, τg, η1, . . . , ηn) ∈ S 2g+n d :

ηi has cyclic structure λi ∀i = 1, . . . , n, g Y i=1 [σi, τi] n Y i=1 ηi= id, hσ1, τ1, . . . , σg, τg, η1, . . . , ηni is transitive on {1, . . . , d}} ) where σ(γj) 2g+n j=1 = σγjσ−1 2g+n j=1 for all γ = (γj) 2g+n j=1 ∈ S(D) and σ ∈ Sd.

4.1 Orbits of the action of S

d

on S(D)

To compute the number of orbits of this action, we need the following:

Lemma 4.1.1 (Burnside Lemma). Let G be a finite group which acts on a set X. Then the number of orbits of the action is

1 |G| X g∈G |Xg| where Xg = {x ∈ X : g · x = x}. 44

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4.1 Orbits of the action of Sd on S(D) 45

Proof. We first observe that

X g∈G |Xg| = X x∈X |Gx|

where Gxis the stabilizer of x in G.

If G · x is the orbit of an element x, we also find

|G · x| = |G| |Gx| whence 1 |G| X g∈G |Xg| = 1 |G| X x∈X |Gx| = X x∈X 1 |G · x|.

Let Ω be the set of orbits. Splitting our summation and recalling that the set of the orbits gives a partition of X, we obtain

X x∈X 1 |G · x| = X A∈Ω X x∈A 1 |G · x| = X A∈Ω X x∈A 1 |A| = X A∈Ω 1

which is in fact the number of orbits.

To compute the number of orbits of the action of Sd on S(D) we need to

characterize the centralizer in Sd of the subgroup generated by an element of

S(D). We need the following:

Definition 4.1.1. A permutation σ ∈ Sd is semiregular if for all i = 1, . . . , n

we have

hσi_i= {id} where hσi_i= {σp∈ hσi : σp_{(i) = i}.}

A permutation σ ∈ Sd is regular if it decomposes into disjoint cycles of the

same length.

We prove the following:

Lemma 4.1.2. If σ is semiregular then it is regular.

Proof. Suppose that the cyclic decomposition of σ contains cycles (a1· · · ap)

and (b1· · · bq) of different lengths 1 ≤ p < q; then σp fixes a1 but not b1 and

this is a contradiction.

We prove that a σ in the centralizer of hσ1, τ1, . . . , σg, τg, η1, . . . , ηni is

regu-lar. We need the following:

Proposition 4.1.3. If G < Sd is transitive on {1, . . . , d} and σ is in the

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4.1 Orbits of the action of Sd on S(D) 46

Proof. By Lemma 4.1.2, it is enough to show that σ is semiregular. If it is not, there exist i ∈ {1, . . . , d} with hσi_i 6= {id}; suppose that σp_{(i) = i and σ}p_{6= id.}

For every τ ∈ G we obtain

τ (i) = τ σp(i) = σpτ (i)

and, from transitivity of G, we have σp_{= id, which is a contradiction.}

So we get the following:

Corollary 4.1.4. Let ω ∈ S(D); then the elements in the centralizer of hωi in Sd are regular.

Hence, from Lemma 4.1.1, we get

|C_∗,++ (D)| = 1 d! X σ∈Sd σ regular |S(D)σ|, (4.1)

so we are interested in the centralizer of regular permutations.

Given ` and m such that `m = d, we now want to characterize the centralizer of a regular permutation which splits into m disjoint cycles of length `. We need the definition of wreath product.

Let C`be the cyclic group Z/`Z; then C`oSmis the wreath product between

C`and Smand it is constructed as follows; consider the group

H =

m

Y

i=1

C_`(i) with the action of Smdefined as

σ((ci)i=1,...,m) = (cσ−1_(i))_i=1,...,m

where σ ∈ Sm. The wreath product is the semidirect product

C`o Sm= H o Sm

with the operation

(a, σ) • (b, τ ) = (aσ(b), στ ).

We show that the wreath product C`o Smcan be seen as a subset of Sd. Let

a = (c1, . . . , cm, aα) ∈ C`o Sm; then we associate to a the permutation σa∈ Sd

defined as follows: if we write i as k` + h with 0 ≤ h < `, then

Surface branched covers and Hurwitz numbers

In Supremae Dignitatis · Universit`

a di Pisa

Surface branched covers

and Hurwitz numbers

Relatore:

Chiar.mo Prof.

Carlo Petronio

Presentata da:

Filippo Sarti

Sessione di Luglio

Anno Accademico 2018/19

Introduction

Contents

Chapter 1

Branched coverings and

notions of equivalence

1.1

Branched coverings

1.2

Notions of equivalence between coverings

1.3

Basic results

Chapter 2

Dessins d’enfant

2.1

From positive marked coverings

to dessins and back

2.2

Unmarked coverings

Γ1

Γ2

Γ3

Γ1

Γ2

Γ3

Γ4

Γ9

Γ5

Γ6

Γ7

Γ8

Chapter 3

Constellations

3.1

From positive marked coverings

to constellations and back

3.2

Notions of equivalence and constellations

3.2.1

Marked coverings

3.2.2

Unmarked positive coverings

3.2.3

Unmarked coverings

Γ1

Γ2

Γ3

Γ4

Γ9

Γ5

Γ6

Γ7

Γ8

Chapter 4

Mednykh’s formula

4.1

Orbits of the action of S

on S(D)