Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.

The value of the cobracket on a loop a ∈ λ(X) is obtained by representing it by an immersed circle α : S 1 → X with transverse self intersections and trivial winding number relative to ξ. Each double point P of α divides it into two loops based at P , which we denote by α ′ P and α ′′ P . Let ǫ P = ±1 be the intersection number of the initial arcs of α ′ P and α ′′ P . The cobracket of a is then defined by where a ′ P and a ′′ P are the classes of α ′ P and α ′′ P , respectively. The powers of the augmentation ideal I of Rπ 1 (M, x) define the I-adic topology on it and induce a topology on Rλ(X). Kawazumi and Kuno [25] showed that δ ξ is continuous in the I-adic topology and thus induces a map δ ξ : Rλ(X) ∧ → Rλ(X) ∧ ⊗ Rλ(X) ∧ on I-adic completions. This and the completed Goldman bracket give Rλ(X) ∧ the structure of an involutive completed Lie bialgebra [25]. Now suppose that X is a smooth affine curve over C or, equivalently, the complement of a finite set D in a compact Riemann surface X. In this case Qλ(X) ∧ has a canonical mixed Hodge structure [10]. Our first main result is that the Turaev cobracket is compatible with this structure. Theorem 1. If ξ is a nowhere vanishing holomorphic vector field on X that is meromorphic on X, then is a morphism of pro-mixed Hodge structures, so that Qλ(X) ∧ ⊗ Q(1) is a complete Lie coalgebra in the category of pro-mixed Hodge structures.
We will call such a framing ξ an algebraic framing. The main result of [17] asserts that is a morphism of mixed Hodge structure (MHS), so that Qλ(X) ∧ ⊗ Q(−1) is a complete Lie algebra in the category of pro-mixed Hodge structures.
Corollary 2. If ξ is an algebraic framing of X, then Qλ(X) ∧ , { , }, δ ξ is a "twisted" completed Lie bialgebra in the category of pro-mixed Hodge structures.
By "twisted" we mean that one has to twist both the bracket and cobracket by Q(±1) to make them morphisms of MHS. There is no one twist of Qλ(X) that makes them simultaneously morphisms of MHS.
Let v be a non-zero tangent vector of X at a point of D. Standard results in Hodge theory (see [17, §10.2]) imply: These isomorphisms are torsors under the prounipotent radical U MT X, v of the Mumford-Tate group of the MHS on Qπ 1 (X, v) ∧ .
Such splitting of the weight filtration are called "Goldman-Turaev" formality isomorphisms in [3].
One can ask whether a fixed affine curve X has an algebraic framing ξ and, if so, how many it has. Such questions can be answered using algebraic geometry. Specifying ξ is equivalent to specifying the meromorphic 1-form ω on X that takes the value 1 on it. This 1-form is holomorphic and nowhere vanishing on X; its divisor is supported on D := X − X. So, up to rescalling, algebraic framings of X correspond to canonical divisors of X supported on D. These (if they exist) form a principal homogeneous space over ΓH 1 (X; Z(1)) := Hom MHS Z(−1), H 1 (X; Z) = {df /f : f ∈ H 0 (X, O × X ) alg }. For each X and general D ⊂ X, this group is trivial, so that an algebraic framing of a general curve X is unique if it exists at all.
As for existence, when D consists of a single point, we can take X to be hyperelliptic and D to consist of a single Weierstrass point. More generally, a result of Kontsevich and Zorich [26] implies that the locus of curves X in the moduli space of curves M g that possess a holomorphic differential ω whose divisor is supported on a set D has codimension max(g − 1 − #D, 0). Similarly in the meromorphic case [4].
Our main application is to the Kashiwara-Vergne problem [3]. Solutions of the Kashiwara-Vergne problem of type (g, n + 1) are automorphisms Φ the complete Hopf algebra Q x 1 , . . . , x g , y 1 , . . . , y g , z 1 , . . . , z n that solve the Kashiwara-Vergne equations. In [3] it is shown that the Kashiwara-Vergne problem admits solutions for all framed surfaces of genus g = 1 and for surfaces of genus 1 with certain, but not all, framings. (See [3, Thm. 6.1].) Solutions of the Kashiwara-Vergne problem correspond to isomorphisms Φ above that induces isomorphisms of the Goldman-Turaev Lie bialgebra with the completion of its associated weight graded. Corollary 3 and [2,Thm. 5] imply that the automorphism Φ constructed from a Hodge splitting of Qπ 1 (X, v) ∧ in [17, §13.4] solves the KV equations. This constructs solutions of the Kashiwara-Vergne problem for all affine curves with an algebraic framing. To link with the results of [3], we show (Proposition 8.2) that the framings in [3] for which the Kashiwara-Vergne problem can be solved are precisely those that can be realized by an algebraic framing of an affine curve. This gives a second and independent proof of their main result and clarifies their conditions on a framing for which the KV problem has a solution. 1 Solutions of the Kashiwara-Vergne (KV) problem for a framed surface of type (g, n + 1) form a torsor under a prounipotent group, denoted KRV d g,n+ 1 in [3], where d ∈ Z n+1 is the vector of local degrees of ξ at the punctures x j ∈ D. 2 This group depends only on the local degrees d and not on other topological invariants of ξ. The Hodge theoretic construction of solutions to the KV problem has the feature that it produces large torsors of solutions. The solutions of the KV problem of type (g, n + 1, d) constructed in this paper form a principal homogeneous space under a group U d g,n+ 1 , which is described explicitly below. It is a subgroup of KRV d g,n+ 1 and we conjecture that it is equal to KRV d g,n+ 1 . Equivalently, we conjecture that all solutions of the Kashiwara-Vergne problem arise from the Hodge theoretic constructions in this paper. 1 To compare the two statements, one should note that if γ j is the boundary of sufficiently small disk in X, centered at x j and, then d j + rot ξ γ j = 1. Note that the boundary orientation conventions used in [1,2,3] differ from those used in this paper. Their adapted framing has the property that d 0 = 2 − 2g and d j = 0 for all j ≥ 1.
2 If d j is the local degree of ξ at x j , then d = (d 0 , . . . , dn). The Hopf Index Theorem implies Let S be a closed oriented surface of genus g and P = {x 0 , . . . , x n } a finite subset. Suppose that v o is a non-zero tangent vector of S anchored at x 0 . Set S = S − P and suppose that ξ o is a framing of S. Each our our solutions to the KV problem arises by choosing a complex structure on (S, P, v o , ξ o ) and choosing a lifting of the canonical central cocharacter G m → π 1 (MHS ss ). (See [17, §10.2].) The Lie algebra u d g,n+ 1 of U d g,n+ 1 has a natural grading for each of our solutions of the KV problem, and the action is compatible with the splittings of the weight filtrations.
In order to state the next theorem, we need to introduce several prounipotent groups. Denote the category of mixed Tate motives unramified over Z by MTM(Z). Denote the Lie algebra of the prounipotent radical of its tannakian fundamental group π 1 (MTM, ω B ) (with respect to the Betti realization ω B ) by K. Its Lie algebra k is non-canonically isomorphic to the free Lie algebra Denote the relative completion of the mapping class group of (S, P, v o ) by G g,n+ 1 and its prounipotent radical by U g,n+ 1 . (See [12] for definitions.) These act on The vector field ξ o determines a homomorphism U g,n+ 1 → H 1 (S) that depends only on the vector d of local degrees of ξ. Denote its kernel by U d g,n+ 1 . 4 Denote by U d g,n+ 1 the subgroup of KRV d g,n+ 1 generated by U g,n+ 1 and U MT X, v . It is not immediately clear that this group is independent of the algebraic structure on (X, v) as the group MT X, v depends non-trivially on the algebraic structure.
Theorem 4. If 2g + n > 1 (i.e., S is hyperbolic), then the group U d g,n+ 1 does not depend on the choice of an algebraic structure (X, D, v, ξ) on (S, P, v o , ξ o ). The group U d g,n+ 1 is normal in U d g,n+ 1 , and there is a canonical surjective group homomorphism K → U d g,n+ 1 /U d g,n+ 1 , where K denotes the prounipotent radical of π 1 (MTM).
This result follows from a more general result, Theorem 8.12, which is proved in Section 9. We expect the homomorphism K → U d g,n+ 1 /U d g,n+ 1 to be an isomorphism. The injectivity of this homomorphism is closely related to Oda's Conjecture [27] (proved in [29]) and should follow from it.
4 Explicit presentations of the Lie algebras of the U g,n+ 1 are known for all n ≥ 0 when g = 2 [12,15,20]; partial presentations (e.g., generating sets) are known when g = 2, [32]. Presentations of the U d g,n+ 1 can be deduced easily from these.
is an isomorphism if and only if the inclusion of π 1 (MTM) into the Grothendieck-Teichmüller group is an isomorphism. In this case, KRV d g,n+ 1 should be a split extension As when proving that the Goldman bracket is a morphism of MHS [17], the proof of Theorem 1 consists in: (i) Finding a homological description of the cobracket δ ξ analogous to the homological description of the Goldman bracket given by §3]. This description gives a factorization of the cobracket. (ii) Giving a de Rham description of the continuous dual of each map in this factorization. (iii) Proving that, for each complex structure on (S, P, v o , ξ o ), each map in this factorization of the dual cobracket is a morphism of MHS. The homological description of the cobracket is established in Sections 4 and 5. This description appears to be new. The de Rham descriptions of the factors of the dual cobracket are given in Section 6. The proof of Theorem 1 is completed in Section 7 where it is shown that each map in the factorization of the cobracket is a morphism of MHS for each choice of a complex structure. The analysis of the group U d g,n+ 1 and the proof of Theorem 4 are given in Sections 8 and 9. This paper is a continuation of [17]. We assume familiarity with the sections of that paper on rational K(π, 1) spaces, iterated integrals, and Hodge theory.
Acknowledgments: I would like to thank Anton Alekseev, Nariya Kawazumi, Yusuke Kuno and Florian Naef for patiently answering my numerous questions about their work.

Notation and Conventions
Suppose that X is a topological space. There are two conventions for multiplying paths. We use the topologist's convention: The product αβ of two paths α, β : [0, 1] → X is defined when α(1) = β(0). The product path traverses α first, then β. We will denote the set of homotopy classes of paths from x to y in X by π(X; x, y). In particular, π 1 (X, x) = π(X; x, x). The fundamental groupoid of X is the category whose objects are x ∈ X and where Hom(x, y) = π(X; x, y).
As in [17], we have attempted to denote complex algebraic and analytic varieties by the roman letters X, Y , etc and arbitrary smooth manifolds (and differentiable spaces) by the letters M , N , etc. This is not always possible. The diagonal in T ×T will be denoted ∆ T .
The singular homology of a smooth manifold M will be computed using the complex C • (M ) of smooth singular chains. The complex C • (M ) will denote its dual, the complex of smooth singular cochains. The de Rham complex of M will be denoted by E • (M ). The integration map E • (M ) → C • (M ; R) is thus a well-defined cochain map.

Local systems and connections.
Here we regard a local system on a manifold N as a locally constant sheaf. We will denote the complex of differential forms on N with values in a local system V of real (or rational) vector spaces by E • (N ; V ). As in [17], we denote the flat vector bundle associated to V by V and The pullback of a local system V over Y × Y along the interchange map τ : Y 2 → Y 2 will be denoted by V op .

2.2.
Cones. Several homological constructions will use cones. Since signs are important, we fix our conventions. The cone of a map φ : It satisfies dz, c = z, ∂c and thus induces a pairing , :

Preliminaries
We recall and elaborate on notation from [17]. Fix a ring k. Typically, this will be Z, Q, R or C. Suppose that M is a smooth manifold, possibly with boundary. All paths [0, 1] → M will be piecewise smooth unless otherwise noted. Denote the space of paths γ : [0, 1] → M by P M . This is endowed with the compact open topology. For each t ∈ [0, 1], one has the map p t : P M → M defined by p t (γ) = γ(t). It is a (Hurewicz) fibration.

3.1.
Fibrations. The most fundamental path fibration is the map Its fiber over (x 0 , x 1 ) is the space P x0,x1 M of paths in M from x 0 to x 1 . When x 0 = x 1 = x, the fiber is the space Λ x M of loops in M based at x. The local system whose fiber over (x 0 , x 1 ) is H 0 (P x0,x1 M ; k) will be denoted by P M . More generally, for (t 1 , . . . , t n ) ∈ [0, 1] n with 0 < t 1 ≤ t 2 ≤ · · · ≤ t n < 1, one has the fibration n j=1 p tj : P M → M n whose fiber over (x 1 , . . . , x n ) is P ,x1 M × P x1,x2 M × · · · × P xn−1,xn M × P xn, M.
Here P ,x M denotes the space of paths terminating at x ∈ M and P x, M denotes the space of paths emanating from x. Since P x, M and P ,x M are contractible, the fiber of the corresponding local system over M n is π * 1,2 P M ⊗ π * 2,3 P M ⊗ · · · ⊗ π n−1,n P M , where π j,k : M n → M × M denotes the projection onto the product of the jth and kth factors.
The "pullback path fibration" obtained by pulling back 3.2. Homology. The following result follows easily from the fact that a noncompact surface is a K(π, 1) and has cohomological dimension 1. Cf. [17,Prop. 3.5].  Proof. Each loop α : S 1 → M induces a map α 2 : where α ′ is the restriction of α to the positively oriented arc in S 1 from θ to φ and α ′′ is its restriction to the arc from φ to θ. This lift does not extend continuously to S 1 × S 1 , except when α is null homotopic.
To extend the lift, we replace S 1 × S 1 by U := [0, 2π] ⊗ S 1 . The map is a quotient map that takes the boundary of U onto the diagonal ∆ S 1 . It induces a homeomorphism (0, 2π) The result follows from the fact that ∂s α = 1 ⊗α −α ⊗ 1.

A Homological Description of the Turaev Cobracket
Throughout this section, M will be a smooth oriented surface, possibly with boundary, and k is arbitrary. Denote space of non-zero tangent vectors of M by to be the map whose restriction to the fiber L M,v over v is defined by The maps ∆ and ι induce a chain map of singular chain complexes. We can therefore form the cone is the section associated to the loop π • α ∈ ΛM that is defined in the proof of Proposition 4.1.
Proposition 5.1. The map that takes the class of a loop α ∈ Λ M to the class of the cycle (s π•α ,α) ∈ C • (∆ * ⊗ ι) defines a homomorphism ). In this case, after applying the Serre spectral sequence to the fibration (4.1), one sees that . This vanishes when M is not a closed surface by Proposition 3.1. Plugging this into the long exact sequence of the cone C • (∆ * ⊗ ι), we obtain the commutative diagram whose bottom row is exact for all non-closed surfaces. For future reference , we note that the existence of the lift ϕ implies that ψ • β CS = 0.

The groups H
where j : M 2 − ∆ M → M 2 is the inclusion. Denote its cohomology groups by H • ∆ (M 2 , T ). They can also be computed by the complex There is a long exact sequence Proof. The long exact sequence comes from the short exact sequence We are interested in the 3 cases: T is empty; T = ∆ M and h is the inclusion; T = M and h is the composition of the projection π with the diagonal map. When T is empty, the Thom isomorphism gives an isomorphism H j (M ) ∼ = H j+2 ∆ (M 2 ). We'll consider the case T = M in the next section. Here we consider the case T = ∆ M .
We will suppose that ξ is a nowhere vanishing vector field on M . The normal bundle of the diagonal ∆ M in M 2 is isomorphic to the tangent bundle T M of M . The exponential map induces a diffeomorphism of a closed disk bundle in T M with a regular neighbourhood N of ∆ M in M 2 . By rescalling ξ, we may assume that exp ξ is mapped into ∂N . We will henceforth regard ξ as the section exp ξ of ∂N . Denote the closed unit ball in R 2 by B. We can choose a trivialization is null homotopic. This condition determines the homotopy class of the trivialization.
The inclusion (N, The Künneth Theorem implies that q * : Proposition 5.4. There is a short exact sequence Proof. This is part of the long exact sequence of the triple (N, ∆ M ∪ ∂N ). Exactness on the left follows from the Künneth Theorem (or the Thom Isomorphism Theorem); exactness on the right follows as This map depends on the homotopy class of the trivialization ξ. Denote the positive integral generator of H 2 (B, ∂B) by τ B . Define The image of τ ξ in H 2 (N, ∂N ) is the Thom class τ M of the tangent bundle of M .
To better understand τ ξ , suppose that γ : Define the rotation number rot ξ (γ) of γ with respect to ξ to be the rotation number of q • γ about 0 ∈ B. Let Γ γ to be the relative 2-cycle that corresponds to the map Give I × S 1 has the product orientation.

5.3.
The class c ξ . In this section, we show that each non-vanishing vector field ξ determines a class c ξ ∈ H 2 ∆ (M 2 , M ). Pairing with this class corresponds to intersecting with the diagonal and is a key component of the homological description of δ ξ .
Lemma 5.6. Each section ξ of M → M determines a class f ξ ∈ H 1 ( M ; Z) whose pullback ξ * f ξ to M vanishes and whose restriction to each fiber M x is the positive integral generator of H 1 ( M x ; Z). It is characterized by these properties.
Proof. This follows from the Künneth Theorem and the fact the section ξ determines a trivialization r : It is unique up to homotopy. Take f ξ to be the pullback of the positive generator of which depends only on the homotopy class of ξ.
Proof. This is part of the long exact sequence in Lemma 5.3. Exactness of the sequence follows from the Thom isomorphism The triviality of the tangent bundle of M implies that the normal bundle of the diagonal in M 2 is trivial, which gives the exactness on the right.
Since H 2 ∆ (M 2 ) is freely generated by the Thom class τ M of M , to construct the lift, it suffices to lift τ M to H 2 ∆ (M 2 , M ). To do this, note that π : The pairing. Here we define a pairing and compute the pairing of c ξ and s α whose value is close to being the value δ ξ (α) of the cobracket . Proposition 5.9. There is a well-defined pairing It is an open cover of M 2 . We can compute the product using U-small chains C U • and cochains C • U via the pairing . This group is naturally isomorphic to H 0 (M ; L M ⊗ L M ) as the homotopy equivalence r : Recall from the introduction (or the next proof) the notation for ǫ P , α ′ P and α ′′ P .
Lemma 5.10. If α : S 1 → M is an immersed circle with transverse self intersections, then where P ranges over the double points of α.
Proof. We use the notation of Section 5.2. Since the map α 2 : As in the introduction, α ′ P denotes the restriction of α to the positively oriented arc in S 1 from θ to φ, and α ′′ P denotes its restriction to the arc from φ to θ. Denote the initial tangent vectors of α ′ P and α ′′ P by v ′ and v ′′ . The intersection number ǫ P is defined by v ′ ∧ v ′′ ∈ ǫ P × (a positive number) × (the orientation of M at P ).
An elementary computation shows that the intersection number of α 2 : The contribution of the double point P to τ ξ , s α is thus It remains to compute the contribution of the strip Γ to τ ξ , s α . The derivativė α : M → T M of α corresponds to a section of the circle bundle ∂N → ∆ M , unique up to homotopy. By the construction preceding Lemma 5.5, this determines a relative chain Γα in (N, ∆ M ∪ ∂N ).
The inverse image of Γ in [0, 1] × S 1 under the map (4.2) is the disjoint union of two strips, Γ 0 , a regular neighbourhood of 0×S 1 , and Γ 2π , a regular neighbourhood of 2π × S 1 .
Give Γ 0 and Γ 2π the orientation induced from S 1 × S 1 . Then, as classes in As observed in the proof of Proposition 4.1, the restriction of s α to Γ 2π is homotopic to 1 ⊗ α, and to Γ 0 is homotopic to α ⊗ 1.
Lemma 5.5 now implies that the contribution to τ ξ , s α from Γ is The result follows by adding the contribution of the strip (5.4) to the sum of the contributions (5.3) of the double points P .
Remark 5.11. By an elementary case of a theorem of Hirsch [21] (that goes back to Whitney [33]), regular homotopy classes of immersed loops in M correspond to homotopy classes of loops in M . As shown in [25], the expression for τ ξ , s α in Lemma 5.10 is constant on regular homotopy classes of immersed circles in M and thus defines a map It is injective as its composition with (Λπ) * is the identity. The image of a free homotopy class of f : S 1 → M corresponds to the regular homotopy class of an immersed circle α with rot ξ (α) = 0 that is freely homotopic to f . The following factorization of δ ξ follows directly from Lemma 5.10.
Theorem 5.12. If ξ is a section of π : M → M , then the diagram commutes.

De Rham Aspects
In this section, in preparation for applying the machinery of Hodge theory in Section 7, we construct de Rham versions of the continuous duals of the maps used in the homological description of the Turaev cobracket given in Section 5. 6.1. Preliminaries. Suppose that N is a smooth manifold with finite first Betti number and that k is a field of characteristic zero. We are especially interested in the case where N is a rational K(π, 1) space.
Denote the local system over N × N whose fiber over (x 0 , x 1 ) isȞ 0 (P x0,x1 N ; k) byP N and its pullback along the interchange map N 2 → N 2 byP op N . Lemma 6.1. Let p : N × N → N be projection onto the first factor. If N is a rational K(π, 1), then there is a natural isomorphism of locally constant sheaves Proof. This follows directly from [17, Cor. 9.2].
Corollary 6.2. If N is a rational K(π, 1), then there is a natural isomorphism Apply the Leray spectral sequence of the projection p : N × N → N . The previous result and the fact that N is a rational K(π, 1) imply that so that the spectral sequence collapses at E 2 .
6.1.1. Differential forms. Now k will be R or C. We regard a local system on N as a locally constant sheaf. We will denote the complex of differential forms on N with values in a local system V of real (or rational) vector spaces by E • (N ; V ).
In [17], we denoted the flat vector bundle associated to V by V and the sheaf of To connect with [17], we point out that the flat vector bundle associated toĽ N is denoted by L N , and the flat vector bundle associated toP N by P N .
SinceP M and L M are local systems of algebras, ∆ and ι induce a DGA homo-

Denote its cohomology groups by
If M is not a closed surface, then there is an exact sequence whereψ is dual to the connecting homomorphism ψ in Remark 5.2.
Proof. The cohomology long exact sequence of the cone is Since M is not closed, it is homotopy equivalent to a wedge of circles and therefore a rational K(π, 1) of cohomological dimension 1. In particular, H 2 (M 2 ;P M ⊗ P The cohomology of this cone is dual to the homology of the cone defined in Section 5. commute. It is defined over k = Q and is dual to the map ϕ (5.2) in the sense that is zero, whereψ is the connecting homomorphism in the long exact sequence (6.1).
Since the diagram . De Rham's Theorem and the 5-lemma imply that it computes H • ∆ (M 2 , M ; k). Lemma 6.6. There is a well-defined product Proof. This result can be proved using differential forms or singular cochains. We will use differential forms. The proof using singular cochains is similar.
Choose regular neighbourhoods U and V of the diagonal ∆ in M 2 , where V ⊂ U , V is closed and U is open. Since ∆ ֒→ U is a homotopy equivalence, every flat section ofĽ M ⊗Ľ M over the diagonal extends uniquely to a flat section ofP M ⊗P op M over U . It follows that restriction to the diagonal induces a quasi-isomorphism Since the inclusion ∆ → V is a homotopy equivalence, the map The cup product pairing (6.2) is induced by the map of complexes . This is a chain map according to the conventions in Section 2.2.
To prove the remaining assertion, suppose that z is represented by [s, u] in On the other hand, since F is locally constant, commutes. The next result follows directly from Theorem 5.12 and the results in Section 6.2.
Proposition 6.7. The mapδ ξ is the continuous dual of δ ξ in the sense that for all f, g ∈Ȟ 0 (ΛM ) and α ∈ ΛM .

Proof of Theorem 1
In this section, k will be Q, R or C, as appropriate, and X will be a smooth affine curve over C. Equivalently, X is the complement X − D of a finite subset D of a compact Riemann surface X. We will also assume that ξ is an algebraic framing of X; that is, ξ is a meromorphic vector field on X whose restriction to X is a nowhere vanishing holomorphic vector field. We prove the theorem by showing that each group in the factorization of given in Section 6.3 has a mixed Hodge structure (MHS) and that each morphism in the factorization is a morphism of MHS. The twist by Q(−1) occurs in the map ⌣ c ξ . Note that the topological factorization of δ ξ in Section 5 implies that all of the maps in the factorization ofδ ξ in Section 6.3 are also defined over Q.
Since X and X are smooth varieties,Ȟ 0 (Λ X) andȞ 0 (ΛX) have natural MHS by [17,Cor. 10.7]. The naturality of the MHS and that fact that ξ is meromorphic on X imply that (Λξ) * :Ȟ 0 (Λ X) →Ȟ 0 (ΛX) is a morphism of MHS. Since the mapĽ X →Ľ X ⊗Ľ X is a direct limit of morphisms of admissible variations of MHS over X, the Theorem of the Fixed Part (alternatively, by a direct argument that uses the construction of these MHS) implies that mult : H 0 (X,Ľ X ) ⊗2 → H 0 (X,Ľ ⊗2 X ) is a morphism of MHS.
To prove that the remaining groups have natural MHS and that the maps between them are morphisms, we need to recall the following standard fact about cones of mixed Hodge complexes, which is implicit in [7].
is a long exact of MHS.

Proposition 7.2. Each group in the diagram
has a natural MHS, and each map is a morphism of MHS.
Proof. The work of Saito [28] implies that if V is an admissible variation of MHS over the complement of a divisor W with normal crossings in a smooth variety Z, then the complex E • (Z log W ; V ) of smooth forms on Z with values in the canonical extension of V to Z and log poles along W is part of a mixed Hodge complex and is naturally quasi-isomorphic to E • (Z − W ; V ). In particular, it computes H • (Z − W ; V ) ⊗ C, together with its Hodge and weight filtrations. The compactification P = P(T X ⊕ O X ) of the tangent bundle T X of X is a compactification of X whose complement W is a divisor with normal crossings. The cone is quasi-isomorphic to E • (X 2 , X;P X ⊗P op X →Ľ X ). Lemma 7.1 implies that it is the complex part of a mixed Hodge complex and that the bottom row of the diagram is an exact sequence of MHS.
The mapβ CS is morphism of MHS by Lemma [17,Lem. 11.1]. The fact that the category of MHS is abelian implies thatφ is a morphism of MHS. Proposition 7.3. The group H • ∆ (X 2 , X) has a natural mixed Hodge structure and c ξ is a Hodge class of type (1, 1).
Let Z be the normal crossings compactification of X constructed in the proof of Proposition 7.2. The commutative diagram of morphisms of complex algebraic maps of DGAs in which each vertical map is a quasi-morphism. Each DGA in this diagram is the complex part of the natural mixed Hodge complex associated to the corresponding variety. The Five Lemma implies that the complex E • ∆ (X 2 , X) is naturally quasi-isomorphic to Lemma 7.1 implies that it is the complex part of a mixed Hodge complex. It follows that H • ∆ (X, X) has a natural MHS and that the exact sequence of Lemma 5.7 0 → H 1 ( X) → H 2 ∆ (X 2 , X) → H 2 ∆ (X 2 ) → 0 is an exact sequence of mixed Hodge structures.
It remains to show that c ξ is a Hodge class that spans a copy of Q(−1). Recall the notation and the construction of c ξ from Section 5.3. In particular, c ξ = π * τ ξ + f ξ . The topological constructions in that section imply that π * τ ξ and f ξ are both defined over Q. We first show that f ξ is a Hodge class.
The framing ξ induces an algebraic isomorphism Denote the corresponding projection X → C * by r. Then r * dt/t spans a copy of Since c ξ is defined over Q, to prove that it is a Hodge class, it suffices to show that it is a real Hodge class. To do this, we use the fact that the MHS on H • ∆ (X 2 , ∆ X ) depends only on X and the normal bundle of ∆ X in X 2 , which is just the (holomorphic) tangent bundle T X of X. This follows from the construction of a (real) mixed Hodge complex for the punctured neighbourhood of one variety in another that was constructed in [8]. That construction implies that the natural isomorphism that is constructed using topology, is an isomorphism of real MHS. Here H • X (T X, X) is defined to be the homology of the complex where the map is restriction to the zero section. The trivialization (7.2) induces a MHS morphism q * : H 2 (C, C * ) → H 2 X (T X, X). The class c ξ is the image of the positive generator τ B of H 2 (C, C * ) ∼ = Z(−1) under the sequence It follows that c ξ is a real (and therefore rational) Hodge class. The final observation is that π * : H 2 ∆ (X 2 , ∆ X ) → H 2 ∆ (X 2 , X) is a morphism of MHS, from which it follows that π * c ξ is a Hodge class.

Torsors of Splittings of the Goldman-Turaev Lie Bialgebra
Suppose that 2g + n > 1, where g and n are non-negative integers. Suppose that S is an (n + 1)-punctured surface of genus g. Write S = S − P , where P = {x 0 , . . . , x n } is a subset of S. Fix a vector d = (d 0 , . . . , d n ) ∈ Z n+1 . Suppose that v o is a non-zero tangent vector of S anchored at the point x 0 and that ξ o is a nowhere vanishing vector field on S with local degree d j at x j . where X is a compact Riemann surface and ξ is a meromorphic vector field.
Complex structures on (S, P, ξ o ) and (S, P ) are defined similarly.
Proposition 8.2. If g = 1, or if g = 1, n > 0, and at least one local degree d j is non-zero, then every framing (S, P, ξ o ) of S has a complex structure.
Note that a once punctured genus g = 1 has only one algebraic framing ξ. This is the restriction of the unique translation invariant framing. It is characterized by the property that rot ξ (γ) = 0 for every simple closed curve γ in the surface.
Proof. This is elementary when g = 0. Suppose now that g ≥ 1. Note that d = 0 except possibly when g = 1 as j d j = 2 − 2g. Regard the tangent bundle T of S as a C ∞ complex line bundle. The section ξ o trivializes it over S. We can extend it to the topological analogue of a meromorphic section of T . This section will be homotopic to the section z → z dj in a punctured neighbourhood of x j . For each complex structure (X, D) on (S, P ), we have the holomorphic line bundle L over X whose sheaf of sections is O X ( d j x j ). It has a meromorphic section s with divisor d j x j . When d = 0, this section trivializes L over X = X − D. 6 There is thus a homotopy class of isomorphisms under which the restriction of s to X is homotopic to ξ o . The framing ξ o of S is algebraic on X if and only if L is isomorphic to T X, the holomorphic tangent bundle of X. Equivalently, it is algebraic if and only if the divisor d j x j is an anti-canonical divisor on X.
To prove the result, we have to prove that there is a point (X, D) in the moduli space M g,n+1 where this is the case. To this end, consider the section F d : (X, D) → K X + n j=0 d j x j ∈ Jac X of the universal jacobian J over M g,n . We have to show that it vanishes at some point of M g,n+1 .
When g ≥ 2, the most direct way to do this is to appeal to [26] when all d j ≥ 0, and [4] in the general case, which implies that there are curves (X, D) for which d j x j is anti-canonical.
To prove the genus 1 case we need to assume that d = 0. In this case, we will show that for every X, there are distinct points {x 0 , . . . , x n } in X such that d j x j is linearly equivalent to 0. Write X = C/Λ, where Λ is a lattice in C. We may assume that each d j = 0. We may also take x 0 to be the identity. Consider the map F d : C n → C defined by Since all components of d are non-zero, ker F d is not contained in any diagonal x j = x k (j = k) or any coordinate hyperplane x j = 0. This implies that there are solutions x = (x 1 , . . . , x n ) of F d (x) = 0 with the x j distinct and non-zero in every neighbourhood of 0. The result follows as the quotient map C n → E n is a biholomorphism in a neighbourhood of 0. Remark 8.3. This result implies that the framings that occur in [3, Thm. 6.1] are precisely those that admit a complex structure. To compare the two statements, one should note that if γ j is the boundary of sufficiently small disk in S, centered at x j and, then d j + rot ξ γ j = 1.
Note that the boundary orientation conventions used in [1,2,3] differ from those used in this paper. Their adapted framing has the property that d 0 = 2 − 2g and d j = 0 for all j ≥ 1. Proof. By [17,Thm. 6], the MHS on Qπ 1 (X, v) ∧ determines a torsor of isomorphisms

Solutions of KV
(g,n+1) d that arise from Hodge theory will be called motivic solutions as they arise from an algebraic structure on (S, P, v o , ξ o ). For a given complex structure φ, the motivic solutions will be a torsor under U MT X, v . All solutions of KV Remark 8.6. The classification [26, Thm. 1, §2.3] of strata of abelian differentials of genus g curves given by Kontsevich and Zorich implies that, when g ≥ 3, there can be more than one Γ g,n+ 1 -orbit of triples (S, P, ξ) with ξ having local degree vector d. A topological classification of orbits of framings is given in [23].
Set H k = H 1 (S; k), where k is a commutative ring. The intersection pairing is a unimodular symplectic form on H Z and thus gives an isomorphism (Poincaré duality) of H Z with its dual Hom Z (H Z , Z). There is affine group Sp(H) whose group of k-rational points is Sp(H k ). The action of Γ g,n+ 1 on S induces a surjective homomorphism ρ : Γ g,n+ 1 → Sp(H Z ).
The Torelli group T g,n+ 1 is, by definition, the kernel of ρ.
Proof. The group Diff + (S, P, v o ) acts on S, the space of non-zero tangent vectors of S. This implies that is an extension of Γ g,n+ 1 -modules. The vector field ξ o induces a splitting s o of this sequence, and thus an isomorphism H 1 ( S; Z) ∼ = H 1 (S; Z) ⊕ Z. There is therefore a homomorphism Γ g,n+ 1 → Aut H 1 (S; Z) ⋉ H 1 (S; Z) whose kernel is the stabilizer of ξ o .
For each x ∈ P , let γ x be an oriented loop in S that bounds a small disk in S centered at x. Observe that H 1 (S) = H 1 (S)/ γ x : x ∈ P . Since Γ g,n+ 1 fixes x ∈ P , it acts trivially on s o (γ x ). This implies that the image of the action lies in the subgroup Aut H 1 (S; Z) ⋉ H Z . Since the mapping class group respects the intersection pairing on H Z , the image of Γ g,n+ 1 lies in the subgroup Sp(H Z ) ⋉ H Z . 7 More generally, Γ g,n+ r is the mapping class group π 0 Diff + (S, P, V ), where P is a finite subset of S with #P = n, and V is a set of r non-zero tangent vectors (or r boundary components) that are anchored at r distinct points, none of which are in P .
One might hope that KRV d g,n+ 1 is the subgroup of Aut Qπ 1 (X, v o ) ∧ generated by U MT X,ξ and the Zariski closure of the image of T d g,n+ 1 . While this might be true when g = 1, it cannot be true when g = 1. (See Remark 8.8.) To circumvent this issue, and to allow the use of Hodge theory, it is better to replace mapping class groups by their relative unipotent completions.
Denote the completion of Γ g,m+ r relative to ρ : Γ g,m+ r → Sp(H Q ) by G g,m+ r . This is an affine Q-group that is an extension where U g,m+ r is prounipotent. There is a Zariski dense homomorphismρ : Γ g,m+ r → G g,m+ r (Q) whose composition with the homomorphism G g,m+ r (Q) → Sp(H Q ) is ρ.
The cohomology groups of U g,m+ r are ind-objects of the category of Sp(H)modules. The homology groups of U g,m+ r are their hom duals and are pro-objects of the category of Sp(H)-modules.
Remark 8.8. The homomorphism T g,m+ r → U g,m+ r (Q) induced byρ has Zariski dense image when g > 1. This follows from the right exactness of relative completion [13,Thm. 3.11] and the vanishing of H 1 (Sp g (Z), V ) for all V when g = 1.
For all m and r, the group U g,m+ r has a natural weight filtration [12] U g,m+ r = W −1 U g,m+ r ⊇ W −2 U g,m+ r ⊇ · · · This induces a weight filtration on its abelianization H 1 (U g,m+ r ). When m = n + 1 and r = 0, it has the property that there are Σ n+1 ×Sp(H) equivariant isomorphisms where V denotes the 3rd fundamental representation of Sp(H) and where Σ n+1 acts on the mapping class group by permuting the points and on H n+1 by permuting the factors. (When g = 1, H n is identified with (u 0 , . . . , u n ) ∈ H n+1 with u 0 +· · ·+u n = 0.) Proposition 8.9. Suppose that 2g + n > 1 and that d ∈ Z n+1 satisfies j d j = 2 − 2g. If g = 0, then U d 0,n+ 1 = U 0,n+ 1 . If g = 1, then U 1,n+ 1 is the kernel of . . . , u n ) = d 1 u 1 + · · · + d n u n . If g > 1, then U d g,n+ 1 is the kernel of Proof. The map Γ g,n+ 1 → Γ g,n+1 is a central extension with kernel Z. The corresponding map G g,n+ 1 → G g,n+1 is a central extension with kernel Q(1). It therefore induces an isomorphism H 1 (U g, Since H 1 (P 1 ) = 0, the homomorphism τ d is trivial when g = 0. This implies that U d 0,n+ 1 = U 0,n+ 1 for all n > 1. Now suppose that g > 0. By a complex structure on the diagram we mean a homotopy class of orientation preserving isomorphisms of the previous diagram with . , x n }, L → X is a holomorphic line bundle of degree 2 − 2g, L ′ is the associated C * bundle, and ξ is a meromorphic section of L over X whose divisor is d j=0 d j x j . 8 Such a complex structure gives a point of the relative Picard scheme Pic C/Mg,n+1 over the universal curve C over M g,n+1 . That is, the choice of a complex structure on the diagram above, determines a section F d of the universal Picard scheme. Denote the divisor classes of degree d by Pic d . Subtracting the section that takes a pointed curve (X, D) to its canonical divisor class K X in Pic 2g−2 X gives a canonical isomorphism of Pic 2−2g C/Mg,n+1 with the universal jacobian J over M g,n+1 . We can, and will, regard F d as a section of the universal jacobian via this isomorphism. is an extension of Γ g,n+1 by H Z . The identity section induces a splitting of this extension and thus a canonical isomorphism The canonical representation Γ g,n+ 1 → Sp(H Z ) induces a homomorphism Every section µ of J over M g,n+ 1 that vanishes at o induces a homomorphism where the right-hand homomorphism is induced by the canonical homomorphism Γ g,n+ 1 → Sp(H Z ). The homologically trivial algebraic cycle where the second homomorphism is multiplication by 2 − 2g on H Z when g > 1 [16, Prop. 11.2] and the identity when g = 1 [16, §12]. When g = 1, the universal jacobian is just the universal elliptic curve. We can take x 0 to be the identity section. Then the normal function of x j − x 0 is the tautological section of the universal elliptic curve corresponding to x j . This section induces projection on to the jth factor, 1 ≤ j ≤ n in (8.3). When g > 1, the section κ j : (X, D) → K X − (2g − 2)x j induces the projection onto the jth factor in (8.3). The result now follows as when g = 1 as K X vanishes, and There is a natural homomorphism Denote the image of U g,n+ 1 under this homomorphism by U g,n+ 1 and the image of Recall that a MHS on an affine Q-group G is, by definition, a MHS on its coordinate ring O(G). Equivalently, a MHS on G is an algebraic action of π 1 (MHS) on G. A homomorphism G 1 → G 2 of affine Q-groups with MHS is a morphism of MHS if it is π 1 (MHS) equivariant.
Proof. The complex structure φ determines a mixed Hodge structure on U g,n+ 1 . This gives an action of π 1 (MHS) on it, so that one has the group π 1 (MHS) ⋉ U g,n+ 1 . The pro-MHS on Qπ 1 (X, v) ∧ corresponds to a homomorphism π 1 (MHS) → Aut Qπ 1 (X, v) ∧ . Since the homomorphism U g,n+ 1 → Aut Qπ 1 (X, v) ∧ is a morphism of MHS, it extends to a homomorphism Its image is U g,n+ 1 . The inner action of π 1 (MHS) on U g,n+ 1 gives it a MHS. The inclusion U g,n+ 1 ֒→ Aut Qπ 1 (X, v) ∧ is π 1 (MHS) invariant, which implies that it is a morphism of MHS.
Remark 8.11. Lest there be any confusion about the Mumford-Tate groups, we point out that the Mumford-Tate group MT X, v of Qπ 1 (X, v) ∧ is canonically isomorphic to the Mumford-Tate group MT λ(X) of Qλ(X) ∧ . This follows from the fact that the Mumford-Tate groups of Qπ 1 (X, v) ∧ and Der Qπ 1 (X, v) ∧ are isomorphic, and because the Lie algebra homomorphism The following theorem is proved in Section 9. It and the previous lemma imply Theorem 4. Theorem 8.12. There is an injective homomorphism U d g,n+ 1 ֒→ KRV d g,n+ 1 of prounipotent Q-groups whose conjugacy class does not depend on the complex structure φ. The group U d g,n+ 1 is a normal subgroup of U d g,n+ 1 . There is a canonical surjection K → U d g,n+ 1 /U d g,n+ 1 , where K is the prounipotent radical of π 1 (MTM).
Remark 8.13. The complex structure on (S, P, v o , ξ o ) defines a C-point, and thus a geometric point, p of the moduli stack M g,n+ 1/Q . Theétale fundamental group π 1 (M g,n+ 1 , p) is an extension where Γ ∧ g,n+ 1 denotes the profinite completion of the mapping class group. For each prime number ℓ, there is an homomorphism πé t 1 (M g,n+ 1 , p) → Sp(H Z ℓ ) ⋉ H Z ℓ . Denote its kernel by πé t 1 (M g,n+ 1 , p) d . There is a homomorphism Using weighted completion [14, §8], one can show that the Zariski closure of the image of φ ℓ is U d g,n+ 1 (Q ℓ ). gives an isomorphism of the Goldman-Turaev Lie bialgebra with its associated weight graded Lie bialgebra, and thus a solution of the KV problem. The group U d g,n+ 1 acts simply transitively on the set of such splittings. Next we show that this action is graded. Denote the Lie algebra of U d g,n+ 1 by u d g,n+ 1 . such that the diagram commutes.
Proof. The complex structure determines an action of π 1 (MHS) on G g,n+ 1 and thus a semi-direct product π 1 (MHS)⋉G g,n+ 1 . The inner action of π 1 (MHS) on this group determines a MHS on its coordinate ring. As pointed out above, the homomorphism π 1 (MHS) ⋉ G g,n+ 1 → Aut Qπ 1 (S, v) ∧ is a morphism of MHS. This implies that its image G g,n+ 1 has a MHS and that the inclusion G g,n+ 1 → Aut Qπ 1 (S, v) ∧ is a morphism of MHS. The complex structure φ determines a MHS on H = H 1 (S), and thus a MHS on Sp(H) ⋉ H. The homomorphism G g,n+ 1 → Sp(H) ⋉ H is a morphism of MHS as it is the monodromy representation associated to the normal function κ d defined in the proof of Proposition 8.9. So its kernel U d g,n+ 1 has a natural MHS and the homomorphism U ∼ = π 1 (M g,n+ 1 , φ o ) and a mixed Hodge structure on the relative completion G g,n+ 1 . This MHS corresponds to an action of π 1 (MHS) on G g,n+ 1 . One can therefore form the semi-direct product π 1 (MHS) ⋉ G g,n+ 1 .
Remark 9.1. Since π 1 (MHS) normalizes G g,n+ 1 and U g,n+ 1 , and since the Mumford-Tate group MT X, v is the image of π 1 (MHS), the Mumford-Tate group normalizes U g,n+ 1 . This implies that U g,n+ 1 is a quotient of U MT X, v ⋉ U g,n+ 1 . In particular, U g,n+ 1 is generated by U MT X, v and U g,n+ 1 .
Remark 9.2. One can argue as in [19] that, if g ≥ 3, then then U MT X, v → U g,n+ 1 is an isomorphism (equivalently, π 1 (MHS) → G g,n+ 1 is surjective) if and only if the Griffiths invariant ν(X) ∈ Ext 1 MHS (Q, P H 3 (Jac X(2))) of the Ceresa cycle in Jac X is non-zero, and if the points κ j := (2g − 2)x j − K X ∈ (Jac X) ⊗ Q, 0 ≤ j ≤ n, are linearly independent over Q. Similarly, one can show that U MT X, v → U d g,n+ 1 is surjective if and only if ν(X) = 0 and the only relation between the κ j in (Jac X)⊗Q is d j κ j = 0. Proof. The first task is to show that the G g,n+ 1 form a local system over M g,n+ 1 . This is not immediately clear, as the size of the Mumford-Tate group depends non-trivially on complex structure on (S, P, v). To this end, let x = (X, D, v) and denote the relative completion of π 1 (M g,n+ 1 , x) by G x . Let y = (Y , E, v ′ ) be another point of M g,n+ 1 and let G y be the relative completion of π 1 (M g,n+ 1 , y). Denote the relative completion of the torsor of paths in M g,n+ 1 from x to y by G x,y . Its coordinate ring has a natural MHS and the multiplication map G x × G x,y → G y is a morphism of MHS [11]. This is equivalent to the statement that the map (π 1 (MHS) ⋉ G x ) × G x,y → π 1 (MHS) ⋉ G y defined by (σ, λ, γ) → (σ, γ −1 λγ) is a π 1 (MHS)-equivariant surjection, where α ∈ π 1 (MHS) acts by α : (σ, λ, γ) → (ασα −1 , α · λ, α · γ) and α : (σ, µ) → (ασα −1 , α · µ).
The diagram (π 1 (MHS) ⋉ G x ) × G x,y / / π 1 (MHS) ⋉ G y Aut Qπ 1 (X, v) ∧ × G x,y / / Aut Qπ 1 (Y, ξ ′ ) ∧ commutes, where Y = Y − E and where the bottom arrow is induced by parallel transport in the local system whose fiber over x is Aut Qπ 1 (X, v) ∧ . This implies that there is a morphism G x ×G x,y → G y that is compatible with path multiplication. It follows that the G x form a local system over M g,n+ 1 . We now prove the remaining assertions. The monodromy action of Γ g,n+ 1 on G g,n+ 1 /G g,n+ 1 is the composite Γ g,n+ 1 → G g,n+ 1 (Q) → Aut G g,n+ 1 /G g,n+ 1 (Q), where the first homomorphism is the canonical map, and the second is induced by conjugation. It is easily seen to be trivial as G g,n+ 1 is normal in G g,n+ 1 .
The coordinate ring of G g,n+ 1 /G g,n+ 1 has a MHS as the inclusion G g,n+ 1 → G g,n+ 1 is π 1 (MHS)-equivariant. This variation has no monodromy, and so is constant by the theorem of the fixed part. Since U g,n+ 1 = U g,n+ 1 ∩ G g,n+ 1 , the map U g,n+ 1 /U g,n+ 1 → G g,n+ 1 /G g,n+ 1 is a π 1 (MHS)-equivariant inclusion. It follows that U g,n+ 1 /U g,n+ 1 is also a constant variation of MHS over M g,n+ 1 .
This map should be an isomorphism and should follow from the proalgebraic analogue of the proof of Oda's Conjecture [29].
Sketch of Proof. Since the variation O( U g,n+ 1 /U g,n+ 1 ) is constant, it extends over the boundary of M g,n+1 . Since the variation of MHS over M g,n+ 1 with fiber u g,n+ 1 is admissible, it has a limit MHS at each tangent vector of the boundary divisor ∆ of M g,n+ 1 . These tangent vectors correspond to first order smoothings of an (n + 1)-pointed stable nodal curve of genus g together with a tangent vector at the initial point. For each such maximally degenerate stable curve 9 (X 0 , P 0 , v 0 ) of type (g, n + 1), Ihara and Nakamura [22] construct a proper flat curve X → Spec Z[[q 1 , . . . , q N ]], N = dim M g,n+1 = 3g + n − 2 with sections x j , 0 ≤ j ≤ n and a non-zero tangent vector field v along x 0 that specialize to the points of P 0 and the tangent vector ξ 0 at q = 0. The projection is smooth away from the divisor q 1 q 2 . . . q N = 0. These are higher genus generalizations of the Tate curve in genus 1.
There is a limit MHS each of corresponding to the tangent vector q := N j=1 ∂/∂q j of M g,n+ 1 at the point corresponding to (X 0 , P 0 , v 0 ). These can be thought of as MHSs on the invariants of (X q , v), where X q denotes the fiber of X over q and X q the corresponding affine curve.
The main result of [18] is that these MHS are Hodge realizations of objects of MTM. This implies that each has an action of π 1 (MTM) and that the action of 9 These correspond to pants decompositions of (S, P, v) and also to the 0-dimensional strata of M g,n+ 1 . π 1 (MHS) on each factors through the canonical surjection π 1 (MHS) → π 1 (MTM). Brown's result [5] asserts that π 1 (MTM) acts faithfully on This implies that it also acts faithfully on Qπ 1 (X q , v) ∧ as (by the construction in [18]), the unipotent path torsor of X q is built up from the path torsors of copies of P 1 − {0, 1, ∞} (and is 6 canonical tangent vectors) in X q . In other words, MT X q , v is naturally isomorphic to π 1 (MTM). This implies that there is a surjective homomorphism h : π 1 (MTM) → U g,n+ 1 /U g,n+ 1 .