This is the accepted version of the journal article: Marín Pérez, David; Mattei, Jean-François; Salem, Eliane. «Topological Mod- uli Space for Germs of Holomorphic Foliations». International Mathematics Research Notices, Vol. 2020, Issue 23 (November 2020), p. 9228-9292. DOI 10.1093/imrn/rny244 This version is available at https://ddd.uab.cat/record/287179 under the terms of the license TOPOLOGICAL MODULI SPACE FOR GERMS OF HOLOMORPHIC FOLIATIONS DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM Abstract. This work deals with the topological classification of germs of singular foliations on (C2 , 0). Working in a suitable class of folia- tions we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topological classes in terms of the coho- mology of a new algebraic object that we call group-graph. This moduli space may be an infinite dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological structure. Contents 1. Introduction 2 2. Statement of results 3 2.1. Marking of a foliation 3 2.2. Globalization of topological equivalences 3 2.3. The Teichmüller space of foliations 5 2.4. Topological moduli space of a marked foliation 5 2.5. The generic case: non-degenerate foliations 6 2.6. Foliations of finite topological type 7 3. Group-graphs 8 4. Automorphism group-graph 12 5. Symmetry group-graph 16 6. Foliations of finite type 22 7. Non-degenerate foliations 27 8. Examples 29 9. Exponential and disconnected group-graphs 32 10. Proof of Theorem D 35 11. Appendix 42 11.1. Step (i): Equality of Camacho-Sad indices 44 11.2. Step (ii): Monodromy and holonomy 45 11.3. Step (iii): Conjugation at non-exceptional cut-components 49 11.4. Step (iv): Construction along the exceptional cut-components 50 Date: July 31, 2020. 2010 Mathematics Subject Classification. Primary 37F75; Secondary 32M25, 32S50, 32S65, 34M. Key words and phrases. Complex dynamical systems, singular foliations, holonomy, topological invariants, Teichmüller space, moduli space, non-abelian cohomology. D. Marín acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through grants MTM2011-26674-C02-01, MTM2015-66165-P and the "María de Maeztu" Programme for Units of Excellence in R&D (MDM-2014-0445). 1 2 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM 11.5. Step (v): Extension and gluing 51 References 51 1. Introduction This work deals with the topological classification of germs of singular fo- liations on (C2 , 0). To every (possibly dicritical) foliation F we can associate the separatrix set SepF , that is the collection of all germs at 0 ∈ C2 of in- variant irreducible analytic curves, called separatrices, its minimal reduction map EF : (MF , EF ) → (C2 , 0), cf. [5], and the marked exceptional divisor EF = (EF , ΣF , ıF ), where ΣF := Sing(F ] ) is the finite set consisting of the singular points of the foliation F ] := EF∗ F and ıF is the intersection pairing of EF = EF−1 (0) in MF . The topological class of SepF is clearly a topological invariant of F. In this paper we will assume that F is a generalized curve, i.e. F ] has no saddle-node singularities. The topological class [EF ] of EF (as a marked intrinsic curve) is then a topological invariant of F because in this situation EF is also the minimal desingularization map of SepF , cf. [3]. We know [18] that under some assumptions the Camacho-Sad indices of F ] at the points of ΣF and the holonomy representations (up to inner auto- morphisms) of every component of EF are also topological invariants of the germ F at 0 ∈ C2 . Our purpose in this work is to describe the set of all other topological invariants and highlight its geometric and algebraic structure. MAIN RESULT. Under generic conditions, (a) there exists an analytic family of foliations parametrized by a fi- nite dimensional space which gives all the topological types once we fix the topological class of the marked exceptional divisor, the Camacho-Sad indices and the holonomy representations; (b) the quotient of this complete family by the topological equivalence relation is naturally isomorphic to the abelian group F ⊕ B ⊕λj=1 (C∗ /αjZ ) ⊕ (C∗ )ν Z ,  where αj ∈ C∗ , F is a finite abelian group, Z is a finite subgroup, B is a direct sum of β totally disconnected subgroups of the uni- tary group U(1) and the natural numbers β, λ, ν only depend on the (combinatorics of the) local types of the singularities inside the exceptional divisor. We will also give an explicit characterization of those foliations satisfying Assertion (a) in the main result above, that we will call finite type foliations. We thank the anonymous reviewers for their careful reading of our man- uscript and their many useful comments and suggestions. TOPOLOGICAL MODULI 3 2. Statement of results 2.1. Marking of a foliation. To give a precise sense to our problem let us call marked divisor any collection E  = (E, Σ, ı) consisting of a compact curve with normal crossings E whose irreducible components are biholomor- phic to P1 , a finite subset Σ of E and a symmetric map ı : Comp(E)2 → Z, Comp(E) denoting the set of irreducible components of E. We will denote by E d ⊂ E the union of the irreducible components of E that do not contain any point of Σ; we call them dicritical components. A marking of a foliation F by E  will be a homeomorphism f : E → EF sending Σ onto ΣF compatible with the intersection pairing: ıF (f (D), f (D0 )) = ı(D, D0 ). In this way, the holonomy representations and the Camacho-Sad indices of all pairs F  := (F, f ) can now be associated to two common sets of indices: the set CE  := Comp(E \ E d ) of irreducible components of E \ E d and the set IE  := {(D, s) ∈ CE  × Σ | s ∈ D} . Indeed, we define  CSF := (CS(F  , D, s))(D,s)∈IE  , CS(F  , D, s) := CS(F ] , f (D), f (s)) ,    ] ḢF := (ḢD F )D∈CE  , F HD := HfF(D) ◦ f∗ : π1 (D \ Σ, ·) −→ Diff(C, 0) , ] where HfF(D) is the F ] -holonomy representation of π1 (f (D) \ ΣF , ·) in the F is its  group Diff(C, 0) of germs of holomorphic automorphisms of (C, 0), ḢD class up to inner automorphisms, f∗ is the isomorphism induced by f at the fundamental groups level and CS(F ] , f (D), f (s)) is the Camacho-Sad index of F ] along f (D) at f (s). Let us denote by Fol(E  ) the set of germs of generalized curves F at 0 ∈ C2 for which there exists a marking f : E → EF of F by E  . Our general goal is to describe a generic subset of the quotient set [Fol(E  )]C 0 of the set Fol(E  ) by the equivalence relation: • F ∼C 0 G if F and G are topologically equivalent as germs at 0 ∈ C2 . 2.2. Globalization of topological equivalences. Consider now the equiv- alence relation: • F ∼E G if F ] and G ] are topologically conjugated, as germs along the exceptional divisors, by a germ of a homeomorphism (MF , EF ) → (MG , EG ) which is holomorphic at each point of ΣF \ N CF , N CF denoting the subset of the singular points of EF , called nodal cor- ners, where the Camacho-Sad index of F ] is a strictly positive real number. Clearly relation ∼E is stronger than ∼C 0 , but they will coincide on a generic class of foliations when E  fulfills the following condition 4 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM (TC) The closure of each connected component of E \ E d contains an irre- ducible component D with card(D ∩Σ) 6= 2, i.e. there is no connected component of E \ E d as in Figure 1. s0 s00 • • ··· • • Figure 1. The only situation excluded by Condition (TC). Every divisor is non-dicritical and the elements of Σ are s0 , s00 and the intersection points of the divisors; dicritical com- ponents may intersect any component. To specify the notion of genericity let us call cut-component of EF any closure C of a connected component of EF \ (EFd ∪ N CF ); if card(D ∩ Σ) ≤ 2 for each D ∈ Comp(C) we will say that C is exceptional. Now consider the following transverse rigidity condition: (TR) Any non-exceptional cut-component of EF contains an irreducible component with topologically rigid1 holonomy group. The Krull-open density in Fol(E  ) of the subset Foltr (E  ) consisting of the foliations F fulfilling Condition (TR) is proven in [10]. Theorem A. If E  satisfies condition (TC) then the relations ∼E and ∼C 0 are equal on Foltr (E  ). In other words [Foltr (E  )]E = [Foltr (E  )]C 0 ⊂ [Fol(E  )]C 0 , [X]E and [X]C 0 denoting the quotient of a subset X ⊂ Fol(E  ) by the relations ∼E and ∼C 0 respectively. This result, proven in Appendix (Theorem 11.4), is an extended version of Main Theorem of [18]. Remark 2.1. Theorem A implies that under the hypothesis (TR) and (TC) the collection of Camacho-Sad indices at the singular points of F ] is a topo- logical invariant of the germ of F at 0. The topological classification of logarithmic foliations obtained by E. Paul shows [28, Théorème 3.5] that Condition (TR) is necessary for this. When Condition (TC) is not sat- isfied it is easy to construct topologically conjugated foliations with same separatrices but different Camacho-Sad indices. Indeed, on a neighborhood of a cut-component described in Figure 1, all the foliations with non-real Camacho-Sad indices are topologically conjugated.  1We recall that a subgroup G of the group Diff(C, 0) of germs of biholomorphisms of C at 0 is called topologically rigid if every topological conjugation between G and another subgroup G0 ⊂ Diff(C, 0) is necessarily conformal. This class contains the non-solvable groups [25] and the non-abelian groups with dense linear part [7, Théorème 2]. TOPOLOGICAL MODULI 5 2.3. The Teichmüller space of foliations. In order to describe [Foltr (E  )]C 0 we consider the set MFol(E  ) of marked by E  foliations F  = (F, f ) and we adapt the equivalence relation ∼E in Fol(E  ) to MFol(E  ) by means of • (F, f ) ∼ (G, g) if there is a germ of homeomorphism Φ : (MF , EF ) → (MG , EG ) that conjugates F ] and G ] , is holomorphic at each point of ΣF \N CF and its restriction to EF is isotopic to g ◦f −1 by an isotopy fixing ΣF . We define the (topological) Teichmüller space as the quotient set Mod(E  ) := MFol(E  )/∼ so that the Forgetful map Mod(E  ) −→ [Fol(E  )]C 0 , [F, f ] 7→ [F]C 0 is well defined. We consider the action ϕ̇ ? [F, f ] := F, f ◦ ϕ−1 , ϕ̇ ∈ Mcg(E  ) , (F, f ) ∈ MFol(E  ) ,   on Mod(E  ) of the Mapping Class Group Mcg(E  ) of E  defined as the group2 of isotopy classes of C 0 -automorphisms of E leaving the set Σ and the inter- section form ı invariant. A direct consequence of Theorem A is: Corollary 2.2. If E  satisfies condition (TC) then the fibers of the Forgetful map over Foltr (E  ) are exactly the orbits of the action of the mapping class group Mcg(E  ). Thus [Foltr (E  )]C 0 ' Modtr (E  )/Mcg(E  ) , where Modtr (E  ) := {[F, f ] ∈ Mod(E  ) | F ∈ Foltr (E  )}. 2.4. Topological moduli space of a marked foliation. In order to de- scribe [Foltr (E  )]C 0 we are led to study Modtr (E  ). In fact, we give a descrip- tion of the whole Mod(E  ) without the assumption (TC) on E  . We obtain it by fixing the Camacho-Sad indices and the holonomy representations. In other words we give a description of each nonempty fiber of the well-defined map He := CS × Ḣ : Mod(E  ) −→ CIE  × ṘE  , [F  ] 7→ (CSF  , ḢF  ) , where ṘE  is the set of conjugacy classes of group morphisms from the free product of the groups π1 (D \ Σ, ·) for all D ∈ CE  with values in Diff(C, 0). Definition 2.3. We call topological moduli space of [F  ] ∈ Mod(E  ) the fiber of H e above H([F e  ]), that is the set n     o Mod([F  ]) := [G  ] ∈ Mod(E  ) | CSF = CSG , ḢF = ḢG . Here the term topological does not refer to the C 0 -equivalence of folia- tions but to the equivalence relation ∼ on the marked foliations. Indeed, as we have seen in Remark 2.1, if E  does not satisfy (TC) or F ∈ Fol(E  ) does not satisfy Condition (TR) then there may exist G ∈ Fol(E  ) such that F ∼C 0 G and [F, f ] 6= [G, g] for any markings f, g. However, the descriptions 2In fact, it is an extension of a finite group by a direct product of pure Artin groups. 6 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM of Mod([F  ]) given in the following results do not assume conditions (TR) or (TC) anymore. We will consider in Section 3 a new algebraic notion, which we call group- graph, that will be the key tool in the whole paper. It allows us, by com- bining Theorems 4.9 and 5.14, to obtain a bijection between this moduli space Mod([F  ]) and the cohomology of a suitable group-graph defined in  Section 5, namely the symmetry group-graph SymF . If we call F  -cut- component of E any inverse image f −1 (C) of a cut-component C of EF and if we denote by AF  the dual graph of the disjoint union of all the F  -cut- components of E, then one can prove: Theorem B. If F  ∈ MFol(E  ) then we have a natural bijection ∼  Mod([F  ])−→H 1 (AF  , SymF ) . Without any other assumption the computation of this cohomological space is difficult and the usefulness of this result is essentially theoreti- cal. However it will allow us in Section 8 to construct examples for which Mod([F  ]) is an infinite dimensional functional space. To get finiteness we shall need to restrict to some Krull open dense subsets of Fol(E  ) by requir-  ing conditions on HF depending only on a finite jet3 of a differential 1-form defining F. 2.5. The generic case: non-degenerate foliations. Let us call singular chain4 of the dual graph of EF any sequence D0 , . . . , D` , ` ≥ 1, of invariant irreducible components of EF such that: a) D0 and D` contain at least 3 singular points of F ] , b) Di ∩ ΣF = {si , si+1 } with si = Di−1 ∩ Di , if 1 ≤ i ≤ ` − 1. At all the points si , 1 ≤ i ≤ ` − 1, F ] has the same property of normalization and we will say that the chain is linearizable, resonant normalizable or non- normalizable, non-resonant, if F ] fulfills this property at these points si . Definition 2.4. A germ of a foliation F is called non-degenerate if it sat- isfies the following properties: ] F ) of any invariant component D of E with (i) the holonomy group Im(HD F card(D ∩ ΣF ) ≥ 3, is non-abelian; (ii) for any singular chain D0 , . . . , D` in EF , the local holonomies of F ] at the singular points si = Di−1 ∩ Di , i = 1, . . . , `, are non-periodic. The subset of Fol(E  ) of all non-degenerate foliations will be denoted by Folnd (E  ). 3It is well known that for p large enough all the foliations defined by a 1-form sharing the same p-jet have the same reduction map, the same singular points on the exceptional divisor, the same Camacho-Sad indices and the same dicritical components. 4Notice that a singular chain may not correspond to a chain of the dual graph of E, in the usual sense. Indeed the interior vertices Di , 0 < i < `, may meet dicritical components and the number of their adjacent edges can be greater than two, and also D0 or D` may have only two adjacent edges. Conversely a chain of the dual graph of E may not be a singular chain because there may exist points of Σ outside the singular locus of E. TOPOLOGICAL MODULI 7 Remark 2.5. Theorem 6.2.1 of [24] claims the Krull open density, in the set of formal 1-forms defining foliations of second kind (in particular non- dicritical generalized curves), of the set of 1-forms defining foliations fulfilling conditions (i) and (ii) of Definition 2.4. The proof, which is given in the formal non-dicritical context, remains valid for all holomorphic foliations that are generalized curves.  Theorem C. Let F  = (F, f ) ∈ MFol(E  ) be a marked foliation with F non-degenerate. Then we have an identification:  . Mod([F  ]) ' F ⊕ B ⊕λj=1 (C∗ /αjZ ) ⊕ (C∗ )ν Z, where αj ∈ C∗ , F is a finite abelian group, Z is a finite subgroup, B is a direct sum of β totally disconnected subgroups of the unitary group U(1) and λ, ν and β are respectively the number of linearizable, resonant normalizable and non-resonant non-linearizable singular chains contained in cut-components of EF , the factor F corresponding to resonant non-normalizable chains. In particular, λ + ν is equal to the codimension τF of F given in Definition 6.9. The naturality of this identification will be explained by Assertion (b) in Theorem D below. 2.6. Foliations of finite topological type. We have seen that Mod([F  ]) is endowed with a very specific structure of topological group of finite dimen- sion if F ∈ Folnd (E  ). However this finiteness property continues to be valid for a larger class of foliations that we shall call finite type foliations. The set Folft (E  ) ⊃ Folnd (E  ) of these foliations is defined in Section 6 and it is opti- mal for finiteness as Example 3 in Section 8 shows. Furthermore we obtain complete families of marked foliations parametrized by finite dimensional spaces, completeness meaning that the family contains all the topological types of marked foliations by E  with prescribed Camacho-Sad indices and holonomies. Such a family of marked foliations is called SL-equisingular de- formation, where SL stands for semi-local. We will give a precise formulation of this notion in Definition 10.2, Step (vi) of Section 10. Theorem D. Let F  = (F, f ) ∈ MFol(E  ) be a marked foliation with F of finite type. Then Mod([F  ]) admits an abelian group structure with identity element [F  ] such that: (a) there is an exact sequence Λ Γ Zp → CτF → Mod([F  ]) → D → 0 (1) where D is a totally disconnected topological abelian group and τF is the codimension of F given by Definition 6.9; (b) given a section i 7→ [Fi , fi ] ∈ Γ−1 (i) of Γ, there is a family parametrized by i ∈ D of SL-equisingular deformations (Fi,t c ) t∈CτF of Fi such that τ c c c being the marking for all t ∈ C F we have [Fi,t , fi,t ] = Λ(t) · [Fi , fi ], fi,t induced by fi and the dot · denoting the operation in the group Mod([F  ]). The superscript c in the deformation stands for complete. The group D will be specified in the proof (Step (i) of Section 10): it is a quotient of a product of a finite family of totally discontinuous subgroups of U(1), that can be 8 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM uncountable. However, let us highlight that D is “generically finite” in the following sense: - there is a subset N of zero measure in the algebraic subset CS(Mod(E  )) of CIE  such that if CS([F  ]) 6∈ N , the formally linearizable singularities of F ] are holomorphically linearizable, and in this case we can prove that D is finite. As a direct consequence of the proof we can see that if H([F e  ]) = H([G e  ])   then the sets Mod([F ]) and Mod([G ]) coincide. However their respective abelian group structures are related by the map µ 7→ γµ where γ = [G  ] ∈ Mod([F  ]). The paper is organized as follows. Section 3 is devoted to general notions about group-graphs and their cohomology. In Section 4 we introduce our first  group-graph AutF over AF  and we construct a natural bijection between  Mod([F  ]) and H 1 (AF  , AutF ). In Section 5 we introduce a simpler group-   graph SymF over AF  having the same cohomology as AutF , obtaining in  this way Theorem B. In Section 6 we prove that the group-graph SymF is  abelian over a subgraph RF  ⊂ AF  . Then the cohomology H 1 (RF  , SymF ) is an abelian group. We also prove Theorem 6.8 asserting that under finite  type hypothesis the cohomology of SymF over AF  and RF  coincide. The- orem C is proven in Section 7. Some applications of Theorems B and C are  discussed in Section 8. In Section 9 we introduce the group-graphs ExpF  and DisF which allow us to compute the continuous and discrete parts of  the cohomology group of SymF in Section 10. This computation jointly with Theorem B will conclude the proof of Theorem D. Finally, Theorem A is a direct consequence of Theorem 11.4 proven in Appendix. 3. Group-graphs In this section we will introduce and study the algebraic notion of group- graph which differs in an essential way from the notion of graph of groups introduced by Serre in [34] and that will be a key tool of this work. Let A be a finite graph with vertex set VeA and edge set EdA . In all the paper the graph we consider will be without loops, i.e. without edges e ∈ EdA with ∂e = {v} for some v ∈ VeA . Denote by IA := {(v, e) ∈ VeA × EdA | v ∈ ∂e} the set of oriented edges of A. Definition 3.1. A group-graph G over A is the data of groups Gv and Ge for each vertex v ∈ VeA and each edge e ∈ EdA , and of group morphisms ρev : Gv → Ge for each (v, e) ∈ IA which are called restriction maps. A morphim α : F → G between group-graphs over the same graph A is given by group morphisms αv : Fv → Gv and αe : Fe → Ge such that the diagram αv Fv −→ Gv e ρv ↓ ↓ ρev αe Fe −→ Ge TOPOLOGICAL MODULI 9 commutes for each (v, e) ∈ IA . A group-graph G is called abelian if all the groups Gv and Ge are abelian. e1 e2 •v1 v•v0 v•v2 Gv1 / Ge o Gv0 / Ge o Gv2 1 2 e3 e4   v•v3 v•v4 Ge 3 Ge 4 O O Gv3 Gv4 Figure 2. Dual tree of the desingularization of the curve (y 2 − x3 )2 − x2 y 3 = 0 in the sense of Definition 4.1 and a group-graph G over it. Remark 3.2. One can define in a natural way the notions of image and kernel of a group-graph morphism α : F → G, which are group-graphs over the same graph. In the abelian case the cokernel can also be defined as a group-graph. We also have an obvious notion of restriction of a group-graph over a graph to a subgraph.  Definition 3.3. Let G be a group-graph over a graph A. The cochain com- plex of G consists of Y Y C 0 (A, G) := Gv and C 1 (A, G) := Gv,e , Gv,e := Ge , v∈VeA (v,e)∈IA jointly with the right action C 0 (A, G) × C 1 (A, G) → C 1 (A, G) given by (gv ) ?G (gv,e ) := (ρev (gv )−1 gv,e ρev0 (gv0 )) where ∂e = {v, v 0 }. The set of 0-cocycles is the subset H 0 (A, G) of C 0 (A, G) of all elements (gv ) satisfying the relations ρev (gv ) = ρev0 (gv0 ) whenever ∂e = {v, v 0 }. Let us consider the set of 1-cocycles Z 1 (A, G) := {(gv,e ) ∈ C 1 (A, G) | gv,e gv0 ,e = 1 when ∂e = {v, v 0 }} ⊂ C 1 (A, G) which is invariant by the action of C 0 (A, G) and its quotient, the 1-cohomology set: H 1 (A, G) := Z 1 (A, G)/C 0 (A, G). Remark 3.4. The sets of cochains C 0 (A, G) and C 1 (A, G) are groups but in general H 0 (A,Q G) and H 1 (A, G) are merely sets (although Z 1 (A, G) is in bijection with Ge which is a group). However, if the group-graph G e∈EdA is abelian then we can consider the group C 2 (A, G) := Q Ge and the e∈EdA morphisms ∂0 : C 0 (A, G) → C 1 (A, G) and ∂1 : C 1 (A, G) → C 2 (A, G) given by ∂ 0 (gv ) := (gv ) ?G (1) = (ρev (gv )−1 ρev0 (gv0 )) and ∂ 1 (gv,e ) = (gv,e gv0 ,e ) 10 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM whenever ∂e = {v, v 0 }. It turns out that ∂ 1 ◦∂ 0 = 1 and we obtain a complex ∂0 ∂1 C ∗ (A, G) : C 0 (A, G) → C 1 (A, G) → C 2 (A, G) → 1 whose cohomology is H 0 (C ∗ (A, G)) = H 0 (A, G) = ker ∂ 0 , H 1 (C ∗ (A, G)) = H 1 (A, G) = Z 1 (A, G)/∂ 0 C 0 (A, G) and H 2 (C ∗ (A, G)) = 1 because Z 1 (A, G) = ker ∂ 1 and coker ∂ 1 = 1.  The following result is straightforward. Lemma 3.5 (Functoriality). Every morphism α : F → G of group-graphs over the same graph A induces well-defined maps αi : H i (A, F ) → H i (A, G), for i = 0, 1, given by α0 (gv ) = (αv (gv )) and α1 ([gv,e ]) = [αe (gv,e )]. Moreover, if F and G are abelian then αi are morphisms. α β A short sequence 1 → F → G → J → 1 of morphisms of group-graphs over the same graph A is exact if for all a ∈ VeA ∪ EdA the corresponding α βa short sequence of groups 1 → Fa →a Ga → Ja → 1 is exact. In the abelian case the complexes of abelian groups considered in Remark 3.4 fit into a short exact sequence 1 → C ∗ (A, F ) → C ∗ (A, G) → C ∗ (A, J) → 1. The following results are classical. Lemma 3.6 (Long exact sequence). If 1 → F → G → J → 1 is a short exact sequence of abelian group-graphs over the same graph A, then there is a long exact sequence 1 → H 0 (A, F ) → H 0 (A, G) → H 0 (A, J) → H 1 (A, F ) → H 1 (A, G) → H 1 (A, J) → 1. Lemma 3.7 (Mayer-Vietoris). Let G be an abelian group-graph over a graph A. If A0 and A1 are subgraphs of A such that A = A0 ∪ A1 then there is an exact sequence 1 → H 0 (A, G) → H 0 (A0 , G) ⊕ H 0 (A1 , G) → H 0 (A0 ∩ A1 , G) → H 1 (A, G) → H 1 (A0 , G) ⊕ H 1 (A1 , G) → H 1 (A0 ∩ A1 , G) → 1. Proof. We have a short exact sequence of complexes of abelian groups 1 → C ∗ (A, G) → C ∗ (A0 , G) ⊕ C ∗ (A1 , G) → C ∗ (A0 ∩ A1 , G) → 1 and we consider the long exact sequence of cohomology.  Definition 3.8. The valency in A or the A-valency of a vertex v of A is the cardinality valA (v) of the set {e ∈ EdA ; v ∈ ∂e}. We say that v is an extremity of A if valA (v) = 1; in that case we will write v ∈ ∂A. A partial dead branch (M, v0 ) of A is the data of a vertex v0 of A called attaching point and a connected subgraph M of A such that: • M contains an extremity v00 of A, • all its vertices are of valency 2 in A, except possibly its extremities that are v00 and v0 . Notice that M is always a chain. When M 6= A and valA (v0 ) ≥ 3 one says that M is a dead branch of A. We define a total order 0 such that the compact disc D2ε ⊂ f (D) defined by |u| ≤ 2ε, is contained in W and in a definition domain of φ. Denote by C the compact annulus contained in D2ε given by ε ≤ |u| ≤ 2ε. By the implicit function theorem, there is a holomorphic function τ defined in an open neighborhood Ω of C, that verifies: (u ◦ φ)(m) = u ◦ ΦZ τ (m) (m) and τ|f (D) = 0 , ΦZ ∞ function t being the flow of the previous vector field Z. Let us take a C α : f (D) → R with compact support in Ω ∩ f (D), that is equal to 1 on a neighborhood of C. The map ξ : m 7→ ξ(m) := ΦZ α(u(m))τ (m) (m) is a C ∞ -diffeomorphism, because its restriction to f (D) is the identity and moreover it is a local diffeomorphism. Indeed using coordinates (u, z) at each point of f (D), with z a local first integral of the foliation, we easily see that the jacobian matrix of ξ is the identity. Clearly χ := φ ◦ ξ −1 coincides with φ on a neighborhood of f (s), it preserves the foliation and it leaves invariant each fiber of u: ∆c := {u = c}, ε ≤ |c| ≤ 2ε . Thus the restriction χ|∆c of χ to ∆c leaves invariant the orbits of the ho- lonomy map of F ] around f (s) represented on ∆c -which is equal to the restriction of ΦZ 2π to ∆c . By Lemma 5.7, χ|∆c is an iteration of this holo- nomy map. Therefore, there exists an integer p ∈ Z such that χ coincides with ΦZ 2iπp on a neighborhood of C. Let us now take a continuous function σ : [0, 2ε] → R+ vanishing on [0, ε] and being equal to 1 on [ 32 ε, 2ε]. The homeomorphism Θ : m 7→ ΦZ2iπpσ(|u(m)|) (m) is the identity on a neighborhood of f (s), it coincides with χ for 32 ε ≤ |u| ≤ 2ε and it leaves F ] invariant. To end the proof, we define the required diffeomorpism Φ as the germ along f (D) of the diffeomorphism equal to Θ−1 ◦ χ when |u| ≤ 2ε and equal to the identity otherwise.    Theorem 5.14. The morphism of group-graphs α : AutF → SymF in- duces a natural bijection  ∼  α1 : H 1 (AF  , AutF ) → H 1 (AF  , SymF ). (5) Proof. The surjectivity of α1 follows easily from the surjectivity of αs :   AutFs → SymFs . For this we fix an orientation ≺ of the union of trees AF  and for (cD,s ) ∈ Z 1 (AF  , SymF ), we choose for each edge s with 22 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM ∂s = {D, D0 }, D ≺ D0 , an element ϕD,s such that αs (ϕD,s ) = cD,s and we F set ϕD0 ,s := ϕ−1 1 D,s . Clearly the family (ϕD,s ) is an element of Z (AF , Aut )  defining a lift of (cD,s ).  To prove the injectivity of α1 we consider [φD,s ], [φeD,s ] ∈ H 1 (AF  , AutF ) such that α1 ([φD,s ]) = [αs (φD,s )] = [αs (φeD,s )] = α1 ([φeD,s ]). Then there is  (gD ) ∈ C 0 (AF  , SymF ) such that  αs (φeD,s ) = ρsD (gD )−1 ◦ αs (φD,s ) ◦ ρsD0 (gD0 ) ∈ SymF s   where s = D ∩ D0 . Let ϕD ∈ AutF F D be extensions of gD ∈ SymD and let us  denote by (ϕD )s ∈ AutFs their germs at s. Then αs (φeD,s ) = αs ((ϕ−1 D )s ) ◦ αs (φD,s ) ◦ αs ((ϕD0 )s )  −1 and there is Fs ∈ FixFs such that φD,s = (ϕD )s ◦ φD,s ◦ (ϕD0 )s ◦ Fs . Now e we choose a map δ : EdAF  → VeAF  such that s ∈ δ(s) for each s ∈ EdAF  and we define F̄D as the composition over the set {s ∈ EdAF  | δ(s) = D} of extensions of Fs to a neighborhood of δ(s) with disjoint supports given by  Proposition 5.13. Finally putting ϕ̄D = ϕD ◦ F̄D ∈ AutFD we have that F φeD,s = ϕ̄−1 D ◦ φD,s ◦ ϕ̄D0 ∈ Auts ,  i.e. [φD,s ] = [φeD,s ] in H 1 (AF  , AutF ).  Proof of Theorem B. It follows immediately from Theorem 4.9 and Theo- rem 5.14.  6. Foliations of finite type In this section we introduce the optimal condition on a germ of singular foliation F in order to have a finite dimensional moduli space Mod([F  ]). We keep all the notations of previous sections. Given a marked foliation F  = (F, f ) and a sheaf Q defined on a neighbor- hood of EF in the ambient space MF of F ] , we can associate a group-graph,   denoted by QF , over the cut-graph AF  as follows: if s ∈ EdAF  then QF s is the stalk of Q at f (s) and if D ∈ VeAF  , then  −1 QF 0 D := H (f (D), ιf (D) Q), ιf (D) being the inclusion map of f (D) in MF and for s ∈ D the morphism  F ρsD : QFD → Qs being the canonical restriction. Definition 6.1. We call group-graph of transverse infinitesimal symmetries  ] ] ] of F the group-graph T F associated to the sheaf T F := B F /X F on MF ] equal to the quotient of the sheaf B F of F ] -basic6 holomorphic vector fields ] tangent to the F ] -invariant components of EF , by the sheaf X F of holomor- phic vector fields tangent to F ] . 6i.e. whose flow leaves F ] invariant. TOPOLOGICAL MODULI 23 Remark 6.2. For each (D, s) ∈ IE  let us consider the local holonomies hD,s as in Definition 5.6. There are linear isomorphisms, depending on the choice of an appropriate compact discs system,  ∼  ∼ TDF −→THD , TsF −→ThD,s , where THD (resp. ThD,s ) is the vector space of germs at f (oD ) of vector fields on the transversal disc ∆D which are invariant by the holonomy group HD of F ] along f (D) (resp. invariant by hD,s ), see [13, 24]. Moreover, if X∆ 0 D denotes the set of germs of vector fields on (∆D , f (oD )) vanishing at f (oD ), then the exponential map exp : X∆ 0 → Diff(∆D , f (oD )) sends ThD,s into D C(hD,s ) and THD into C(HD ).  Now we define a coloring on AF  by saying: (1) D ∈ VeAF  is green if the holonomy group HD is finite, (2) s ∈ EdAF  is green if for each D ∈ ∂s the holonomy map hD,s is periodic, (3) D ∈ VeAF  or s ∈ EdAF  are red otherwise.  Let us denote by J F the group-graph of holomorphic first integrals asso- ciated to the sheaf of germs of holomorphic first integrals of F ] . Because F ] does not have saddle-node singularities, an element a ∈ VeAF  ∪ EdAF   is green iff JaF 6= C, see [21]. Notice that if an edge s = D ∩ D0 of AF  is red then the vertices D and D0 are also red. We can therefore consider the following: Definition 6.3. The set of red elements of AF  is a subgraph called red graph of F  and denoted by RF  . Proposition 6.4. Let s and D ∈ ∂s be a green edge and a green vertex of AF  . Then the following properties are equivalent: (1) the holonomy group HD is generated by hD,s ;  (2) JDF → JsF  is surjective;   (3) TDF → TsF is surjective;  F (4) SymF D → Syms is surjective. Proof. Let D be a green vertex of AF  and let z : (∆D , f (oD )) → C be a linearizing coordinate of the holonomy group HD ⊂ Diff(C, 0) which is finite. For each singular point s of D (necessarily a green edge of AF  ) there is nD,s ∈ N such that the local holonomies hD,s given in Definition 5.6 are hD,s (z) = ζD,s z for some primitive nD,s -root of unity ζD,s . Let us denote by nD ∈ N the least common multiple of {nD,s | s ∈ D ∩ Σ} and by ζD a primitive nD -root of unity. Because a first integral is completely determined  by its restriction to the transversal ∆D , we can consider JDF as subrings of  C{z}. In the same way, by extending the elements of JsF along the compact  discs used in Definition 5.6 to define hD,s , we can also consider JsF as a subring of C{z}. With these identifications and using Remark 6.2, we have the following well known equalities and isomorphisms:    JDF = C{z nD } , TDF ' THD = JDF z∂z , C(HD ) = {z(α + ND ) | α ∈ C∗ } ,    JsF = C{z nD,s } , TsF ' ThD,s = JsF z∂z , C(hD,s ) = {z(α+Ns ) | α ∈ C∗ } , 24 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM   with ND ⊂ JDF and Ns ⊂ JsF being the maximal ideals. Furthermore HD is cyclic, generated by ζD z. The required equivalences follow immediately.  If B is a nonempty connected subgraph of a connected component AiF  of AF  , then for every vertex D ∈ / B of AiF  or D ∈ ∂B there is a unique geodesic [D, B] ⊂ AiF  joining D to ∂B. We define the following pre-order relation on the set of vertices of the closure of AiF  \ B by means of D0 ≤ D ⇐⇒ D0 ∈ [D, B]. Definition 6.5. We say that B is repulsive in AiF  if for each edge s = D∩D0  F of AiF  \ B with D0 ≤ D, the restriction map SymF D → Syms is surjective. Remark 6.6. Notice that when B is repulsive there is a filtration B0 = B ⊂ B1 ⊂ · · · ⊂ Bk = AiF  such that Bj−1 is obtained from Bj by pruning a  SymF -repulsive partial dead branch Mj of Bj , cf. Definitions 3.9 and 3.10.  A change of marking induces an isomorphism of colored graphs compatible with the repulsiveness property that gives sense to the following definition: Definition 6.7. The foliation F is of finite type if for each connected com- ponent AiF  of AF  we have: either the subgraph AiF  ∩ RF  is nonempty, connected and repulsive, or it is empty and there exists a green repulsive vertex in AiF  . As we have already pointed out, this finiteness property does not depend on the marking. In fact thanks to Theorem 11.4 of Appendix, under (TR) and (TC) conditions, it only depends on the topological class of the germ F at 0 ∈ C2 and it is fulfilled by all the foliations G with [G  ] ∈ Mod([F  ]) as soon as it holds for one of them. Theorem 6.8. If F is of finite type then the extension by 1 map   Z 1 (RF  , SymF ) → Z 1 (AF  , SymF ) induces a bijection   H 1 (AF  , SymF ) ' H 1 (RF  , SymF ) . (6) Proof. We apply Pruning Theorem 3.11 to each partial dead branch Mj considered in Remark 6.6, see also Remark 3.12.  When the foliation F is of finite type let us now give the precise value of the integer τF in the statement of Theorem D in Section 2.6. Let us consider the following subgraph R0F  whose • vertices correspond by f to the invariant irreducible components of EF whose holonomy groups do not leave invariant a non-trivial vector field, • edges correspond by f to the resonant non-normalizable or non- resonant non-linearizable singularities of F ] . As before changes of marking induce isomorphisms between the graphs R0F  ; that allows us to put: TOPOLOGICAL MODULI 25 Definition 6.9. If F is of finite type we call codimension of F the integer τF := rank H1 (RF  /R0F  , Z), where RF  /R0F  is the quotient graph obtained from RF  by collapsing R0F  to a single vertex.  We will highlight now a group structure on H 1 (RF  , SymF ) when F is of finite type. Let us choose an arbitrary map s 7→ Ds from EdRF  to VeRF   with Ds ∈ ∂s. Since H 1 (RF  , SymF ) is the quotient of  M  M C(hD ,s ) Z 1 (RF  , SymF ) ' SymF s ' s hhDs ,s i s∈EdRF  s∈EdRF    by C 0 (RF  , SymF ) = SymF L D , we must pay attention to the central- D∈VeRF  izers C(h) of the local holonomy transformations h = hD,s ∈ Diff(∆D , f (oD )). Trivially h is of one and only one following type: (P ) periodic; (L1 ) linearizable and non-periodic; (L0 ) formally linearizable but non-linearizable; (R1 ) resonant non-linearizable but normalizable; (R0 ) resonant non-linearizable and non-normalizable. Classically, in the first three cases there exists a (only formal, in the case (L0 )) local coordinate u on (∆D , f (oD )), such that u ◦ h = αu, with α ∈ C∗ . In these situations h = exp X , with X := log(α)u∂u . In the resonant cases (R0 ) and (R1 ) there exists a coordinate u on ∆D , only formal in the case (R0 ), up+1 such that h = `r ◦ exp t0 X, X := 1+λu p ∂u , where p + 1 is the contact order of k 0 k h with the identity when h (0) = 1, ` is the formal diffeomorphism defined by u ◦ ` := e2iπ/p u, h0 (0) = e2iπr/p , r ∈ {0, 1, . . . , p − 1}, λ ∈ C, t0 ∈ C∗ and we can choose t0 = 1, (remark that ` and exp X commute). In all cases u is unique up to multiplication by an element of C∗ . Let us denote by C(h) b the centralizer of h inside the group Diff(∆ d D , f (oD )) of formal diffeomorphisms of (∆, f (oD )). Clearly C(h) = C(h)∩Diff(∆D , f (oD )). b As in Remark 6.2 let us denote by Th the space of germs of holomorphic vec- tor fields on (∆, f (oD )) invariant by h. The following result contains several well-known facts. Proposition 6.10. According to the type of h ∈ Diff(∆D , f (oD )) we have: (P ) C(h) = {g ∈ Diff(∆D , f (oD )) | u ◦ g = u(α + F (uq )), α ∈ C∗ , F ∈ uC{u}}; C(h)/hhi ' Diff(C, 0); Th = C{uq }u∂u ; (L) C(h) b d D , f (oD )) | u ◦ g = λg, λ ∈ C∗ } = exp CX ' C∗ ; = {g ∈ Diff(∆ (R) C(h) b n = {` ◦ exp tX, n ∈ Z/pZ, t ∈ C} = h`i ⊕ exp CX ' Z/pZ ⊕ C. Moreover: (L1 ) Th = CX and C(h)/ exp Th = {1}; (R1 ) Th = CX, C(h)/ exp Th ' Z/pZ and C(h)/hh, exp Th i ' Z/hp, ri; (L0 ) Th = {0}; C(h) = {g ∈ Diff(∆D , f (oD )) | u ◦ g = λu, λ ∈ D} ' D, where D := {λ ∈ C∗ | u−1 ◦ (λu) is convergent} is a totally discontinuous subgroup of U(1), that can be uncountable [29]; 26 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM (R0 ) Th = {0}, there exists m ∈ N∗ such that the sequence α β t 1 0 → Z −→ C(h) −→ Z/pZ , α(t) = exp X, β(g) = log g 0 (0) m 2iπ is exact and C(h)/hhi is finite. Proof. The periodic case has already been described in the proof of Proposi- tion 6.4 except the isomorphism C(h)/hhi ' Diff(C, 0) which follows eas- ily from the fact that every g ∈ Diff(C, 0) commuting with a rotation 1 z 7→ e2iπ/q z is equal to (g [ (z q )) q for a unique g [ ∈ Diff(C, 0). The descrip- tion of the formal centralizers is given for instance in [13, Proposition 1.3.2] or [5, p. 150], where it is also shown that C(h) = C(h) b in the normaliz- ∂ able cases (L ) and (R ). In addition, in the case (L1 ), Th = Cu ∂u 1 1 and exp : Th → C(h) can be canonically identified to the surjective morphism C → C∗ given by µ 7→ eµ . In the case (R1 ), Th is equal to CX and C(h)/hh, exp CXi ' (Z/pZ ⊕ C)/hṙ ⊕ 1, 0 ⊕ Ci ' Z/hp, ri, where ṙ is the class of r modulo p. Thanks to the description of C(h), b in the 0 case (R ) the kernel of β consists of the convergent elements of the flow of X and by the Écalle-Liverpool Theorem, see for instance [13, Corollary 2.8.2] or [11, Theorem 4] and [1, Theorems 7 and 10], it is equal to α(Z) for a suitable m ∈ N∗ . Indeed, Theorem 7 of [1] claims that if exp tX is convergent for |t| ≤ p then exp tX is bounded in |t| ≤ p and |z| ≤ ρ; consequently in d this case X = dt t=0 exp(tX) would be convergent. Thus, the set {t ∈ C : exp tX is convergent}, which contains Z, is different from C. By Theorem 10 of [1] this set is a lattice whose dimension is necessarily 1 by Theorem 4 of [11]. Finally in the case (L0 ), let g be an element of C(h) = C(h) b ∩ Diff(C, 0). Then u ◦ g = λu with |λ| = 1. Indeed if |λ| = 6 1, g would be linearizable in a convergent coordinate and h, that commutes with g, would be also lin- earizable in the same coordinate, contradicting the assumption (R0 ). On the other hand, D is a totally discontinuous subgroup of U(1) because otherwise D = U(1) and h would be linearizable.  It follows from this proposition and from Remark 3.4, Theorem 6.11. Given a ∈ VeAF  ∪ EdAF  we have the equivalences:   a is red ⇐⇒ SymF a is abelian ⇐⇒ dimC Ta F <∞  a ∈ VeR0  ∪ EdR0  ⇐⇒ TaF = 0 . F F  Furthermore the abelian structure of the group-graph SymF over RF  in-  duces an abelian group structure on H 1 (RF  , SymF ). Remark 6.12. By following the natural bijections (2), (5) and (6) provided by Theorems 4.9, 5.14 and 6.8 respectively, one can check that if H([F e  ]) = H([G e    ]) then Mod([F ]) and Mod([G ]) coincide as sets, but their respective abelian group structures are related by the map µ 7→ γµ where γ = [G  ] ∈ Mod([F  ]). Indeed, the bijection (2) is given by iF  which sends [F  ] to  [(id)] ∈ H 1 (AF  , AutF ).  TOPOLOGICAL MODULI 27 We end this section by proving some properties of centralizers which will be useful in the sequel. Lemma 6.13. If g and h are non-periodic and g ∈ C(h), then C(g) = C(h). Proof. The group C(h) in case (L) only depends on the formal coordinate u that linearizes h. Similarly in case (R) all the non-periodic elements of C(h) have the same invariants p and λ, and if we fix these invariants, the centralizers of resonant diffeomorphisms only depend on the normalizing coordinate u. Thus the lemma follows from the fact that in both cases all the non-periodic elements of a centralizer can be linearized or normalized by using the same coordinate.  Lemma 6.14. Let H be a finitely generated subgroup of Diff(C, 0). Suppose that H and its centralizer are infinite. Then H is abelian, it contains a non-periodic element h and H ⊂ C(H) = C(h). Proof. We will first prove that H is abelian by contradiction. Take a non- trivial element f ∈ [H, H] which is tangent to the identity: f (z) = z +ck z k + · · · with k ≥ 2 and ck 6= 0. Then f n (z) = z + nck z k + · · · is the identity only for n = 0. Thus f is non-periodic and C(H) ⊂ C(f ). By the description of C(f ) given in Proposition 6.10 there is a non-periodic element g in the infinite subgroup C(H) ⊂ C(f ). The fact that g ∈ C(H) is equivalent to the inclusion H ⊂ C(g). As C(g) ⊂ C(g) b is abelian by Proposition 6.10 we get a contradiction. Since H = hh1 , . . . , hk i is abelian, there is a non-periodic generator hi , otherwise H would be finite. Let h ∈ H be any non-periodic element. By Proposition 6.10, the group C(h) ⊂ C(h) b is abelian. Since H is abelian we have H ⊂ C(H) ⊂ C(h). Since C(h) is abelian every element of C(h) commutes with H and consequently C(h) ⊂ C(H). Hence H ⊂ C(H) = C(h).  It follows immediately: Proposition 6.15. Under the hypothesis of the previous lemma, all the non- periodic elements of H are of the same type (L1 ), (L0 ), (R1 ) or (R0 ). 7. Non-degenerate foliations A F  -singular chain C of E is a sequence D0 , . . . , D`C of irreducible com- ponents of E defining a connected subgraph s1 s`C (C) •D0 •D1 · · · ··· •D`C (7) of AF  such that the singular valency (cf. Definition 5.4) of its extremities D0 and D`C is at least three, and that of the others, called interior vertices, is exactly two. If `C = 1, then C consists of only two adjacent divisors of s1 singular valency at least three: •D0 •D1 . The image by the marking  map f of a F -singular chain of E is a singular chain of EF as considered in Section 2.5. Proposition 7.1. Let F  be a non-degenerate marked foliation. Then F is of finite type. Moreover, the union R̆F  of all F  -singular chains of E 28 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM is a subgraph of RF  such that for each connected component AiF  of AF  , R̆F  ∩ AiF  is connected and repulsive in RF  . Hence   H 1 (RF  , SymF ) ' H 1 (R̆F  , SymF ) . In order to simplify the notations in the two proofs below, we will write  R, R̆, Sym, instead of RF  , R̆F  , SymF . Proof of Proposition 7.1. Clearly R̆ is connected in each connected compo- nent Ai of the cut-graph A and the closure of R\ R̆ in R is exactly the union of all connected subgraphs C denoted as in (7) but with singular valencies sat- isfying valΣ (D0 ) ≥ 3, valΣ (Dj ) = 2 for 0 < j < `C and valΣ (D`C ) = 1 or 2. By definition of the group-graph Sym, for j ≥ 1 the morphisms s ρDjj : SymDj → Symsj are bijective and R̆ is repulsive in R. Since F is non-degenerate all the vertices and edges of R̆ are red; thus R ∩ Ai is also re- pulsive and connected. By using Pruning Theorem 3.11 we obtain the group isomorphism H 1 (R, Sym) ' H 1 (R̆, Sym).  Proof of Theorem C. To have uniqueness of the numbering in the notation (7) of a singular chain C, we fix in the sequel an extremity D̆ of R̆ and we prescript that D0 belongs to the geodesic joining D`C to D̆. We will say that D0 , resp. s`C , is the initial vertex, resp. terminal edge of C. s s For an interior vertex Dj of C the morphisms ρDjj and ρDj+1 j are bijective and by composition they induce isomorphisms ξDj : Syms1 ' SymDj , ξsj : Syms1 ' Symsj , 0 < j < `C . Let us consider the subgroups: • Ze1 (R̆, Sym) ⊂ Z 1 (R̆, Sym) of the 1-cocycles (φD,a )(D,a)∈IR̆ such that φD,a = 1 if a is not the terminal edge of some singular chain, • Ce 0 (R̆, Sym) ⊂ C 0 (R̆, Sym) of the 0-cochains (φD )D∈Ve such that R̆ φD = ξD ◦ ρsD10 (φD0 ) for all interior vertices D of any singular chain, D0 denoting its initial vertex. Notice that the coboundary morphism ∂ 0 defined in Remark 3.4 maps C e 0 (R̆, Sym) 1 into Ze (R̆, Sym), allowing us to define the group e 1 (R̆, Sym) := coker(∂ 0 : C H e 0 (R̆, Sym) −→ Z e1 (R, Sym)) . We easily see that each element of H 1 (R̆, Sym) can be represented by a e1 (R̆, Sym). We deduce that the morphism cocycle belonging to Z e 1 (R̆, Sym) −→ H 1 (R̆, Sym) τ :H induced by the inclusion Ze1 (R̆, Sym) ⊂ Z 1 (R̆, Sym) is surjective. On the other hand if a cochain c0 := (φD )D∈VeR̆ ∈ C 0 (R̆, Sym) satisfies ∂ 0 (c0 ) ∈ Ze1 (R̆, Sym), then for each singular chain C of lenght `C ≥ 2 denoted as in (7) we have the equalities s s ρDjj (φDj ) = ρDjj−1 (φDj−1 ) , j < `C . It follows that c0 ∈ C e 0 (R̆, Sym). Therefore ker(τ ) is trivial and τ is an isomorphism. To achieve the proof of Theorem C, let us first notice that TOPOLOGICAL MODULI 29 the group C e 0 (R̆, Sym) is finite. Indeed it is isomorphic to the direct sum of all centralizers of holonomy groups associated to the vertices of R having singular valency at least three. These holonomy groups being non-abelian by the non-degeneracy assumption, thanks to Lemma 6.14 their centralizers are finite. On the other hand, Proposition 6.10 gives a decomposition of Ze1 (R̆, Sym) as F ⊕ B ⊕λj=1 (C∗ /αjZ ) ⊕ (C∗ )ν ; that completes the proof of Theorem C.  8. Examples Before proving Theorem D in full generality let us motivate its statement by computing the moduli space of some non-trivial examples using the iden-  tification Mod([F  ]) ' H 1 (RF  , SymF ). • Example 0: a logarithmic generic multicusp Let L be the germ at 0 ∈ C2 of the logarithmic foliation defined by the meromorphic form p p X d(y 2 + ai x3 ) d(y − x) X d(x2 + bi y 3 ) ω := αi 2 +δ + βi 2 , y + ai x3 y−x x + bi y 3 i=1 i=1 with ai , bi ∈ C mutually ∗ Pp distinct. We normalize the coefficients αi , βi , δ ∈ C by requiring δ + 2 i=1 (αi + βi ) = 1. To simplify the exposition we suppose that E is equal to the exceptional divisor EL , Σ = Sing(L] ), the marking being the identity map and L = (L, idEL ). Clearly E is formed by five irreducible components, its dual tree is equal to the cut-graph AL s0∞ s00 s00 0 s00 ∞ (AL ) •C 0 •D0 •D •D00 •C 00 valΣ (C 0 ) = valΣ (C 00 ) = 1 , valΣ (D0 ) = valΣ (D00 ) = p + 2 , valΣ (D) = 3 , where valΣ is the singular valency introduced in Definition 5.4. The cut graph AL decomposes into one singular chain that is the red part RL of the graph s00 s00 0 (RL ) •D0 •D •D00 s0∞ s00 ∞ and two dead branches •C 0 •D0 and •D00 •C 00 necessarily green.   L L Thus the restriction morphisms SymL L C 0 → Syms0∞ , SymC 00 → Syms00 ∞ are 1 L surjective and by Pruning Theorem 3.11, the group H (AL , Sym ) is iso-   morphic to H 1 (RL , SymL ). On R = RL the morphism ∂ 0 defined in Re- mark 3.4 decomposes, with additive notations on abelian groups, as  C 0 (R, SymL ) = C(HD0 ) ⊕ C(HD ) ⊕ C(HD00 ) ξ4 ∂0  ξ1 " { ξ2 ξ3 # z    Z 1 (R, SymL ) = SymL s00 ⊕ SymL s00 0 ∂ 0 (c1 ⊕ c2 ⊕ c3 ) = (ξ1 (c1 ) + ξ2 (c2 )) ⊕ (ξ3 (c2 ) + ξ4 (c3 )) . On the other hand, Sing(L] ) ∩ D0 , resp. Sing(L] ) ∩ D00 , is formed by s0∞ , s00 , resp. s00∞ , s000 , and the attaching points s0i , resp. s00i , of the strict transforms 30 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM of the curve {y 2 + ai x3 = 0}, resp. {x2 + bi y 3 = 0}, i = 1, . . . , p; and Sing(L] ) ∩ D is formed by s00 , s000 and the attaching point s1 of the strict transform of {y − x = 0}. The Camacho-Sad indices of L] at these points are CS(D0 , s0∞ ) = CS(D00 , s00∞ ) = −1/2 , CS(D, s1 ) = −δ , CS(D0 , s00 ) = −e α, CS(D00 , s000 ) = −βe , CS(D0 , s0j ) = −αj α e , CS(D00 , s00j ) = −βj βe , e := (1 + pi=1 αi )−1 /2, βe := (1 + pi=1 βi )−1 /2. Assuming that p ≥ 3, P P with α we choose αi , βi and δ sufficiently generic so that no Camacho-Sad index is a real number, except at the points s0∞ and s00∞ . All the singularities of L] are linearizable and according to Proposition 6.10 the centralizers C(HD0 ), C(HD ), C(HD00 ) are isomorphic to C∗ = C/2πiZ. Using Remark 5.9 we obtain: C 2iπZ ⊕ C 2iπZ ⊕ C 2iπZ ξ1 && xx ξ2 ξ3 && xx ξ4 C/2iπ(Z + α eZ) ⊕ C/2iπ(Z + βZ) e moreover ξ2 and ξ3 are induced by the identity map, but ξ1 is induced by z 7→ z/eα and ξ4 by z 7→ z/β.e It immediately follows that ∂ 0 is surjective  and H 1 (R, SymL ) = 0. We conclude that Mod([L ]) = {[L ]}. Thus, L is topologically SL-rigid which means that if a foliation F has the same separa- trices, Camacho-Sad indices and holonomies than L then F is topologically conjugated to L. It is worth to notice that the converse is not true, as [28, Théorème 3.5] shows, because condition (TR) is not satisfied. Next Examples 1–4 will be suitable perturbations of this logarithmic foli- ation L that we have considered in Example 0. • Example 1: non-degenerate multicusps Let F1 be a foliation with the same reduction as the previous logarithmic foliation L, same Camacho-Sad indices but such that the holonomy groups along D, D0 and D00 are non-abelian. Such a foliation can be obtained by perturbing the holonomy groups of D, D0 and D00 as in [22] and by using the realization theorem of [15]. In this case it is well-known that F1 satisfies condition (TR) and therefore we can compute Mod([F1 ]) by identifying it  with H 1 (R, SymF1 ). According to Lemma 6.14 the centralizers of HD0 , HD F and HD00 are finite groups F10 , F1 and F100 respectively, however Syms01 and 0 F   Syms001 remain isomorphic to SymL s00 and SymL s00 because the singularities of 0 0 F1] at s00 and s000 are again linearizable. F10 ⊕ F1 ⊕ F100 ξ1 && xx ξ2 ξ3 && xx ξ4 C/2iπ(Z + α eZ) ⊕ C/2iπ(Z + βZ) e It follows: TOPOLOGICAL MODULI 31 - Mod([F1 ]) is a finite quotient of a product of two elliptic curves. • Example 2: partially degenerate multicusps Let F2 be a perturbation of the logarithmic foliation L with same Camacho- Sad indices, with non-abelian holonomy groups along D0 and D00 (having finite centralizers F20 and F200 ), but such that there is a biholomorphism be- tween neighborhoods of D that conjugates F2 to L. The existence of F2 is guaranteed by the same process as in Example 1, but without perturbing the holonomy group of D. We have F20 ⊕ C/2iπZ ⊕ F200 ξ1 && ww ξ2 ξ3 '' xx ξ4 C/2iπ(Z + α eZ) ⊕ C/2iπ(Z + βZ) e where again ξ2 and ξ3 are induced by the identity map. We easily obtain the exact sequence K −→ C/2πi(Z + α e −→ Mod([F  ]) −→ 0 , eZ + βZ) 2 K being finite. If 1, α e, βe are Z-independent then  - Mod([F2 ]) is not a finite quotient of a product of elliptic curves; in particular, in the statement of Theorem D we cannot replace Zp by a finite group in the exact sequence (1). • Example 3: infinite type multicusps First we choose the coefficients αi , βi , δ in the expression of the 1-form ω, so that CS(D0 , s00 ) ∈ Z<0 , the other Camacho-Sad indices being in C \ R, except for the points s0∞ and s00∞ . At s00 the foliation L] possesses now a germ of holomorphic first integral, the local holonomy is a periodic rota-  tion, thus SymL s00 is isomorphic to Diff(C, 0). Then, as before we perform a perturbation F3 of L changing only the holonomy groups HD0 and HD that become non-abelian, but without changing HD00 nor the local analytic F types at any singular point. For such foliation F3 the group Syms03 is always 0 F F3 isomorphic to Syms03 ' Diff(C, 0). The group-graph Sym is not abelian 0 and its cohomology is no longer given by the cokernel of a morphism ∂ 0 . However F F ξ4 : Sym 300 ' C/2πiZ −→ Sym 003 ' C/2πi(Z + βZ) D s0 e is always a submersion. Thus performing a new pruning we have an isomor-   s00 phism H 1 (RF3 , SymF3 ) ' H 1 (R0 , SymF3 ) with R0 := •D0 •D . The centralizers of HD0 and HD being finite by Lemma 6.14, we obtain: - Mod([F3 ]) is a quotient of Diff(C, 0) by the action of a finite group and RF3 is not connected. • Example 4: Cremer multicusps By gluing techniques and thanks to realization Theorem [31] and Pérez Marco’s results [29] we can build a foliation F4 with same separatrices, thus same reduction, as in previous examples, with non-abelian holonomy groups HD0 , HD , HD00 , but whose local holonomies at s00 and s000 are Cremer with uncountable centralizers. In this case 32 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM - Mod([F4 ]) is a finite quotient of a product of two uncountable totally discontinuous subgroups of U(1). • Example 5: non-degenerate foliations with a single separatrix. For such a foliation F5 , after pruning all dead branches of the dual graph of EF5 , the obtained graph is the red graph RF5 which is reduced to a geo- s1 s` desic segment •D0 ··· •D` . All Camacho-Sad indices are ratio- nal numbers. The singular chains in RF5 are in two categories: the normaliz- able chains whose edges s correspond to normalizable resonant singularities F of F5] and the non-normalizable chains. For the first one the group Syms 5 is isomorphic to C∗ and for non-normalizable chains it is isomorphic to Z/ms Z for a suitable ms ∈ N. It follows: - Mod([F5 ]) ' ( µi=1 Z/mi Z ⊕ C∗ν )/Z, with Z a finite subgroup, µ, L resp ν, the number of non-normalizable, resp. normalizable singular chains; furthermore µ + ν is equal to the number of Puiseux pairs of the unique separatrix. Another specificity of this foliation F5 is that the mapping class group of EF 5 is trivial because every singular point of EF5 is fixed7 by Mcg(EF 5 ) and the pure mapping class group of P1 punctured at three points is trivial [9, Proposition 2.3]. From Corollary 2.2 we obtain that Mod([F5 ]) ⊂ Mod(EF 5 ) ' [Foltr (EF 5 )]C 0 . • Example 6: some topologically SL-rigid foliations. Whenever for a marked foliation F  the red part of any cut-component of AF  is reduced to one vertex, the moduli space Mod([F  ]) is reduced to one element. In particular this is the case for: - any non-dicritical foliation reduced after only one blow-up, its separatrices being smooth curves mutually transversal, or more generally any topolog- ically quasi-homogeneous foliation, see [14], - absolutely dicritical foliations of Cano-Corral [6], - dicritical foliations that are non-singular after one blow-up, see [2] and [27]. 9. Exponential and disconnected group-graphs We keep all notations used in Section 6. For technical reasons the last group-graphs that we must consider will be defined uniquely over the red    graph RF  ⊂ AF  . Recall that X F , B F and T F denote the group-graphs ] ] ] ] ] over AF  associated to the sheaves X F , B F and T F = B F /X F of tangent, basic and transverse holomorphic vector fields for F ] , respectively.   Lemma 9.1. For s ∈ EdAF  the exponential map exp : BsF → AutF s induces a well-defined map expF  F  → SymF  . s : Ts s   Proof. We must prove that exp(Z + X) ≡ exp(Z) modulo FixF F s if Z ∈ Bs F  and X ∈ Xs . For that it suffices to show that for each neighborhood V of f (s) there is another neighborhood U of f (s) such that for each p ∈ U 7Each element of the mapping class group of E  preserves dead branches so it must fix F5 every singular point except maybe the attaching points of the two dead branches of the extremity divisor of singular valency 3. But these two points have different Camacho-Sad indices, as it can be easily deduced from [12, p. 164]. TOPOLOGICAL MODULI 33 ] the curve α ∈ [0, 1] 7→ exp(Z + αX)(p) is contained in a leaf of F|V . We choose U ⊂ V such that the map φ : U × D2 × D2 → V given by φ(p, t, α) = exp(t(Z + αX))(p) is well-defined. Fix p ∈ U and take a local holomorphic first integral F defined in a neighborhood W of p. If t is small enough then φ(p, t, α) ∈ W and     ∂ ∂ ∂ ∂ F (φ(p, t, α)) = F (φ(p, t, α)) ∂t ∂α ∂α ∂t ∂ = ((Z + αX)(F ) ◦ φ(p, t, α)) ∂α ∂ = [X(F ) ◦ φ(p, t, α)] φ(p, t, α) = 0 ∂α because X is tangent to F ] . Since φ(p, 0, α) = p does not depend on α we ∂ obtain that ∂α F (φ(p, t, α)) = 0 for t small enough. As φ is holomorphic, we ] conclude that the curve α 7→ φ(p, 1, α) is contained in a leaf of F|V .   Remark 9.2. It can be checked that under the identifications TsF ' ThD,s  and C(hD,s )/hhD,s i ' SymF s given by Remark 6.2 and Corollary 5.8, the  morphism expF s coincides with the composition of the restriction ThD,s → C(hD,s ) of the exponential map on the transverse section ∆D and the quo- tient map C(hD,s ) → C(hD,s )/hhD,s i.   Motivated by the above remark, for D ∈ VeAF  we define the map expF D : F  F F  exp F TD → SymD as the composition TD ' THD → C(HD ) → SymD . From Remark 9.2 it is clear that the following diagram is commutative:  expF TDF  D / SymF  (8) D ρsD ρsD    expF TsF  s / SymF  s  the vertical maps being the restriction maps of the group-graphs T F and  SymF , written with the same notation. Although the exponential map exp : C{z}z∂z → Diff(∆D , f (oD )) ' Diff(C, 0), z : (∆D , f (oD )) → (C, 0) being a germ of coordinate, is not a morphism of groups, its restriction to a subspace of complex dimension ≤ 1 is. On the   other hand it is well-known that dimC TsF ≤ 1 if the stalk JsF of the sheaf   of first integrals is equal to C. Since TDF ⊂ TsF for s ∈ D we deduce that   expF F D and exps define a morphism    expF : T F → SymF of abelian group-graphs over RF  .  Definition 9.3. The group-graph over RF  image of expF is called the  exponential group-graph of F  . We denote it by ExpF . 34 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM At this point it is clear that the subset of RF  consisting of all a ∈  VeAF  ∪ EdAF  such that ExpF 0 a = 0 is just the subgraph RF  of RF  pre- viously defined in Section 6 and characterized by the second equivalence in Theorem 6.11. Let us denote by R1F  the completion of RF  \ R0F  , i.e. the minimal subgraph of RF  containing RF  \ R0F  .  Lemma 9.4. If (D, s) ∈ IRF  and TDF 6= 0, then the restriction map ρ0 sD :   TDF → TsF is an isomorphism and all the red singular points in D share the same character resonant non-linearizable or linearizable; we will say that D is resonant or linearizable according to the case. Furthermore, the isomorphism  class of the group ExpFD is given by the following table  ExpFD valΣ (D) ≤ 2 valΣ (D) ≥ 3 D resonant non-linearizable C/Z C (9) D linearizable C∗ /αZ C∗   The restriction morphism ρsD : ExpF F D → Exps is surjective and  Z if valΣ (D) ≥ 3, ker ρsD ' 0 if valΣ (D) ≤ 2. Proof. The homogeneity of singular types in D is given by Proposition 6.15. Notice that if a basic vector field for F ] defined on a connected open set U ⊂ MF is tangent to the foliation in a neighborhood of a point of U then it is tangent to the foliation on the whole U . Using this fact it is easy to see that if W is a connected subset of a F ] -invariant component of EF , ] ] the stalk maps T F (W ) → TmF , m ∈ W are injective. Taking W = D   we deduce that the restriction map ρ0 sD : TDF → TsF is injective. Since   s   dim TDF = dim TsF = 1, ρ0 D is an isomorphism. In fact, TsF ' TDF can be identified to a line CX in the space of germs of vector fields on (∆D , f (oD )). Then Table (9) follows from Proposition 6.10. On the other hand expF  F  → ExpF  being surjective by definition s : Ts s and the restriction map ρ0 sD being an isomorphism, it follows that ρsD ◦  F 0s expF s D = exps ◦ρ D is surjective. Therefore ρD is also surjective. The last assertion follows from Remark 9.2 and the commutativity of the diagram (8).   Thanks to Theorem 6.11, H 1 (RF  , SymF ) is an abelian group and the   natural inclusion ExpF → SymF is an injective morphism of abelian group- graphs over RF  . Definition 9.5. The quotient group-graph over RF     DisF := SymF /ExpF  given by DisF F F a := Syma /Expa for every a ∈ VeRF  ∪ EdRF  , is called the disconnected group-graph of F  . Obviously, we have over RF  the short exact sequence of abelian group- graphs:    0 → ExpF → SymF → DisF → 0. (10) The name “disconnected” is explained by the following proposition. Proposition 9.6. For s ∈ EdRF  and D ∈ VeRF  we have that TOPOLOGICAL MODULI 35  (1) the abelian group DissF ' C(hD,s )/hhD,s , exp(ThD,s )i is: (a) trivial if f (s) is a linearizable (non-periodic) singularity (b) a finite abelian group if f (s) is a resonant (non-periodic) singu- larity, cyclic in the normalizable case; (c) a cyclic quotient of a totally disconnected subgroup of U(1) if f (s) is a non-resonant non-linearizable singularity; F (2) the abelian group DisD is (a) infinite and finitely generated if HD is abelian and all the red singularities on f (D) are resonant non-normalizable; (b) a cyclic quotient of a totally disconnected subgroup of U(1) if HD is abelian and all the red singularities on f (D) are non-resonant non-linearizable; (c) finite in all the remaining cases;  F (3) DisF s and DisD are finite if s ∈ EdR1  and D ∈ VeR1  . F F Proof. Assertions (1) result directly from Proposition 6.10. To obtain Assertions (2) we can suppose that the singular valency of D  F is at least three, otherwise DisFD = Diss for s ∈ D ∩ Σ and Assertions (2) result again directly from Proposition 6.10. Now let us suppose also that  DisFD -thus also C(HD )- is infinite. Because D is red, HD is infinite and it follows from Lemmas 6.13 and 6.14 that the set H 0 of all non-periodic elements of HD is nonempty and for every h ∈ H 0 we have C(HD ) = C(h),  therefore DisFD = C(h)/exp(Th ). By Proposition 6.10 the only case where this group is infinite is when h is resonant non-normalizable or non-resonant non-linearizable. To see that these two possibilities correspond to the cases (2a) and (2b) above, it is enough to notice that the local holonomies hD,s , s ∈ D ∩ Σ, that generate HD , cannot be all periodic (otherwise by abelianity HD would be finite), and to use Proposition 6.15. Assertion (3) follows immediately from Assertions (1) and (2) except for  DisF 1 0 D when D is a common vertex of RF  and RF  . In this case although  ExpF 0 D = 0, at the meeting points s of D with components D of RF  we 1 F F have Exps 6= 0 because ExpD0 6= 0. Therefore D does not correspond to  case (2a) nor case (2b) and DisF D is finite according to (2c).  10. Proof of Theorem D In order to simplify the notations in the proofs below, we will write again    A, R, Aut, Sym, Exp, Dis and τ , instead of AF  , RF  , AutF , SymF , ExpF ,  DisF and τF . We will also write R0 and R1 instead of R0F  , R1F  . We have already shown that there are “natural” bijections: (2) (5) (6) Mod([F  ]) ' H 1 (A, Aut) ' H 1 (A, Sym) ' H 1 (R, Sym) , (11) the bijection (6) being only valid when F is of finite type. Moreover Sym is an abelian group-graph over R and consequently H 1 (R, Sym) is an abelian group, cf. Theorem 6.11. Recall that R0 is the subgraph of R constituted by all the vertices and edges b satisfying Expb = 0 and R1 is the completion of R \ R0 . The rest of the proof is divided in several steps: 36 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM (i) The abelian group H 1 (R, Sym) fits into an exact sequence 0 → F → H 1 (R, Exp) → H 1 (R, Sym) → D → 0, where F is a finite abelian group and D is a totally disconnected topological abelian group. (ii) We have group isomorphisms M H 1 (R, Exp) ' H 1 (R1 , Exp) ' H 1 (Zα , Exp), α Zα where each zone Zα is the completion of a S with R1 =: α∈π0 (R\R0 ) connected component of R \ R0 = R1 \ R0 . (iii) To simplify the computation of the cohomology groups H 1 (Zα , Exp) we modify each zone (not reduced to a single vertex) without changing the number of its extremities nor its cohomology, by adding a vertex and an edge, for each of its extremities. The modified zones fulfill the following property: (∗) each extremity of Zα is joined by its edge to a vertex of valency 2 in Zα . (iv) We decompose each modified zone Z as Z = Z0 ∪ Z1 where Z0 is either empty or a disjoint union of n + 1 ≥ 1 segments •Di0 •Di 0 with Di ∈ R0 , Di ∈ R1 and Z0 ∩ Z1 = {D0 , . . . , Dn }. We prove that n H 1 (Z, Exp) is trivial if Z0 is empty, and it is the quotient of L ExpDi i=1 by a finitely generated subgroup if Z0 6= ∅. (v) Since ExpDi is isomorphic to C or C/Z ' C∗ or C∗ /αZ by Lemma 9.4, we can construct a morphism Λ : Cτ → Mod([F  ]) with totally dis- connected cokernel and finitely generated kernel. (vi) We specify the notion of semi-local-equisingularity, denoted by SL- equisingularity. This notion was introduced in [24] for germs of de- formations and in this paper we adapt it to the context of a global parameter space. U satisfying The- (vii) We construct SL-equisingular families of foliations Ft,i orem D. Step (i). Consider the long exact sequences χ · · · → H 0 (R, Dis) → H 1 (R, Exp) → H 1 (R, Sym) → H 1 (R, Dis) → 0 (12) and χ1 · · · →H 0 (R1 , Dis) → H 1 (R1 , Exp) → H 1 (R1 , Sym) → H 1 (R1 , Dis) → 0 associated by Lemma 3.6 to the short exact sequence (10) of abelian group- graphs. By the first part of Proposition 9.6, Z 1 (R, Dis) is a finite product of totally disconnected subgroups of U(1) and H 1 (R, Dis) is thus a totally disconnected abelian topological group. Moreover when all the singularities of the foliation are resonant or linearizable, the case 1c) is excluded and Z 1 (R, Dis) is finite. In order to conclude this step it only remains to prove that ker χ is finite. Let us notice that H 0 (R0 ∩R1 , Exp) = 0, H 1 (R0 , Exp) = 0 TOPOLOGICAL MODULI 37 and H 1 (R0 ∩ R1 , Exp) = 0. By applying Mayer-Vietoris Lemma 3.7 to the union R = R0 ∪ R1 we obtain the following commutative diagram with exact rows 0 / H 1 (R, Exp) / 0 ⊕ H 1 (R1 , Exp) /0 χ 0⊕χ1   ··· / H 1 (R, Sym) / H 1 (R0 , Sym) ⊕ H 1 (R1 , Sym) / ··· (13) Thus ker χ is isomorphic to a subgroup of ker χ1 and it is sufficient to prove that H 0 (R1 , DisF ) is finite. But using the second part of Proposition 9.6 we obtain that C 0 (R1 , DisF ) is finite. Step (ii). Diagram (13), coming from the Mayer-Vietoris sequence, gives us the isomorphism H 1 (R, Exp) ' H 1 (R1 , Exp) . (14) Clearly R1 is a finite union of zones Zα . Moreover, Zα ∩ Zβ is either empty or a single vertex of R0 . Therefore H 0 (Zα ∩ Zβ , Exp) = 0 and H 1 (Zα ∩ Zβ , Exp) = 0. By applying recursively Mayer-Vietoris Lemma 3.7 we deduce that M H i (R1 , Exp) ' H i (Zα , Exp) , i = 0, 1 . (15) α∈π0 (R\R0 ) Step (iii). If a zone Zα is reduced to a single vertex then H 1 (Zα , Exp) is clearly trivial. If this is not the case, we modify Zα in the following way: if v 0 is an extremity of Zα and v 00 ∈ VeZα is the unique vertex joined to v 0 by e0 e00 e an edge e0 , we replace the segment •v00 •v0 by •v00 •v •v0 . We also extend the group-graph Exp to the new edges and vertices by defining 00 Expe00 := Expe := Expv := Expe0 , ρev := ρev := idExpe0 , (16) 00 0 0 ρev00 = ρev00 , ρev0 = ρev0 . We call this operation the blow-up of the edge e0 . By performing these blow- ups for each extremity of Zα we get a new graph Z eα called a modified zone. eα fulfills property (∗) of (iii). Clearly Z By doing this process on each zone, we get a modified graph R e endowed with a group-graph still denoted by Exp. We define now a contraction map: [ Z 1 (R, e Exp) → Z 1 (R, Exp) , c = (φve ) 7→ c[ = (φ[ve ) where φ[ve = φve if e is not produced by a blow-up, and φv00 e0 := φv00 e00 φve =: e00 e e0 φv−1 0 e0 if •v 00 •v •v0 is given by the blow-up of •v00 •v0 . It is easy to see that this map induces group isomorphisms H 1 (Z eα , Exp) ' H 1 (Zα , Exp) , H 1 (R, e Exp) ' H 1 (R, Exp) . (17) 38 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM Step (iv). Fix Z = Z eα a modified zone of R1 . Let Z1 be the maximal subgraph of Z with all vertices and edges b satisfying Expb 6= 0. Denote by Z0 the completion of Z \ Z1 . Clearly Z = Z0 ∪ Z1 and Z0 is either empty or a disjoint union of n + 1 ≥ 1 segments •D0 •D with ExpD0 = 0, ExpD 6= 0, D0 being an extremity of Z and valZ (D) = 2. Notice that H 1 (Z1 , Exp) = 0. Indeed the restriction morphisms of the group-graph Exp over Z1 are surjective by Lemma 9.4. We apply recursively Pruning Theorem 3.11 and we conclude by Remark 3.12 that H 1 (Z, Exp) = 0 if Z0 is empty. Now suppose that Z0 6= ∅. We will apply Mayer-Vietoris Lemma 3.7 to Z = Z0 ∪ Z1 . Using again Lemma 9.4 we see that H 1 (Z0 , Exp) = 0 and H 0 (Z0 , Exp) = 0 by construction of the modified zones. We obtain the exact sequence σ δ H 0 (Z1 , Exp) −→ α H 0 (Z0 ∩ Z1 , Exp) → α H 1 (Z, Exp) → 0, (18) σα being the restriction map and δα the connecting map. In the sequel we will choose one vertex D0 in Z0 ∩ Z1 = {D0 , . . . , Dn } and we will call the remaining vertices D1 , . . . , Dn the active vertices of the zone Z. Lemma 10.1. The projection πα : H 0 (Z1 , Exp) → H 0 (D0 , Exp) ' ExpD0 , (yD )D 7→ yD0 is surjective and its kernel is a finitely generated abelian group. Proof. From Lemma 9.4 for each (D, s) ∈ IZ1 the restriction map ρsD is surjective with kernel 0 or Z. Since D0 ∈ Z0 ∩ Z1 then valZ (D0 ) = 2 by Step (iv). Let s0 be the edge of Z passing through D0 and let D00 be the other vertex of s0 . Let si , i = 1, . . . , `, be the edges such that D00 ∈ ∂si . For each y0 ∈ ExpD0 there are yi ∈ Expsi , i = 1, . . . , `, such that ρsDi i (yi ) = ρsDi 0 (y0 ), see Figure 3. Moreover, the different choices of yi are parametrized by Zk with k ≤ `. By induction we easily deduce that πα is surjective and its kernel is finitely generated.  Denote by ρ : H 0 (Z0 ∩ Z1 , Exp) → H 0 (D0 , Exp) ' ExpD0 the projection map. Then we have the following morphism of exact sequences σα δα H 0 (Z1 , Exp) / H 0 (Z0 ∩ Z1 , Exp) / H 1 (Z, Exp) /0 πα ρ    0 / Exp id / Exp /0 D0 D0 Ln Z0 ∩ Z1 only contains the vertices Since Ln Di we have that H 0 (Z0 ∩ Z1 , Exp) = i=0 ExpDi and ker ρ is just i=1 ExpDi . Because πα is surjective, by applying the snake lemma we obtain the following exact sequence: n δα M ker(πα ) → ExpDi → H 1 (Z, Exp) → coker(πα ) . (19) i=1 Since πα is surjective and ker(πα ) is finitely generated, thanks to Lemma 10.1 we get that H 1 (Z, Exp) is a quotient of Cn by a finitely generated subgroup. TOPOLOGICAL MODULI 39 ExpD0 s ρD0 0  Exps0 OO s0 ρ 0 D0 s1 ExpD00 s` ρ 0 ρ 0 D0 D0 vv (( Exps1 ··· Exps` OO OO s s ρD1 ρD` 1 ` ExpD1 ··· ExpD` || ··· "" ··· }} ··· !! Figure 3. Schematic diagram for the group-graph Exp re- stricted to Z1 . Step (v). First we notice that the number of active vertices of a modified zone Z = Z eα is equal to the rank of the homology groups H1 (Z/Z0 , Z) ' α α H1 (Z /(Z ∩ R0 ), Z) of the corresponding quotient graphs. We easily deduce that the number of all active vertices ar for all the modified zones is equal to the rank τ := rank H1 (R/R0 , Z) introduced in Definition 6.9. Each active vertex ar , r = 1, . . . , τ , belonging to some modified zone, is sr produced by the blow-up of an edge •vr00 •vr0 , vr0 being an extremity of the zone. By construction, Expar = Expsr , cf. (16). With this identification and thanks to the isomorphisms (14), (15) and (17) it can be easily checked, using the proof of Lemma 3.7, that the map δ = ⊕α δα given by the con- necting maps δα of the Mayer-Vietoris exact sequences (18) is the surjective morphism τ M τ M δ : Expar = Expsr 3 (ϕr ) 7−→ [(φve )] ∈ H 1 (R, Exp) r=1 r=1 with φvr00 sr := ϕr , φvr0 sr := ϕr−1 and φve is trivial otherwise. Now for each r = 1, . . . , τ we choose a local holomorphic basic vector field Xr transverse to the foliation F ] and defined on a neighborhood of f (sr ) in the ambient space of F ] . We define the group morphism Λ : Cτ → Mod([F  ]) of Theorem D, as the composition τ ξ M δ χ (11) Cτ  Expsr  H 1 (R, Exp) → H 1 (R, Sym) ' Mod([F  ]) , r=1 40 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM • • • where ξ(t) = (expt1 X1 , . . . , exptτ Xτ ), exptr Xr denotes the class of exp tr Xr in Autsr /Fixsr = Symsr and χ is induced by the natural inclusion of group- graphs Exp ,→ Sym, see (10). The last bijection (11) induces an abelian group structure on Mod([F  ]). Moreover, if we define D := H 1 (R, Dis) and Γ : Mod([F  ]) → D as the composition of the isomorphism (11) and the last arrow in the sequence (12), then the sequence Λ Γ Cτ → Mod([F  ]) → D → 0 is exact. It remains to check that ker Λ = ker(χ ◦ δ ◦ ξ) is finitely gener- ated. Since ker χ is finite by step (i) and ker δ is finitely generated thanks to Lemma 10.1 and (19), it suffices to see that ker ξ is also finitely generated. In fact, we will conclude by proving that the kernel of each group morphism • ξr : C → Symsr , t 7→ exp tr Xr is finitely generated. Since Remark 9.2 allows us to work on a transversal, we can use Proposition 6.10 to describe ker(ξr ) as the kernel of the morphism C → C(hr )/hhr i given by t 7→ [exp tXr ], which is finitely generated thanks to Lemma 9.4. Step (vi). Let P be a complex connected manifold and t0 a point of P . A deformation of F with parameter space the manifold P pointed at t0 , is a germ along all {0} × P of a 1-dimensional holomorphic singular foliation FP defined on an open neighborhood of {0} × P in C2 × P , which is tangent to the fibers of the projection πP : C2 × P → P and such that F is equal to ∼ the restriction of FP to C2 × {t0 }, with the identification C2 −→C2 × {t0 }, (x, y) 7→ (x, y, t0 ). We say that FP is equireducible if there exists a map EFP : M → C2 × P obtained by composition of blow-up maps Ej : Mj+1 → Mj fulfilling: (1) each center of blow-up Cj ⊂ Mj of Ej is biholomorphic to P by the map πj := πP ◦ E0 ◦ E1 ◦ · · · ◦ Ej−1 : Mj → P , (2) the singular locus of the foliation EF∗ P FP is smooth, contained in the exceptional divisor EFP := EF−1P ({0} × P ) and the restriction of πP ◦ EFP to each of its connected component is a biholomorphism onto P , (3) the restriction of EFP to Mt := (πP ◦ EFP )−1 (t) is exactly the min- imal reduction map of the foliation Ft on C2 × {t} induced by FP ; Notice that EFP is a topological product over P , i.e. there is a homeomor- ∼ phism ΦP : Mt0 × P −→M such that πP ◦ ΦP is the second projection map. By identifying C2 × {t0 } with C2 , each marking f : E → EFt0 of Ft0 by E  extends via ΦP to markings ft : E → EFt ⊂ Mt of Ft , t ∈ P , defining in this way a map P → Mod([F  ]) , t 7→ [Ft , ft ] . On the other hand, given a point t0 ∈ P , for each base point oD in a com- ponent D of E introduced at the beginning of Section 5, let us choose a (1 + dim P )-dimensional submanifold ∆P,D of M, transverse to ft0 (D) at the point ft0 (oD ). The representation of FP -holonomy of the leaf ft0 (D \ Σ) defines a representation HP,Dt0 of the fundamental group π1 (D \ Σ, oD ) in the group Diff(∆P,D , ft0 (oD )) of germs of holomorphic automorphisms of (∆P,D , ft0 (oD )). Definition 10.2. We say that FP is SL-equisingular at a point t0 of P if TOPOLOGICAL MODULI 41 (1) FP is equireducible, (2) for t ∈ P sufficienty close to t0 and for each (D, s) ∈ VeAF  × EdAF  , D ∈ ∂s, the Camacho-Sad indices CS(Et∗ Ft , ft (D), ft (s)) do not depend on t; ∼ (3) there is a germ of biholomorphism ψ : (∆P,D , ft0 (oD ))−→(C×P, (0, t0 )) such that (a) the composition of ψ with the second projection C × P → P is equal to πP ◦ EFP restricted to ∆P,D ; (b) for all γ ∈ π1 (D \ Σ, oD ) the biholomorphism (z, t) 7→ ψ ◦ t0 (γ) ◦ ψ −1 (z, t) does not depend on t. HP,D We say that FP is SL-equisingular if it is SL-equisingular at each point of P . Step (vii). We now do a construction similar to the one given in the proof of Theorem 4.9 after a suitable choice of cocycles in Z 1 (A, Aut). Consider the elements [Fi , fi ] of Mod([F  ]), i ∈ D, given in the statement of Theorem D. By isomorphism (11) they are represented by cocycles ci = (ciD,s ) ∈ Z 1 (R, Sym). Now we fix an orientation ≺ of A and as in the proof of Theorem 5.14, we lift this cocycle to a cocycle (ϕiD,s ) ∈ Z 1 (R, Aut) and we continue to denote by s1 , . . . , sτ the edges associated to the active vertices a1 , . . . , aτ chosen in step (v). We define then (ϕi,t 1 D,s ) ∈ Z (A, Aut) by setting   id if s ∈ EdA \ EdR , ϕi,t D,s = ϕiD,s if s ∈ EdR \ {s1 , . . . , sτ } ,  i ϕD,s ◦ exp tj Xj if s = sj ∈ {s1 , . . . , sτ } , ϕi,t i,t −1 D0 ,s = (ϕD,s ) , for s ∈ EdA with ∂s = {D, D0 } and D ≺ D0 , where exp tj Xj is defined in Step (v). Using these cocycles, for each i ∈ D we glue suitable neighborhoods WD of D × Cτ inside MF × Cτ obtaining (i) a manifold Mi endowed with a submersion map πi onto Cτ , (ii) a normal crossing divisor such that the restriction of πi to each irre- ducible component is a locally trivial fibration with fiber P1 and the restriction of πi to the singular locus of the divisor is a covering over Cτ , (iii) and a foliation by curves tangent to the fibers of the submersion πi and to the divisor. By the same arguments used in Theorem 4.9 we obtain an open neighborhood of {0} × Cτ in C2 × Cτ and on this neighborhood an holomorphic vector field defining a one-dimensional equireducible foliation tangent to the fibers of the projection onto Cτ , whose singular locus is {0} × Cτ . By construction, after equireduction the exceptional divisor, as an intrinsic analytic space, is holomorphicaly trivial over Cτ and along each of its irreducible components the reduced foliation is holomorphically trivial. Hence we have obtained a c of F , see [23], and biholomorphisms h SL-equisingular deformation Fi,t i i,t : ∼ EFi −→EFi,t , i ∈ D. The superscript c stands for complete. We define the c ∼ c : E −→E c by f c := h ◦ f . markings fi,t Fi,t i,t i,t i 42 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM Notice that by the construction of Λ in step (v), Λ(t) is represented in H 1 (A, Aut) by the following 1-cocycle with support in R1 :   id if s ∈ EdA \ EdR , atD,s := id if s ∈ EdR \ {s1 , . . . , sτ } , exp tj Xj if s = sj .  Thanks to Theorems 6.8 and 6.11 we have in H 1 (R, Sym) the equality h i,t i   h t i • • (ϕD,s ) = ci · (aD,s ) , i,t •t where ϕD,s and aD,s denote the classes of ϕi,t • t D,s and aD,s in Syms . The abelian group structure on Mod([F  ]) being induced by the one on H 1 (R, SymF ) by (11), the previous equality proves that in Mod([F  ]) we have c c [Fi,t , fi,t ] = Λ(t) · [Fi , fi ] , i ∈ D. 11. Appendix Let us denote by Br ⊂ C2 the closed ball {|x|2 + |y|2 ≤ r}. For a curve S 3 0 in C2 let us call Milnor ball any ball B = BR such that S ∩ B \ {0} is regular and meets transversely each sphere ∂Br , 0 < r ≤ R. We fix a germ F at 0 ∈ C2 of a singular holomorphic foliation. Definition 11.1. A germ of an invariant curve S at 0 ∈ C2 will be called F-appropriate if S is invariant by F, contains all the isolated separatrices8 and its strict transform by the reduction of F meets any dicritical component D with card(D ∩ Sing(EF )) = 1. The following incompressibility property is proved under some additional assumptions in [16], [19] and an optimal version was obtained by L. Teyssier in [35]: Theorem 11.2. Let F be a generalized curve and let S be an F-appropriate curve in a Milnor ball B. Then there exists a fundamental system (Un )n∈N of open neighborhoods of S in B such that for each n ∈ N (1) the inclusion map Un ,→ B induces an isomorphism between the fundamental groups of Un \ S and B \ S; (2) for each leaf L of the foliation F|(Un \S) the inclusion map L ,→ Un \S induces an injective morphism π1 (L, ·) ,→ π1 (Un \ S, ·); (3) there is a finite union of curves in Un \ S whose preimage Ω in the universal covering U en of Un \ S is a disjoint union of holomorphically embedded open discs Ωα such that each leaf L of the foliation Fen induced by F on U en meets Ω and card(L ∩ Ωα ) ≤ 1 for any α. Remark 11.3. Two direct consequences of this result are the simple con- nectedness of the leaves of the foliation Fen and a structure of (non-Hausdorff) Riemann surface on the leaf space Q eF , an atlas being given by the transver- Un sals Ωα .  Previous Theorem 11.2 will allow us to extend Theorem 1.6 of [18] with weaker assumptions: 8i.e. their strict transforms meet invariant components of the exceptional divisor. TOPOLOGICAL MODULI 43 Theorem 11.4. Let F and G be two topologically equivalent germs of folia- tions at 0 ∈ C2 and suppose the existence of a germ of homeomorphism that conjugates them: ψ : (C2 , 0) → (C2 , 0) , ψ∗G = F . (20) If F is a generalized curve fulfilling Conditions (TC) and (TR) stated in Section 2, then there exists a germ of homeomorphism φ : (C2 , 0) → (C2 , 0) such that: (1) the lifting EG−1 ◦ φ ◦ EF of φ through the reduction maps of F and G extends to the exceptional divisor as a germ of homeomorphism Φ : (MF , EF ) → (MG , EG ) along the exceptional divisors; (2) Φ is holomorphic at each singular point of F ] := EF∗ F which is not a nodal corner; (3) Φ is transversely holomorphic at each point of the exceptional divisor which is regular for F ] and not contained in a dicritical component. The rest of this appendix is devoted to the proof of this theorem, which is similar to that of Main Theorem of [18]. The new difficulties lie in the fact that ψ may not be transversely holomorphic on a whole neighborhood of 0, and this for three reasons: • on the union of leaves meeting a dicritical component of EF there are C 0 -automorphisms of the foliation F which are not transversally holomorphic; • there are also such automorphisms near nodal singular points with support in nodal separators, cf. [18, page 406]; • exceptional cut-components9 introduced in Section 2.2 of EF are not excluded by Condition (TC) and at such a component C the prop- erty (TR) is ineffective. Indeed, every irreducible component of C contains at most two singular points, therefore its holonomy group, being monogenous or trivial, cannot be topologically rigid. To prove Theorem 11.4 we will proceed in five steps: (i) we prove that the one-to-one correspondence induced by ψ between the irreducible components of EF and those of EG and between the singular points of both foliations F ] and G ] preserves the Camacho-Sad indices; (ii) we recall the notion of monodromy and two key results given in [18] that remain valid in our more general setting; (iii) we construct a conjugation Φ0 between F ] and G ] along the non-excep- tional cut-components of EF , except at the nodal corners and at the intersection with the dicritical components; (iv) we construct a conjugation Φ00 along the exceptional cut-components, except at the nodal corners and at the intersection with the dicritical components; (v) we extend and glue Φ0 and Φ00 at the nodal corners and along the dicritical components. 9which cannot exist if F is non-dicritical by Theorem 11.5 below. 44 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM 11.1. Step (i): Equality of Camacho-Sad indices. Since F and there- fore G are generalized curves [3], there is a unique one-to-one correspondence D 7→ D0 and s 7→ s0 (21) between the irreducible components of the exceptional divisors EF and EG , the strict transforms of the isolated separatrices of F and G, and between the points of Sing(EF ) ∪ Sing(F ] ) and Sing(EG ) ∪ Sing(G ] ), such that: • if s is the intersection point of an isolated separatrix S with EF , then s0 is the intersection point of ψ(S) with EG , • we have equalities of intersection numbers (D10 , D20 ) = (D1 , D2 ) for D1 7→ D10 , D2 7→ D20 . (22) Indeed the reduction map of a generalized curve foliation coincides with the reduction map of the curve formed by all its isolated separatrices and two dicritical separatrices for each dicritical component of the exceptional divi- sor. Thus the above properties follow from classical topological properties for germs of curves. Let us point out that property (2) in Theorem 11.4 implies equality of Camacho-Sad indices of these foliations. These equalities will be strongly used in the proof of the above theorem and in fact we need to prove them first. We will use the following result [4, Theorem 9 and Remark 11]: Theorem 11.5. For any cut-component there is a strict transform of an isolated separatrix which meets it at a non-nodal singular point. Lemma 11.6. Under the assumptions of Theorem 11.4, F fulfilling again Conditions (TR) and (TC), if s ∈ D ⊂ EF and s0 ∈ D0 ⊂ EG correspond by (21), then CS(F ] , D, s) = CS(G ] , D0 , s0 ) . (23) As discussed in Remark 2.1, Conditions (TR) and (TC) are necessary to get this lemma. Proof. The induction process given in [18, §7.3], which is based on the Camacho-Sad Index Formula, proves that equalities (23) hold at all singu- larities in a cut-component C if they are satisfied at every intersection point s of the strict transform of a separatrix with C. The existence of such a point s being assured by Theorem 11.5, we distinguish three possibilities. a) λ := CS(F ] , D, s) is an irrational real number. If λ is positive, s is a nodal singular point, and equality (23) was obtained by R. Rosas in [33, Proposition 13]. Another proof is given in [19, Theorem 1.12] that remains valid for λ < 0. b) C is non-exceptional and s is not a nodal singular point. Then by using (TR) and thanks to an extended version [18] of the rigidity theorem in [32], ψ is transversely holomorphic on the image by EF of a neighborhood of C and specifically at the points of the separatrix S. In this case the proof of equality (23) given in [18, §2] remains valid. c) C is exceptional. Then C is a union of components of EF , C = D1 ∪· · ·∪D` , ` ≥ 1, Di meeting Di+1 in one point si , and Di ∩ Dj = ∅ if |i − j| = 6 1. TOPOLOGICAL MODULI 45 Perhaps C meets several dicritical components of EF , but we only have two possibilities fulfilling Assumption (TC): (i) s ∈ D1 and the other singular points of F ] belonging to C are s1 , . . . , s`−1 , see Figure 4; (ii) s ∈ D1 , D` contains a nodal singular point s` 6= s`−1 , the other singular points of F ] belonging to C being s1 , . . . , s`−1 , see Figure 5. Indeed, Theorem 11.5 implies that singularity s is not nodal. In case (ci) using the classical index formula, we see that CS(F ] , D1 , s) is given by a continuous fraction whose coefficients are the self-intersections (Di , Di ), i = 1, . . . , `; thus (23) follows from (22). In the same way we obtain in case (cii) that CS(F ] , D1 , s) is an irrational (negative) real number, but this case was already examined in case (a).  s s1 s`−1 • • ··· • D1 D2 D` Figure 4. Situation (ci) with two dicritical components. s s1 s`−1 s` • • ··· • • • D1 D2 D` Figure 5. Situation (cii) with a dicritical component and a nodal singularity s` . Since the dicriticity of an irreducible component D can be characterized by the vanishing of the Camacho-Sad indices along all the adjacent components at their intersection points with D, we have: Corollary 11.7. The image by correspondence (21) of a dicritical compo- nent, an exceptional cut-component, a non-exceptional cut-component of F is respectively a dicritical component, an exceptional cut-component, a non- exceptional cut-component of G. 11.2. Step (ii): Monodromy and holonomy. Let us now fix a F-appropr- iate curve S, Milnor balls B and B 0 for S and the G-appropriate curve S 0 := ψ(S), where ψ is given by (20). Let us also fix e −→ B \ S , q:B q0 : B e 0 −→ B 0 \ S 0 universal coverings of B \ S and B 0 \ S 0 respectively. In all the sequel we adopt the following conventions and notations: (a) for A ⊂ B, resp. A0 ⊂ B 0 , we write e := q −1 (A \ S) , A f0 := q 0−1 (A0 \ S 0 ) , resp. A 46 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM (b) we suppose that B 0 contains ψ(B), we choose a lifting ψe : B e0 e → B of ψ and we denote the isomorphisms induced by ψe between the deck transformation groups of these coverings by ∼ ψe∗ : Γ−→Γ0 , ψe∗ (f ) := ψe ◦ f ◦ ψe−1 , Γ := Autq (B) e , Γ0 := Autq0 (B e0) , (c) we fix fundamental systems of open neighborhoods of S in B, resp. S 0 in B 0 , (Un )n∈N , (Un0 )n∈N , with ψ(Un ) ⊂ Un0 , fulfilling Properties (1)-(3) of Theorem 11.2, (d) QeF := U en /Fe e denotes the leaf space of the restriction to U en of the Un | Un foliation Fe := q ∗ F, endowed with its structure of Riemann surface, cf. Remark 11.3, (e) via EF : BF → B and EG : BG0 → B 0 , the reduction maps of F and G, we perform the following identifications BF \ S ' B \ S , BG0 \ S 0 ' B 0 \ S 0 , S := EF−1 (S) , S 0 := EG−1 (S 0 ) , (f) we consider B e and Bf0 as universal coverings of BF \ S and B 0 \ S 0 G respectively and, with these identifications, for A ⊂ BF or A0 ⊂ BG0 we write: e := q −1 (A \ S) , A A e0 := q 0−1 (A0 \ S 0 ) . Definition 11.8. The monodromy of F is the morphism between Γ and the group of automorphisms of inverse systems MF eF eF S : Γ → AutAn (Q∞ ) ⊂ AutTop (Q∞ ) , eF := (Q Q∞ eF ) Un n∈N , ←− ←−− given by the actions of Γ on the leaf spaces (f, L) 7→ f (L), f ∈ Γ, L ∈ QeF , Un with An ←− the category of pro-objects associated to the category of analytic spaces, and Top the category of pro-objets associated to the category of topo- ←−− logical spaces and continuous maps. The monodromy MGS 0 : Γ0 → AutAn (Q eG ) of G is defined in the same way. ∞ ←− Remark 11.9. The lifting ψe of ψ fixed in (b) induces an isomorphism eF ∼ eG hψe : Q∞ −→Q∞ in the category Top, given by the maps Q eF → Q eG 0 that associate to each Un Un ←−− leaf L of Fe|Uen the leaf of Ge|Ue 0 containing ψ(L). e When EF contains an ex- n ceptional cut-component hψe may not be N -analytic in the sense of [18, page 416] and we must extend to our context the notion of geometric conjugation of monodromy introduced in [18].  To any subset A in B meeting S we associate the following object and morphism in the category Top: ←−− • (A, ∞) := (A ∩ Un )n∈N , e e TOPOLOGICAL MODULI 47 • τAe : (A, e ∞) → Q eF is induced by the family of maps τ n : A e ∩ Un → ∞ A e QeF that associate to any point m in A e ∩ Un the leaf containing m Un of the foliation Fe|Uen , n ∈ N, • Γ e is the group of automorphisms f of (A, A,∞ e ∞) such that q ◦ f = q. For A0 ⊂ B 0 meeting S 0 , (A f0 , ∞), τ e0 and Γf0 are similarly defined. A A ,∞ G Definition 11.10. A geometric C 0 -conjugation between MF S and MS 0 is a pair (g, h) formed by: ∼ • an isomorphism of groups g : Γ−→Γ0 induced by the lifting ∼ f0 e ∞)−→( ge : (B, B , ∞) , q 0 ◦ ge = g ◦ q , ∼ of a germ of homeomorphism along the separatices g : (B, S)−→(B 0 , S 0 ) preserving the orientations of B, B 0 , S and S 0 (but not necessarily the foliations), eF ∼ eG • an isomorphism h : Q ∞ −→Q∞ in the category Top, such that h∗ ◦ ←−− G MF S = M S 0 ◦ g, where h∗ is the group morphism eF ) → AutTop (Q h∗ : AutTop (Q eG ) , ϕ 7→ h ◦ ϕ ◦ h−1 . ∞ ∞ ←−− ←−− We also say that (g, ge, h) represents the conjugation (g, h). Remark 11.11. The lifting ψe of ψ choosen in (b) induces a geometric C 0 - G conjugation (ψe∗ , hψe) between MF S and MS 0 , that is represented by (ψ, ψ, hψ e e), with hψe as in Remark 11.9.  Now for each invariant component D of S we will denote by SD the set of singular points of F ] belonging to D, except the nodal singularities which are attaching points of strict transforms of separatrices. Let us choose: • a good fibration ρD transverse to D, cf. Definition 4.5; thus if m ∈ D is not a singular point of S we can write without ambiguity: ∆m = ρ−1 D (m) ; (24) • a collection (ZD,s )s∈SD of holomorphically embedded compact discs in D, centered at the points s, without pairwise intersection. We also require that ZD,s = D when D is the strict transform of an isolated separatrix. Definition 11.12. The following compact sets: (i) ZD := D \ ∪s∈SD ZD,s , with D an invariant component of EF , (ii) Zs := ZD1 ,s ∪ ZD2 ,s , with s the intersection point of two invariant components of S, will be called elementary pieces of S. We perform for G similar choices of good fibrations ρD0 and embedded discs defining similarly elementary pieces Zs0 0 and ZD 0 of S 0 . 0 Theorem A of [17] gives immediately: Lemma 11.13. For any geometric C 0 -conjugation (g, h) between MF S and G MS 0 there exists a representation (g, ge, h) and a choice of elementary pieces of S and S 0 such that: 48 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM (1) g lifts through the reduction maps EF and EG to a germ of homeomor- phism g ] : (BF , S) → (BG0 , S 0 ), i.e. EG ◦ g ] = g ◦ EF , and, for any elementary piece ZD , D ∈ Comp(EF ) invariant, and Zs , s ∈ Sing(S), it satisfies: (2) g ] (ZD ) = Zg0 ] (D) and g ] is compatible with the transverse fibrations over ZD , i.e. (ρg] (D) ◦ g ] )|ρ−1 (ZD ) = (g ] ◦ ρD )|ρ−1 (ZD ) , D D (3) g ] (Zs ) = Zg0 ] (s) , g ] is holomorphic on a neigborhood of Zs and is com- patible with the good fibration over ∂Zs . Definition 11.14. Let V ⊂ (B \ S) and V 0 ⊂ (B 0 \ S 0 ) be subsets whose closures meet respectively S and S 0 . A C 0 -realization of (g, h) over V and V 0 e) of a homeomorphism χ : V → V 0 and a lifting of it (in the is the data (χ, χ category Top) ←−− ∼ f0 χe : (Ve , ∞)−→(V , ∞) , q 0 ◦ χe= χ◦q, such that the following diagrams commute: χ (Ve , ∞) e / (V f0 , ∞) Γ ι / Γe V ,∞ τVe τVe 0 g χ e∗     eF h /Q eG ι0 / Γ0 Q∞ ∞ Γ0 f0 ,∞ V When χ can be lifted via EF and EG to an open neighborhood of a subset Z e) is a C 0 -realization of (g, h) along Z. of EF , we will say that (χ, χ Remark 11.15. If V is an open set then the commutativity of the first diagram implies that χ conjugates the foliations F|V and G|V 0 .  We now fix a geometric C 0 -conjugation (g, h) between the monodromies G MFS and MS 0 and a representation (g, ge, h) of it fulfilling Assertion (1) of Lemma 11.13. Let D be an invariant component of S and let m0 ∈ D \ Sing(F ] ). We will use the notations: D0 := g ] (D) , m00 := g ] (m0 ) , D∗ := D \ Sing(F ] ), D0∗ := D0 \ Sing(G ] ) . Let us fix now (∆, m0 ), resp. (∆0 , m00 ), a germ at m0 , resp. m00 , of an holomorphically embedded disc transversal to the foliation. We also denote by F ] G ] HD : π1 (D∗ , m0 ) → Diff(∆, m0 ) , HD 0∗ 0 0 0 0 : π1 (D , m0 ) → Diff(∆ , m0 ) , the holonomy representations of the foliation F ] , resp. G ] , associated to the leaves D∗ and D0∗ . Theorem 11.16. Let χ : (∆, m0 ) → (∆0 , m00 ) be a germ of homeomor- phism and χ e ∞) → (∆ e : (∆, e 0 , ∞) be a lifting of it, such that (χ, χ e) is a 0 0 realization over ∆ and ∆ of the geometric C -conjugation (g, h) between the G ] monodromies MF S and MS 0 . Then χ and the group isomorphism g∗ from π1 (D∗ , m0 ) to π1 (D0∗ , m00 ) induced by g ] define a conjugation between HD F] G] and HD 0 , i.e. G ] F ] −1 HD ] 0 (g∗ (γ̇)) = χ ◦ HD (γ̇) ◦ χ , γ̇ ∈ π1 (D∗ , m0 ) . TOPOLOGICAL MODULI 49 Proof. The proof of [18, Theorem 4.3.1] remains literally valid for this C 0 - version.  Let us assume now that the previously fixed representation (g, ge, h) of (g, h) satisfies all the properties (1)-(3) of Lemma 11.13 and that the point m0 is in the boundary ∂Z of an elementary piece Z such that either Z = ZD with D an invariant component of EF , or Z = Zs with s a non-nodal singularity. We can suppose that ∆ = ∆m0 and ∆0 = ∆0m0 , cf. Notation (24). 0 Lemma 11.17. With the notations introduced above and under the hypoth- esis of Theorem 11.16, let us also suppose that the following additional as- sumptions are satisfied: • χ e and ge induce the same map from π0 (∆) e to π0 (∆ e 0 ), • χ is holomorphic, • if Z = Zs , the Camacho-Sad indices of F ] at s along D and of G ] at g ] (s) along g ] (D) are equal. Then there exists a C 0 -realization (Φ, Φ), e Φ : V → V0, f0 , ∞) , e : (Ve , ∞) → (V Φ of (g, h) along Z, such that: (1) the restriction of Φ to EF ∩ V is equal to g ] , its restriction to (∆, m0 ) is equal to χ and that of Φ e ∞) is equal to χ e to (∆, e, (2) on a neighborhood of ∂Z ∩ EF , Φ is compatible10 with the good fibra- tions previously choosen; moreover the restriction of Φ to each fiber ∆m , m ∈ ∂Z ∩ EF , is holomorphic, cf. Notation (24); (3) for every m ∈ ∂Z ∩ EF , the restrictions Φ e e and ge e induce the |∆ m |∆ m same map π0 (∆ e 0 ] ). e m ) → π0 (∆ g (m) Proof. It is literally the same proof as that of [18, Lemma 8.3.2] but using Theorem 11.16 above instead of [18, Theorem 4.3.1].  11.3. Step (iii): Conjugation at non-exceptional cut-components. We keep the notations and conventions (a)-(f) introduced at the beginning of Section 11.2. We will also use the notation ∆m , resp. ∆0m0 , introduced at (24) for the fibers of the good fibrations at regular points of S, resp. S 0 . Using Theorem 11.5, in each non-exceptional cut-component C ⊂ EF , let us choose a non-nodal singular point sC ∈ Sing(F ] ) where the strict transform XC of an isolated separatrix meets C. Let us consider the following filtration of S: C0 ⊂ C1 · · · ⊂ Ck ⊂ S defined by: • C0 is the union of all elementary pieces ZsC for any cut-component C of EF , • each Zj := Cj \ Cj−1 , j = 1, . . . , k, is the union of all (disjoint) elementary pieces meeting Cj−1 , but not contained in Cj−1 . 10i.e. ρ0 ◦ Φ(m) = Φ ◦ ρ(m) if ρ(m) ∈ ∂Z, ρ and ρ0 denoting the good fibrations of the corresponding components of the exceptional divisors. 50 DAVID MARÍN, JEAN-FRANÇOIS MATTEI AND ÉLIANE SALEM • S \ Ck is the union of the dicritical components, the strict trans- forms of the components of S meeting a dicritical component and the elementary pieces associated to nodal corners. Lemma 11.18 ([18], §8.4). There exists a representation (g, ge, h) of (g, h) satisfying properties (1)-(3) of Lemma 11.13, such that moreover the restric- tions of ge to ∆ and the lifting ψe of ψ chosen in (b) induce the same map from π0 (∆) e to π0 (∆e 0 ). Let us choose (g, ge, h) given by this lemma. Notice that according to Lemma 11.6, the points s0C 0 = g ] (sC ) are not nodal singular points of G ] and we can consider the filtration of S 0 C00 ⊂ C10 · · · ⊂ Ck0 ⊂ S 0 , Cj0 := g ] (Cj ) . According to Lemma 11.13, C00 is a union of elementary pieces of S 0 , and for j = 1, . . . , k, Zj0 := Cj0 \ Cj−1 0 = g ] (Zj ) is a union of disjoint elementary pieces not contained in Cj−1 0 but meeting 0 e j ) along every Zj Cj−1 . By induction we will now define a realization (Φj , Φ of the conjugation (ψe∗ , hψe), j = 0, . . . , k. • (Φ0 , Φ e 0 ) is defined along any connected component Zs of C0 as the real- C ization of (ψe∗ , hψe) obtained by: first choosing a point mC in the boundary of XC , then modifying ψ near mC by performing a foliated isotopy such that ψ(∆mC ) = ∆0ψ(mC ) and finally applying the extension Lemma 11.17 with Z = Zs , ∆ = ∆m , ∆0 = g ] (∆), χ = ψ|∆ and χ C C e = ψe e ; |∆ • (Φj , Φ e j ) is defined along each connected component Z of Zj as the real- ization of (ψe∗ , hψe) obtained by: first choosing a point mZ in the (unique) component of ∂Z contained in Cj−1 , then applying Lemma 11.17 with Z = Z, ∆ = ∆mZ , ∆0 = g ] (∆), and χ, χ e being respectively the restric- tions of Φj−1 and Φ e j−1 to ∆ and ∆. e According to Remark 11.15, Φj conjugates the germ of F ] along Zj to the germ of G ] along Zj0 . To achieve this step and to obtain a conjugation Φ0 between F ] and G ] on a neighborhood of Ck it suffices to note that for j = 1, . . . , k and for each connected component Θ of Zj−1 ∩ Zj , Φj−1 and Φj are necessary equal when we restrict them to the real hypersurfaces ρ−1 D (Θ), D being the component of S containing Θ. This fact results from the property (2) in Lemma 11.17 and the uniqueness of the extension of a conjugation between the holonomies to a conjugation of the corresponding foliations preserving transversal fibrations. 11.4. Step (iv): Construction along the exceptional cut-components. Let C be an exceptional cut-component of EF and let us keep the notations introduced in the proof of Lemma 11.6: C = D1 ∪ · · · ∪ D` has two possible configurations (ci) and (cii), see Figures 4 and 5. In the first case (ci), D` contains only one singular point of the foliation, hence there is a holomorphic first integral defined on a neighborhood of C and TOPOLOGICAL MODULI 51 specifically the foliation is linearizable at each singular point. Thus by equal- ity of Camacho-Sad indices given by Lemma 11.6 the foliations considered are locally holomorphically conjugated at the singular points corresponding by (21). The equalities (22) of self-intersections of the components of C and C 0 allow us to glue these conjugacies and to obtain a homeomorphism de- fined on a neighborhood of C. We leave the details of this construction to the reader. The situation in case (cii) is similar: we have again equality of Camacho- Sad indices and therefore local conjugacies, and then equality of self-inter- sections allowing to glue and to obtain a global C 0 -conjugation. 11.5. Step (v): Extension and gluing. On the elementary pieces Zs corresponding to a nodal corner s, we perform the gluing of the homeomor- phisms already constructed by the process described in [18, §8.5]. It remains to extend the obtained homeomorphisms to the dicritical components. No- tice that in all the above constructions the homeomorphisms can be built by respecting the dicritical components meeting their domains of definition. Finally we arrive at the following situation described in [19, page 147]: • we identify tubular neighborhoods of the dicritical components D ⊂ EF and D0 ⊂ EG corresponding by (21), with the same tubular neigh- borhood of the zero section of the normal bundle of D; it is possible because D and D0 have same negative self-intersection; • the corresponding foliations are identified with the natural normal fiber bundle; • we have to extend to the whole D a continuous map f from a union K of disjoint closed discs to the group Aut0 (C, 0) of germs of homeomor- phisms of (C, 0). 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Société Mathématique de France (1977) 189 pp. [35] L. Teyssier, Germes de feuilletages présentables du plan complexe. Bull. Braz. Math. Soc. (N.S.) 46 (2015), no. 2, 275–329. [36] T. Yagasaki, Homotopy types of homeomorphism groups of noncompact 2-manifolds. Topology Appl. 108 (2000), no. 2, 123–136. TOPOLOGICAL MODULI 53 Departament de Matemàtiques, Universitat Autònoma de Barcelona, E- 08193 Bellaterra (Barcelona), Spain E-mail address: davidmp@mat.uab.es Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, Route de Narbonne, F-31062 Toulouse Cedex 9, France E-mail address: jean-francois.mattei@math.univ-toulouse.fr Sorbonne Université, Université Paris Diderot, CNRS, Institut de Math- ématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France E-mail address: eliane.salem@imj-prg.fr