Anderson Vera

ALTERNATIVE VERSIONS OF THE JOHNSON HOMOMORPHISMS AND THE LMO FUNCTOR

Keywords: 2010 Mathematics Subject Classification. 57M27, 57M05, 57S05 3-manifolds, mapping class group, Torelli group, Johnson homomorphisms, Lagrangian mapping class group, LMO invariant, LMO functor, Kontsevich integral

Let Σ be a compact connected oriented surface with one boundary component and let M denote the mapping class group of Σ. By considering the action of M on the fundamental group of Σ it is possible to define different filtrations of M together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of M introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by Σ. We shall call it the "alternative Johnson filtration", and the corresponding homomorphisms are referred to as "alternative Johnson homomorphisms". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of 3-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor.

Let Σ be a compact connected oriented surface with one boundary component and let M denote the mapping class group of Σ, that is, the group of isotopy classes of orientation-preserving self-homeomorphisms of Σ fixing the boundary pointwise. The group M is not only an important object in the study of the topology of surfaces but also plays an important role in the study of 3-manifolds, Teichmüller spaces, topological quantum field theories, among other branches of mathematics.

A natural way to study M is to analyse the way it acts on other objects. For instance, we can consider the action on the first homology group H := H 1 (Σ; Z) of Σ. This action gives rise to the so-called symplectic representation

σ : M -→ Sp(H, ω),
where ω : H ⊗ H → Z is the intersection form of Σ. The homomorphism σ is surjective but it is far from being injective. Its kernel is known as the Torelli group of Σ, denoted by I. Hence we have the short exact sequence

(1.1) 1 -→ I ⊂ --→ M σ --→ Sp(H, ω) -→ 1.
We can see that, in order to understand the algebraic structure of M, the Torelli group I deserves significant attention because, in a certain way, it is the part of M that is beyond linear algebra (at least with respect to the symplectic representation).

More interestingly, we can consider the action of M on the fundamental group π := π 1 (Σ, * ) for a fixed point * ∈ ∂Σ. This way we obtain an injective homomorphism ρ : M -→ Aut(π), which is known as the Dehn-Nielsen-Baer representation and whose image is the subgroup of automorphisms of π that fix the homotopy class of the boundary of Σ.

Johnson-type filtrations. As stepwise approximations of ρ, we can consider the action of M on the nilpotent quotients of π ρ m : M -→ Aut(π/Γ m+1 π), where Γ 1 π := π and Γ m+1 π := [π, Γ m π] for m ≥ 1, define the lower central series of π. This is the approach pursued by D. Johnson [START_REF] Johnson | A survey of the Torelli group[END_REF] and S. Morita [START_REF] Morita | Abelian quotients of subgroups of the mapping class group of surfaces[END_REF]. This approach allows to define the Johnson filtration

(1.2) M ⊇ I = J 1 M ⊇ J 2 M ⊇ J 3 M ⊇ • • •
where J m M := ker(ρ m ). Now, there is a deep interaction between the study of 3-manifolds and that of the mapping class group. For instance through Heegaard splittings, that is, by gluing two handlebodies via an element of the mapping class group of their common boundary. Thus, if we are interested in this interaction, it is natural to consider the surface Σ as the boundary of a handlebody V . Let ι : Σ → V denote the induced inclusion and let B := H 1 (V ; Z) and π := π 1 (V, ι( * )). Let A and A be the subgroups ker(H ι * -→ B) and ker(π ι # -→ π ), where ι * and ι # are the induced maps by ι in homology and homotopy, respectively. The Lagrangian mapping class group of Σ is the group

L = {f ∈ M | f * (A) ⊆ A}.
By considering a descending series (K m ) m≥1 of normal subgroups of π (different from the lower central series) K. Habiro and G. Massuyeau introduced in [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF] a filtration of the Lagrangian mapping class group L:

(1.3)
L ⊇ I a = J a 1 M ⊇ J a 2 M ⊇ J a 3 M ⊇ • • • that we call the alternative Johnson filtration. We call the first term I a := J a 1 M of this filtration the alternative Torelli group. Notice that I a is a normal subgroup of L but it is not normal in M. Roughly speaking, the group K m consists of commutators of π of weight m, where the elements of A are considered to have weight 2, for instance

K 1 = π, K 2 = A • Γ 2 π, K 3 = [A, π] • Γ 3 π
and so on. The alternative Johnson filtration will be our main object of study in Section 4. Besides, in [START_REF] Levine | Homology cylinders: an enlargement of the mapping class group[END_REF][START_REF] Levine | The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres[END_REF] J. Levine defined a different filtration of L by considering the lower central series of π , and whose first term is the Lagrangian Torelli group

I L = {f ∈ L | f * | A = Id A }: (1.4) L ⊇ I L = J L 1 M ⊇ J L 2 M ⊇ J L 3 M ⊇ • •
• we call this filtration the Johnson-Levine filtration. The group I L is normal in L but not in M.

We refer to the Johnson filtration, the alternative Johnson filtration and the Johnson-Levine filtration as Johnson-type filtrations. Notice that unlike the Johnson filtration the alternative Johnson filtration takes into account a handlebody. Besides, the intersection of all terms in the alternative Johnson filtration is the identity of M as in the case of the Johnson filtration. But this is not the case for the Johnson-Levine filtration. One of the main purposes of this paper is the study of the alternative Johnson filtration and its relation with the other two filtrations. Proposition 4.9 and Proposition 4. [START_REF] Habiro | Claspers and finite type invariants of links[END_REF] give the following result.

Theorem A. The alternative Johnson filtration satisfies the following properties.

(i) m≥1 J a m M = {Id Σ }. (ii) For all k ≥ 1 the group J a k M is residually nilpotent, that is, m Γ m J a k M = {Id Σ }. Besides, for every m ≥ 1, we have (iii) J a 2m M ⊆ J m M.

(iv) J m M ⊆ J a m-1 M.

(v) J a m M ⊆ J L m+1 M. In particular, the Johnson filtration and the alternative Johnson filtration are cofinal.

Johnson-type homomorphisms. Each term of the Johnson-type filtrations comes with a homomorphism whose kernel is the next subgroup in the filtration. We refer to these homomorphisms as Johnson-type homomorphisms. The Johnson homomorphisms are important tools to understand the structure of the Torelli group and the topology of homology 3-spheres [START_REF] Johnson | The structure of the Torelli group. III. The abelianization of T . Topology[END_REF][START_REF] Morita | Casson's invariant for homology 3-spheres and characteristic classes of surface bundles[END_REF][START_REF] Morita | On the structure of the Torelli group and the Casson invariant[END_REF][START_REF] Morita | Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles[END_REF]. Let us review the target spaces of these homomorphisms. For an abelian group G, we denote by Lie(G) = m≥1 Lie m (G) the graded Lie algebra freely generated by G in degree 1.

The m-th Johnson homomorphism τ m is defined on J m M and it takes values in the group Der m (Lie(H)) of degree m derivations of Lie(H). Consider the element Ω ∈ Lie 2 (H) determined by the intersection form ω : H ⊗ H → Z. A symplectic derivation d of Lie(H) is a derivation satisfying d(Ω) = 0. S. Morita shows in [START_REF] Morita | Abelian quotients of subgroups of the mapping class group of surfaces[END_REF] that for h ∈ J m M, the morphism τ m (h) defines a symplectic derivation of Lie(H). The group of symplectic degree m derivations of Lie(H) can be canonically identified with the kernel D m (H) of the Lie bracket [ , ] : H ⊗ Lie m+1 (H) → Lie m+2 (H). This way, for m ≥ 1 we have homomorphisms For the alternative Johnson homomorphisms [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF], consider the graded Lie algebra Lie(B; A) freely generated by B in degree 1 and A in degree 2. The m-th alternative Johnson homomorphism τ a m : J a m M → Der m (Lie(B; A)) is defined on J a m M and it takes values in the group Der m (Lie(B; A)) of degree m derivations of Lie(B; A). Similarly to the case of Lie(H), we define a notion of symplectic derivation of Lie(B; A) by considering the element Ω ∈ Lie 3 (B; A) defined by the intersection form of the handlebody V . Theorem 5.9 and Proposition 5.11 give the following result.

τ m : J m M -→ D m (H).
Theorem B. Let m ≥ 1 and h ∈ J a m M. Then (i) The morphism τ a m (h) defines a degree m symplectic derivation of Lie(B; A). (ii) The morphism τ L m+1 (h) is determined by the morphism τ a m (h). Property (ii) in Theorem B can be expressed more precisely by the commutativity of the diagram

J a m M ⊂ / / τ a m J L m+1 M τ L m+1 D m (B; A) ι * / / D m+1 (B),
for m ≥ 1, where the inclusion J a m M ⊆ J L m+1 M is assured by Theorem A (v). The homomorphism ι * : D m (B; A) → D m+1 (B) is induced by the map ι * : H → B. Property (i) in Theorem B allows to define a diagrammatic version of the alternative Johnson homomorphisms so that we are able to study their relation to the LMO functor. This is the second main purpose of this paper. Before we proceed with a description of our results in this setting, let us state another result in the context of the alternative Johnson homomorphisms. In [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF], K. Habiro and G. Massuyeau consider a group homomorphism τ a 0 : L → Aut(Lie(B; A)), which we call the 0-th alternative Johnson homomorphism, and whose kernel is the alternative Torelli group I a . In subsection 5.3 we prove the following.

Theorem C. The homomorphism τ a 0 : L → Aut(Lie(B; A)) can be equivalently described as a group homomorphism τ a 0 : L -→ Aut(B) Hom(A, Λ 2 B) for a certain action of Aut(B) on Hom(A, Λ 2 B). The kernel of τ a 0 is the second term J L 2 M of the Johnson-Levine filtration. In particular we have I a = J a 1 M = J L 2 M. Moreover, we explicitly describe the image G := τ a 0 (L) and then we obtain the short exact sequence

(1.5) 1 -→ I a ⊂ --→ L τ a 0 --→ G -→ 1.
This short exact sequence has a similar role, in the context of the alternative Johnson homomorphisms, to that of the short exact sequence (1.1) in the context of the Johnson homomorphisms. This is because in [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF] the authors prove that the alternative Johnson homomorphisms satisfy an equivariant property with respect to the homomorphism τ a 0 , which is the analogue of the Sp-equivariant property of the Johnson homomorphisms. Hence the short exact sequence (1.5) can be important for a further development of the study of the alternative Johnson filtration.

Relation with the LMO functor. After the discovery of the Jones polynomial and the advent of many new invariants, the so-called quantum invariants, of links and 3-manifolds, it became necessary to "organize" these invariants. The theory of finitetype (Vassiliev-Goussarov) invariants in the case of links and the theory of finite-type (Goussarov-Habiro) invariants in the case of 3-manifolds, provide an efficient way to do this task. An important success was achieved with the introduction of the Kontsevich integral for links [START_REF] Kontsevich | Vassiliev's knot invariants[END_REF][START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF] and the Le-Murakami-Othsuki invariant for 3-manifolds [START_REF] Thang | On a universal perturbative invariant of 3-manifolds[END_REF], because they are universal among rational finite-type invariants. Roughly speaking, this property says that every Q-valued finite-type invariant is determined by the Kontsevich integral in the case of links or by the LMO invariant in the case of homology 3-spheres.

The LMO invariant was extended to a TQFT (Topological quantum field theory) in [START_REF] Murakami | Topological quantum field theory for the universal quantum invariant[END_REF][START_REF] Cheptea | A TQFT associated to the LMO invariant of three-dimensional manifolds[END_REF][START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF]. We follow the work of D. Cheptea, K. Habiro and G. Massuyeau in [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF], where they extend the LMO invariant to a functor Z : LCob q → ts A, called the LMO functor, from the category of Lagrangian cobordisms (cobordisms satisfying a homological condition) between bordered surfaces to a category of Jacobi diagrams (uni-trivalent graphs up to some relations). See Figure 1.1 for some examples of Jacobi diagrams. There is still a lack of understanding of the topological information encoded by the LMO functor. One reason for this is that the construction of the LMO functor takes several steps and also uses several combinatorial operations on the space of Jacobi diagrams. This motivates the search of topological interpretations of some reductions of the LMO functor through known invariants, some results in this direction were obtained in [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF][START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF][START_REF] Vera | Johnson-Levine homomorphisms and the tree reduction of the LMO functor[END_REF]. The second main purpose of this paper is to give a topological interpretation of the tree reduction of the LMO functor through the alternative Johnson homomorphisms.

A homology cobordism of Σ is a homeomorphism class of pairs (M, m) where M is a compact oriented 3-manifold and m : Denote by T Y,a m (B ⊕ A) the space generated by tree-like Jacobi diagrams colored by B⊕A with at least one trivalent vertex and with a-deg = m. For a Lagrangian cobordism M let Z t (M ) denote the reduction of Z(M ) modulo looped diagrams, that is, diagrams with a non-contractible connected component. See Figure 1.1 (c) for an example of a looped diagram. This way, Z t (M ) consists only of tree-like Jacobi diagrams. The first step to relate the alternative Johnson homomorphisms with the LMO functor is given in Theorem 6.5 where we prove the following.

∂(Σ × [-1, 1]) → ∂M is
a-deg = m in Z t (M ) for M ∈ F a m C. Then Z Y,a
m is a monoid homomorphism. In Theorem 6.14 and Theorem 6.16 we prove the following.

Theorem E. Let m ≥ 1 and f ∈ J a m M.
Then the m-th alternative Johnson homomorphism can be read in the tree-reduction of the LMO functor.

More precisely, we prove that for h ∈ J a m M with m ≥ 2, the value Z Y,a m (c(h)) coincides (up to a sign) with the diagrammatic version of τ a m (h). For h ∈ J a 1 M, we show that τ a 1 (h) is given by Z Y,a 1 (c(h)) together with the diagrams without trivalent vertices in Z(c(h)) of a-deg= 1. The techniques for the proof of Theorem E in the case m = 1 (Theorem 6.14) and m ≥ 2 (Theorem 6.16) are different. For m = 1 we need to do some explicit computations of the LMO functor and a comparison between the first alternative Johnson homomorphism and the first Johnson homomorphism. For m ≥ 2, the key point is the fact that the LMO functor defines an alternative symplectic expansion of π. To show this, we use a result of Massuyeau [START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF] where he proves that the LMO functor defines a symplectic expansion of π.

Theorem D and Theorem E provide a new reading grid of the tree reduction of the LMO functor by the alternative degree. Theorem E follows the same spirit of a result of D. Cheptea, K. Habiro and G. Massuyeau in [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF] and of the author in [START_REF] Vera | Johnson-Levine homomorphisms and the tree reduction of the LMO functor[END_REF] where they prove that the Johnson homomorphisms and the Johnson-Levine homormophisms, respectively, can be read in the tree-reduction of the LMO functor.

Notice that Theorem D holds in the context of homology cobordisms, as do the results that we use to prove Theorem E. This suggests that the alernative Johnson homomorphisms and Theorem E could be generalized to the setting of homology cobordisms, but we have not explored this issue so far.

The organization of the paper is as follows. In Section 2 we review the definition of several spaces of Jacobi diagrams and some operations on them as well as some explicit computations. Section 3 deals with the Kontsevich integral and the LMO functor, in particular we do some explicit computations that are needed in the following sections. Section 4 and Section 5 provide a detailed exposition of the alternative Johnson filtration and the alternative Johnson homomorphisms, in particular we prove Theorem A, B and C. Finally, Section 6 is devoted to the topological interpretation of the LMO functor through the alternative Johnson homomorphisms, in particular we prove Theorem D and E.

Reading guide. The reader more interested in the mapping class group could skip Section 2 and Section 3 and go directly to Section 4 and Section 5 (skipping subsection 5.4) referring to the previous sections only when needed. The reader familiar with the LMO functor and more interested in the topological interpretation of its tree reduction through the alternative Johnson homomorphisms can go directly to Section 3. Then go to subsection 4.3 and subsection 5.2 to the necessary definitions to read Section 6.

Notations and conventions. All subscripts appearing in this work are non-negative integers. When we write m ≥ 0 or m ≥ 1 we always mean that m is an integer. We use the blackboard framing convention on all drawings of knotted objects. We usually abbreviate simple closed curve as scc. By a Dehn twist we mean a left-handed Dehn twist.

Acknowledgements. I am deeply grateful to my advisor Gwénaël Massuyeau for his encouragement, helpful advice and careful reading. I thank sincerely Takuya Sakasai for helpful and stimulating discussions, in particular for explaining to me Remark 4.15.

Spaces of Jacobi diagrams and their operations

In this section we review several spaces of diagrams which are the target spaces of the Kontsevich integral, LMO functor and Jonhson-type homomorphisms. We refer to [START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF][START_REF] Ohtsuki | Quantum invariants[END_REF] for a detailed discussion on the subject. Throughout this section let X denote a compact oriented 1-manifold (possibly empty) whose connected components are ordered and let C denote a finite set (possibly empty).

2.1.

Generalities. A vertex-oriented unitrivalent graph is a finite graph whose vertices are univalent (legs) or trivalent, and such that for each trivalent vertex the set of halfedges incident to it is cyclically ordered.

A Jacobi diagram on (X, C) is a vertex-oriented unitrivalent graph whose legs are either embedded in the interior of X or are colored by the Q-vector space generated by C. Two Jacobi diagrams are considered to be the same if there is an orientation-preserving homeomorphism between them respecting the order of the connected components, the vertex orientation of the trivalent vertices and the colorings of the legs. For drawings of Jacobi diagrams we use solid lines to represent X, dashed lines to represent the unitrivalent graph and we assume that the orientation of trivalent vertices is counterclockwise. See Figure 2.1 for some examples. The space of Jacobi diagrams on (X, C) is the Q-vector space: If X is not empty, it is well known that, for diagrams D ∈ A(X, C) such that every connected component of D has at least one leg attached to X, the STU relation implies the AS and IHX relations, see [START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF]Theorem 6]. We can also define the space A(X, G) for any finitely generated free abelian group G as A(X, G) = A(X, C), where C is any finite set of free generators of G. If X or C is empty we drop it from the notation. For D ∈ A(X, C) we define the internal degree, the external degree and total degree; This way, the space A(X, C) is graded with the total degree. We still denote by A(X, C) its degree completion. 2.3 (b). The space of tree-like Jacobi diagrams colored by C, denoted by A t (C), is the quotient of A(C) by the subspace generated by looped diagrams. The space of connected tree-like Jacobi diagrams colored by C, denoted by A t,c (C), is the subspace of A t (C) spanned by connected Jacobi diagrams in A t (C). For instance the spaces A t,c (G), for G some particular abelian groups, are the target of the diagrammatic versions of the Johnson-type homomorphisms. See Figure 2.3 (c) for an example of a connected tree-like Jacobi digram. Doubling and orientation-reversal operations. Suppose that we can decompose the 1-manifold X as X = X ↓ for a chosen oriented interval component of X, here X can be empty. Then given a Jacobi diagram D on X ↓ it is possible to obtain new Jacobi diagrams ∆(D) on X ↓= X ↓↓ and S(D) on X ↑. Let us represent the Jacobi diagram D as Then ∆(D) is defined in Figure 2.4, where we use the box notation to denote the sum over all the possible ways of gluing the legs of D attached to the grey box to the two intervals involved in the grey box, in particular if there are k legs attached to the grey box, there will be 2 k terms in the sum. Besides, the Jacobi digram S(D) is given in Figure 2.5. To sum up, we have maps

A(X, C) = Vect Q {Jacobi diagrams on (X, C)} STU,
(2.1) ∆ : A(↓ X ) -→ A(↓↓ X ) and S : A(↓ X) -→ A(↑ X),
called doubling map and orientation reversal map, respectively. Observe that even if we use the same notation for the doubling map and the coproduct, the respective meaning can be deduced from the context. Symmetrization map. Let us recall the diagrammatic version of the Poincaré-Birkhoff-Witt isomorphism. We follow [START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF][START_REF] Chmutov | Introduction to Vassiliev knot invariants[END_REF] in our exposition. Let D be a Jacobi diagram on (X, C {s}) we could glue all the s-colored legs of D to an interval ↑ s (labelled by s) in order to obtain a Jacobi diagram on (X ↑ s , C), i.e. there would not be any s-colored leg left. But there are many ways of doing this gluing, so we consider the arithmetic mean of all the possible ways of gluing the s-colored legs of D to the interval ↑ s . This way we obtain a well defined vector space isomorphism

(2.2) χ s : A(X, C {s}) -→ A(X ↑ s , C),
called symmetrization map. It is not difficult to show that the map (2.2) is well defined, but it is more laborious to show that it is bijective, see [START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF]Theorem 8]. If S = {s 1 , . . . , s l }, it is possible to define, in a similar way, a vector space isomorphism

χ S : A(X, C S) -→ A(X ↑ S , C),
where

↑ S =↑ s 1 • • • ↑ s l . More precisely, χ S = χ s l • • • • • χ s 1 .
Example 2.4. Fix r ∈ S. Denote by H(r) the subspace of A(S) generated by Jacobi diagrams with at least one component that is looped or that possesses at least two r-colored legs. Similarly, denote by H(↑ r ) the subspace of A(↑ S ) generated by Jacobi diagrams with at least one dashed component that is looped or that possesses at least two legs attached to ↑ r . Bar-Natan shows in [2, Theorem 1] that χ(H(r)) = H(↑ r ).

The inverse of the symmetrization map is constructed recursively. Since we will use this inverse, let us review the definition. Let D be a Jacobi diagram on (X ↑ s , C) with n legs attached to ↑ s . Label these legs from 1 to n following the orientation of ↑ s . For a permutation ς ∈ S n , there is a way of obtaining a Jacobi digram ςD on (X ↑ s , C) by acting on the legs. For instance if ς = (123) we have:

σ n (D) =      D + 1 n! ς∈Sn σ n-1 (D -ςD), if l = n, σ n-1 (D), if l < n.

Then the map

σ : A(X ↑ s , C) -→ A(X, C {s}),
defined by σ(D) = σ n (D) is well-defined and it is the inverse of the symmetrization map.

Example 2.6.

Example 2.7.

Example 2.8.

In the last equality we used Example 2.7.

Example 2.9. We are usually interested in the reduction modulo looped diagrams. We use the symbol ≡ to indicate an equality modulo looped diagrams. Using the previous examples, it is possible to show

Here the square brackets stand for an exponential, more precisely

The Kontsevich integral and the LMO functor

In this section we review the combinatorial definition of the Kontsevich integral from [START_REF] Ohtsuki | Quantum invariants[END_REF][START_REF] Quoc | The LMO invariant[END_REF]. We also recall the construction of the LMO functor following [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF]. We focus on particular examples, which will play an important role in the next sections, rather that in a detailed exposition on the subject.

3.1. Kontsevich Integral. Let us start by recalling some basic notions. Consider the cube [-1, 1] 3 ⊆ R 3 with coordinates (x, y, z). A framed tangle in [-1, 1] 3 is a compact oriented framed 1-manifold T properly embedded in [-1, 1] 3 such that the boundary ∂T (the endpoints of T ) is uniformly distributed along {0} × [-1, 1] × {±1} and the framing on the endpoints of T is the vector (0, 1, 0). We draw diagrams of framed tangles using the blackboard framing convention. Let T be a framed tangle. Denote by ∂ t T the endpoints of T lying in {0} × [-1, 1] × {+1}, we call ∂ t T the top boundary of T . Similarly, ∂ b T of T denotes the bottom boundary.

We can associate words w t (T ) and w b (T ) on {+, -} to ∂ t T and ∂ b T as follows. To an endpoint of T we associate + if the orientation of T goes downwards at that endpoint, and -if the orientation of T goes upwards at that endpoint. The words w t (T ) and w b (T ) are obtained by reading the corresponding signs in the positive direction of the y coordinate. See Figure (3.1) (a) for an example of a tangle with its corresponding words.

We consider non-associative words on {+, -}, that is, words on {+, -} together with a parenthesization (formally an element of the free magma generated by {+, -}). For instance ((+-)+) and (+(-+)) are the two possible non-associative words obtained from the word + -+. From now on we omit the outer parentheses. A q-tangle is a framed tangle whose top and bottom words are endowed with a parenthesization. See To define the Kontsevich integral it is necessary to fix a particular element Φ ∈ A(↓↓↓) called an associator. The element Φ is an exponential series of Jacobi diagrams satisfying several conditions, among these, one "pentagon" and two "hexagon" equations; see [38, (6.11)-(6.13)]. From now on we fix an even Drinfeld associator Φ. In low degree we have:

Here 1 means ↓↓↓. The Kontsevich integral is defined so that:

(3.1) Z(T 1 • T 2 ) = Z(T 1 ) • Z(T 2 ), Z(T 1 ⊗ T 2 ) = Z(T 1 ) ⊗ Z(T 2 );
where the composite Z(T 1 ) • Z(T 2 ) and the tensor product Z(T 1 ) ⊗ Z(T 2 ) are defined by vertical and horizontal juxtaposition of Jacobi diagrams. Now every q-tangle can be expressed as the composition of tensor products of some elementary q-tangles, so it is enough to define the Kontsevich integral on these q-tangles. Set where S 2 is the orientation-reversal map applied to the second interval. The Kontsevich integral is defined on the elementary q-tangles as follows:

and for elementary q-tangles of the form where the thick lines represent a trivial tangle and the black dots some non-associative words on {+, -}, the Kontsevich integral is defined by using the doubling and orientation reversal maps, see subsection 2.2, for instance

Here the subscripts indicate the interval to which the operation is applied. It is known that Z is well defined and is an isotopy invariant of q-tangles, see [START_REF] Quoc | Representation of the category of tangles by Kontsevich's iterated integral[END_REF][START_REF] Quoc | The universal Vassiliev-Kontsevich invariant for framed oriented links[END_REF]. For a q-tangle T , we denote by Z t (T ) the reduction of Z(T ) modulo looped diagrams, see Example 2.3. The LMO functor. This subsection is devoted to a brief description of the LMO functor Z : LCob q → ts A and principally to explicit computations which will be useful in the following sections. We refer to [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF] for more details. Throughout this subsection we denote by Σ g,1 a compact connected oriented surface of genus g with one boundary component for each non-negative integer g, see Figure 3.2.

Homology cobordisms and their bottom-top tangle presentation. Let us start with some preliminaries. A homology cobordism of Σ g,1 is the equivalence class of a pair M = (M, m), where M is a compact connected oriented 3-manifold and M and M by using the map m + • (m -) -1 . This composition is associative and has as identity element the equivalence class of the trivial cobordism (Σ g,1 × [-1, 1], Id). Denote by C g,1 the monoid of homology cobordisms of Σ g,1 . This notion plays an important role in the theory of finite-type invariants as shown independently by M. Goussarov in [START_REF] Goussarov | Finite type invariants and n-equivalence of 3-manifolds[END_REF] and K. Habiro in [START_REF] Habiro | Claspers and finite type invariants of links[END_REF].

m : ∂(Σ g,1 × [-1, 1]) → ∂M is
Example 3.5. Denote by M g,1 the mapping class group of Σ g,1 , i.e. the group of isotopy classes of orientation-preserving homeomorphisms of Σ g,1 that fix the boundary ∂Σ g,1 pointwise. This group can be embedded into C g,1 by associating to any h ∈ M g,1 the homology cobordism, called mapping cylinder,

c(h) = (Σ g,1 × [-1, 1], m h ), where m h : ∂(Σ g,1 × [-1, 1]) → ∂(Σ g,1 × [-1, 1]
) is the orientation-preserving homeomorphism defined by m h (x, 1) = (h(x), 1) and m h (x, t) = (x, t) for t = 1. This way we have an injective map c :

M g,1 → C g,1 . The submonoid c(M g,1 ) is precisely the group of invertible elements of C g,1 , see [14, Proposition 2.4].
There is a more general notion of cobordism. For g, f ≥ 0 let C g f denote the compact oriented 3-manifold obtained from [-1, 1] 3 by adding g (respectively

f ) 1-handles along [-1, 1] × [-1, 1] × {+1} (respectively along [-1, 1] × [-1, 1] × {-1}), uniformly in the y direction. A cobordism from Σ g,1 to Σ f,1
is the homeomorphism class relative to the boundary of a pair (M, m), where M is a compact connected oriented 3-manifold and m : ∂C g f → ∂M is an orientation-preserving homeomorphism. Given a homology cobordism (M, m) of Σ g,1 ; or more generally a cobordism from Σ g,1 to Σ f,1 . We can associate a particular kind of tangle whose components split in f bottom components and g top components (they are called bottom-top tangles in [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF]). The association is defined as follows. First fix a system of meridians and parallels {α i , β i } on Σ g,1 for each non-negative integer g as shown in Figure 3.2.

Figure 3.2. System of meridians and parallels {α

i , β i } on Σ g,1 .
Then attach g 2-handles (or f in the case of a cobordism from Σ g,1 to Σ f,1 ) on the bottom surface of M by sending the cores of the 2-handles to the curves m -(α i ). In the same way, attach g 2-handles on the top surface of M by sending the cores to the curves m + (β i ). This way we obtain a compact connected oriented 3-manifold B and an orientation-preserving homeomorphism b :

∂([-1, 1] 3 ) → ∂B. The pair B = (B, b)
together with the cocores of the 2-handles, determine a tangle γ in B. We call the homeomorphism class relative to the boundary of the pair (B, γ), still denoted in the same way, the bottom-top tangle presentation of (M, m). Following the positive direction of the y coordinate, we label the bottom components of γ with 1 -, . . ., f -and the top components with 1 + , . . ., g + , respectively. This procedure is sketched in Example 3.6. Lagrangian Cobordisms. Let us now roughly describe the source category LCob of the LMO functor. For each non-negative integer g, let H g = H 1 (Σ g,1 ; Z) be the first homology group of Σ g,1 with integer coefficients, and ω : H g ⊗ H g → Z the intersection form. Denote by A g the subgroup of H g generated by the homology classes of the meridians {α i }. This is a Lagrangian subgroup of H g with respect to the intersection form. Let V g be a handlebody of genus g obtained from Σ g,1 by attaching g 2-handles by sending the cores of the 2-handles to the meridians α i 's, in particular the curves α i bound pairwise disjoint disks in V g . We also see V g as a cobordism from Σ g,1 to Σ 0,1 , see Figure 3.4. Thus we can also see

A g as A g = ker(H g → H 1 (V g ; Z)). Figure 3.4. Handlebody V g as a cobordism from Σ g,1 to Σ 0,1 . Definition 3.7. [5, Definitions 2.4 and 2.6] A cobordism (M, m) from Σ g,1 to Σ f,1 is said to be Lagrangian if it satisfies: • H 1 (M ; Z) = m -, * (A f ) + m +, * (H g ), • m +, * (A g ) ⊆ m -, * (A f ) in H 1 (M ; Z). Moreover, (M, m) is said to be special Lagrangian if it additionally satisfies V f • M = V g as cobordisms.
Let M be a Lagrangian cobordism and (B, γ) its bottom-top tangle presentation. It follows, from a Mayer-Vietoris argument, that B is a homology cube, i.e. B has the same homology groups as the standard cube [-1, 1] 3 , see [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF]Lemma 2.12]. Notice that the definition of q-tangle in [-1, 1] 3 given in subsection 3.1 extends naturally to q-tangles in homology cubes.

Let us now define the category LCob. The objects of LCob are the non-negative integers and the set of morphisms LCob(g, f ) from g to f are Lagrangian cobordisms from Σ g,1 to Σ f,1 . Denote by s LCob(g, f ) the morphisms from g to f which are special Lagrangian.

Example 3.8. Let h ∈ M g,1 . Then the mapping cylinder c(h) is Lagrangian if and only if h(A g ) ⊆ A g . Moreover, c(h) is special Lagrangian if and only if h can be extended to a self-homeomorphism of the handlebody V g .

Let us consider some particular cases of the mapping cylinders described in Example 3.8. Let γ be a simple closed curve on Σ g,1 and denote by t γ the (left) Dehn twist along γ. Recall that the mapping cylinder c(t γ ) can be obtained from the trivial cobordism Σ g,1 × [-1, 1] by performing a surgery along a (-1)-framed knot in a neighbourhood of a push-off of the curve γ in Σ g,1 × [-1, 1], see for instance [START_REF] Ohtsuki | Quantum invariants[END_REF]Lemma 8.5]. In particular we can obtain the bottom-top tangle presentation of c(t γ ) from that of Σ g,1 × [-1, 1], see Examples 3.9, 3.10 and 3.11.

Example 3.9. Let t α i be the Dehn twist along a meridian curve α i . Then c(t α i ) ∈ s LCob(g, g). Example 3.12. Let N i be the cobordism from Σ g,1 to Σ g+1,1 with the bottom-top tangle presentation shown in Figure 3.8. Then N i is a special Lagrangian cobordism. The label r on the first (from left to right) bottom component stands for root. This is because from these cobordisms we will obtain, via the LMO functor, rooted trees with root r that we will interpret as Lie commutators. See subsection 6.3.

Top-substantial Jacobi diagrams. Let us now describe the target category ts A of the LMO functor. The objects of the category ts A are the non-negative integers. The set of morphisms from g to f is the subspace ts A(g, f ) of diagrams in A( g + f -) (see Example 2.2) without struts whose both ends are colored by elements of g + . These kind of Jacobi diagrams are called top-substantial.

If D ∈ ts A(g, f ) and E ∈ ts A(h, g) the composition D • E = D |j + →j * , E |j -→j * g *
is the element in ts A(h, f ) given by the sum of Jacobi diagrams obtained by considering all the possible ways of gluing the g + -colored legs of D with the g --colored legs of E. A schematic description is shown in Figure 3.9 (a). The identity morphism in ts A(g, g) is shown in Figure 3.9 (b). Sketch of the construction of the LMO functor. The definition of the LMO functor uses the Kontsevich integral which is defined for q-tangles. Because of this, it is necessary to modify the objects of LCob to obtain the category LCob q : instead of non-negative integers, the objects of LCob q are non-associative words in the single letter • and if u and v are non-associative words in • of length g and f respectively, a morphism from u to v is a Lagrangian cobordism from Σ g,1 to Σ f,1 .

Roughly speaking, the LMO functor Z : LCob q → ts A is defined as follows. Let M ∈ LCob q (u, v), where u and v are two non-associative words in •. Let (B, γ ) be the bottom-top tangle presentation of M . By performing the change • → (+-) in u and v we obtain words w t (γ ) and w b (γ ) on {+, -} together with some parenthesizations. Hence γ is a q-tangle in the homology cube B. Next, take a surgery presentation of (B, γ ), that is, a framed link

L ⊆ int([-1, 1] 3 ) and a tangle γ in [-1, 1] 3 such that surgery along L carries ([-1, 1] 3 , γ) to (B, γ ). Set w t (γ) = w t (γ ) and w b (γ) = w b (γ ). Hence L ∪ γ is a q-tangle in [-1, 1] 3 . Now, consider the Kontsevich integral of L ∪ γ,
which gives a series of a kind of Jacobi diagrams. To get rid of the ambiguity in the surgery presentation, it is necessary to use some combinatorial operations on the space of diagrams. Among these operations there is the so-called Aarhus integral (see [START_REF] Bar-Natan | The Aarhus integral of rational homology 3-spheres. I. A highly non trivial flat connection on S 3[END_REF][START_REF] Bar-Natan | The Aarhus integral of rational homology 3-spheres. II. Invariance and universality[END_REF]), which is a kind of formal Gaussian integration on the space of diagrams. We then arrive to ts A. Finally, to obtain the functoriality, it is necessary to do a normalization.

Recall that the definition of the Kontsevich integral requires the choice of a Drinfeld associator, and the bottom-top tangle presentation requires the choice of a system of meridians and parallels. Thus, the LMO functor also depends on these choices.

We are especially interested in the LMO functor for special Lagrangian cobordisms. For these kind of cobordisms the LMO functor can be computed from the Kontsevich integral and the symmetrization map as is assured by a result of Cheptea, Habiro and Massuyeau. We state the result for our particular case. Convention 3.13. From now on, we endow Lagrangian cobordisms with the righthanded non-associative word (

• • • • (•(••)) • • • ) in
the letter • unless we say otherwise. This way we will always be in the context of the category LCob q . Lemma 3.14. [5, Lemma 5.5] Let M ∈ LCob q (u, v), where u and v are non-associative words in the letter • of length g and f , respectively. Suppose that the bottom-top tangle presentation of M is as in Figure 3.10, where T is a tangle in [-1, 1] 3 . Endow T with the non-associative words w t (T ) = u /• →(+-) and w b (T ) = v /• →(+-) . Then the value of the LMO functor Z(M ) can be computed from the value of the Kontsevich integral Z(T ) as shown in Figure 3.11.

Let (M, m) be a homology cobordism and (B, γ) its bottom-top tangle presentation. Define the linking matrix of (M, m), denoted Lk(M ), as the linking matrix of the link

(3.2) Z s (M ) = Lk(M ) 2 .
The colors 1 + , . . . , g + and 1 -, . . . , f -in the series of Jacobi diagrams Z(M ) refer to the curves m + (β 1 ),. . ., m + (β g ) and m -(α 1 ), . . . , m -(α f ) on the top and bottom surfaces of M respectively. Example 3.16. Let us consider the special Lagrangian cobordism c(t α i ), from Example 3.9, equipped with non-associative words as in Convention 3.13. By Lemma 3.14 and the functoriality of Z (see Equation (3.1)), to compute Z t (c(t α i )) in low degrees we need to first compute in low degrees, which we already computed in Example 3.2. Therefore From Example 2.9, we conclude which shows that there are no terms of i-deg = 1 in Z Y,t (c(t α i )).

Example 3.17. Consider the special Lagrangian cobordism c(t α 12 ) from Example 3.10, equipped with non-associative words as in Convention 3.13. By Lemma 3.14, to compute Z t (c(t α 12 )) in low degrees, we need to first compute the tree-like part in the Kontsevich integral of the q-tangle by the functoriality of Z, see (3.1), we have to compute the low degree terms of which was computed in Example 3. Hence,

Z t (N 1 ) = Z t (P •,•,w ) • ( Z t (T ) ⊗ Id g-1 )
. Now, by the functoriality of Z we have

Z(P •,•,w ) |r →0 = ∅ ⊗ Id g and Z(P •,•,w ) |1 -→0 = Id 1 ⊗ ∅ ⊗ Id g-1 , therefore Z Y (P •,•,w ) |r →0 = ∅ and Z Y (P •,•,w ) |1 -→0 = ∅.
This way, each one of the connected diagrams appearing in Z Y (P •,•,w ) has at least one r-colored leg and at least one 1 --colored leg. Hence, each one of the connected diagrams in Z t (N 1 ) coming from Z Y (P •,•,w ) has at least one r-colored leg and at least one 1 --colored leg. We are interested in the low degree terms of Z t (N 1 ) mod H(r). By Lemma 3.14, we need to compute the low degree terms of which we already computed in Example 3.4. Whence we obtain We conclude that each of the terms with i-deg = 1 in Z(N 1 ) mod H(r) has one r-colored and one 1 --colored leg. In a similar way, it can be shown for 1 ≤ i ≤ g that each of the terms with i-deg = 1 in Z(N i ) mod H(r) has one r-colored leg and one i --colored leg.

Johnson-type filtrations

As in subsection 3.2, we denote by Σ g,1 a compact connected oriented surface of genus g with one boundary component. Let M g,1 denote the mapping class group of Σ g,1 . We will often omit the subscripts g and 1 of our notation unless there is ambiguity, then we will usually write Σ and M instead of Σ g,1 and M g,1 .

4.1. Preliminaries. Let us fix a base point * ∈ ∂Σ and set π = π 1 (Σ, * ) and H = H 1 (Σ, Z), finally denote by ab : π → H the abelianization map. Notice that the intersection form ω : H ⊗H → Z is a symplectic form on H. The elements of M preserve ∂Σ, in particular they preserve * , therefore we have a well defined group homomorphism:

(4.1) ρ : M -→ Aut(π),
which sends h ∈ M to the induced map h # on π. It is well known that the map ρ is injective and it is called the Dehn-Nielsen-Baer representation of M. On the other hand, since the elements of M are orientation-preserving, their induced maps on H preserve the intersection form. This way we have a well defined surjective group homomorphism:

(4.2) σ : M -→ Sp(H) = {f ∈ Aut(H) | ∀x, y ∈ H, ω(f (x), f (y)) = ω(x, y)},
that sends h ∈ M to the induced map h * on H. The map σ is called the symplectic representation of M and it is far from being injective, its kernel is known as the Torelli group of Σ, which is denoted by I (or I g,1 ), so 1 also shows the fixed system of meridians and parallels of Σ used in subsection 3.2. Moreover we suppose that the images ι(α i ) of the meridians α i , under the embedding ι, bound pairwise disjoint disks in V . Set H = H 1 (V ; Z) and π = π 1 (V, ι( * )) and denote by ab : π → H the abelianization map. Consider the following subgroups of π and H that arise when looking at the induced maps by ι in homotopy and in homology:

I = I g,1 = ker(σ) = {h ∈ M | h * = Id H }. (4.3)
(4.4) A = ker(ι * : H → H ) and A = ker(ι # : π → π ).
We also consider the following subgroup of π:

(4.5) K 2 = ker(π ι # -→ π ab' -→ H ) = A • Γ 2 π.
The subgroup A ≤ H is a Lagrangian subgroup of H with respect to the intersection form on H and it is the group that appears in the definition of Lagrangian cobordisms in the previous section. We may think of K 2 as the subgroup of π generated by commutators of weight 2, where the elements of π are considered to have weight 1 and the elements of A are considered to have weight 2. The subgroups A, A and K 2 allow us to define some important subgroups of the mapping class group M.

Definition 4.1. The Lagrangian mapping class group of Σ, denoted by L (or L g,1 ) is defined as follows:

(4.6) L = L g,1 = {f ∈ M g,1 | f * (A) ⊆ A}.
We are mainly interested in three particular subgroups of L, one of these is the Torelli group, see equation (4.3).

Definition 4.2. The Lagrangian Torelli group of Σ, denoted by I L (or I L g,1 ), is defined as follows:

(4.7)

I L = I L g,1 = {h ∈ L | h * | A = Id A }.
The groups L and I L appear in the works [START_REF] Levine | Homology cylinders: an enlargement of the mapping class group[END_REF][START_REF] Levine | The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres[END_REF] of J. Levine in connection with the theory of finite-type invariants of homology 3-spheres. From an algebraic point of view these groups were studied by S. Hirose in [START_REF] Hirose | The action of the handlebody group on the first homology group of the surface[END_REF], where he found a generating system for L and by T. Sakasai in [START_REF] Sakasai | Lagrangian mapping class groups from a group homological point of view[END_REF], where he computed H 1 (L; Z) and H 1 (I L ; Z). Definition 4.3. The alternative Torelli group of Σ, denoted by I a (or I a g,1 ), is defined as follows:

(4.8)

I a = I a g,1 =    for x ∈ π : h # (x)x -1 ∈ K 2 h ∈ L and for y ∈ K 2 : h # (y)y -1 ∈ Γ 3 π • [π, A] =: K 3    .
Notice that the definition of I a involves the group

K 3 = Γ 3 π • [π, A] = [[π, π], π] • [π, A],
which we see as the subgroup of π generated by commutators of weight 3. Like the Lagrangian Torelli group, the group I a appears in [START_REF] Levine | Homology cylinders: an enlargement of the mapping class group[END_REF][START_REF] Levine | The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres[END_REF][START_REF] Garoufalidis | Finite type 3-manifold invariants, the mapping class group and blinks[END_REF] in connection with the theory of finite-type invariants but with a different definition: the second term of the Johnson-Levine filtration. Definition 4.3 comes from [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF], see Proposition for the equivalence of the two definitions. J. Levine shows in [29, Proposition 4.1] that I a is generated by Dehn twists along simple closed curves (scc) whose homology class belongs to A. Equivalently, I a is generated by Dehn twists along scc's which bound a surface in the handlebody V . This is the definition of I a given in [START_REF] Levine | The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres[END_REF][START_REF] Garoufalidis | Finite type 3-manifold invariants, the mapping class group and blinks[END_REF].

From the above definitions it follows that I ⊆ I L ⊆ L and I a ⊆ I L ⊆ L. But I a ⊆ I and I ⊆ I a . We shall call here the groups I, I L and I a Torelli-type groups. In contrast with I, the groups I L and I a are not normal in M, but they are normal in L. where all the inclusions are proper. Besides, J. Levine proved in [28, Theorem 2] that (4.9)

I ∩ I a = K • [I, I a ],
where K is the Johnson kernel. D. Johnson proved in [START_REF] Johnson | The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves[END_REF] that K is generated by BSCC maps (bounding scc's), that is, Dehn twists along scc's which are null-homologous in Σ.

G = G 1 ≥ G 2 ≥ • • • ≥ G m ≥ G m+1 ≥ • • • of subgroups of G such that [G i , G j ] ⊆ G i+j for i, j ≥ 1.
We are interested in N -series of the group π = π(Σ, * ). A first example of an N -series of π is the lower central series (Γ k π) k≥1 . We consider an N -series of π in which the subgroup A plays a special role.

Set K 1 = π and K 2 = A • Γ 2 π as defined in Equation (4.5). Let (K m ) m≥1 be the smallest N -series of π starting with these

K 1 and K 2 , that is, if (G i ) m≥1 is any N -series of π with G 1 = K 1 and G 2 = K 2 then K m ⊆ G m for every m ≥ 1. More precisely, for every m ≥ 3 we have (4.10) K m = [K m-1 , K 1 ] • [K m-2 , K 2 ].
In particular

K 3 = Γ 3 π • [π, A]
is the group that we used in the definition of the alternative Torelli group, see (4.8). We can think of K m as the subgroup of π generated by commutators of weight m, where the elements of π have weight 1 and the elements of A have weight 2. By induction on m ≥ 1 we have

(4.11) Γ m π ⊆ K m ⊆ Γ m/2 π,
where m/2 denotes the least integer greater than or equal to m/2.

Restricting the Dehn-Nielsen-Baer representation (4.1) to the Lagrangian mapping class group we get an action of L on K 1 = π. We denote the action of h ∈ L on x ∈ π by h x.

Hence h x = ρ(h)(x) = h # (x). Lemma 4.6. For every h ∈ L we have h (K 2 ) = K 2 .
Proof. It is enough to show h (K 2 ) ⊆ K 2 for every h ∈ L. Let h ∈ L and x ∈ K 2 = ker(ab ι # ). Hence 0 = ab ι # (x) = ι * (ab(x)), so ab(x) ∈ A and then h * (ab(x)) ∈ A.

Therefore ab ι # h # (x) = ι * (ab(h # (x))) = ι * (h * (ab(x))) = 0, that is, h # (x) ∈ K 2 .
It follows from Equality (4.10) and Lemma 4.6, by induction, that h (K m ) = K m for every m ≥ 1 and h ∈ L. From the general setting in [15, Section 3.4 and Section 10.2] we have a decreasing sequence (4.12)

L = J a 0 M ⊇ J a 1 M ⊇ J a 2 M ⊇ • • • ⊇ J a m M ⊇ J a m+1 M ⊇ • • • of subgroups of M satisfying: (4.13) [J a l M, J a m M] ⊆ J a
l+m M for all l, m ≥ 0. In our case, the m-th term in this decreasing sequence is given by αg belong to J a 2 M. We will show this explicitly in Examples 5.7 and 5.8. In particular t δ and t t -1 αg belong to I ∩ I a .

(4.14) J a m M = J a m M g,1 =    for x ∈ π : h # (x)x -1 ∈ K 1+m h ∈ L and for y ∈ K 2 : h # (y)y -1 ∈ K 2+m    .
(i) m≥0 J a m M = {Id Σ }. (ii) For all k ≥ 1 the group J a k M is residually nilpotent, that is, m Γ m J a k M = {Id Σ }.
Proof. In order to prove (i), recall that

K m ⊆ Γ m/2 π for m ≥ 1. Consider h ∈ L such that h ∈ J a m M for all m ≥ 0. Let x ∈ π, thus ∀m ≥ 1, h # (x)x -1 ∈ K m+1 ⊆ Γ (m+1)/2 π. Therefore h # (x)x -1 ∈ Γ k π for all k ≥ 1. Since π is residually nilpotent, we have that h # = ρ(h) = Id π .
In view of the injectivity of the Dehn-Nielsen-Baer representation ρ we conclude that h = Id Σ , so we have (i). Now, let us see (ii). Fix k ≥ 1, from (4.13) it follows, by induction on m, that Γ m J a k M ⊆ J a m M for all m ≥ 1. Therefore by (i) we obtain m≥1 Γ m J a k M = {Id Σ }.

The Johnson filtration satisfies similar properties to those stated in the above proposition. Let us briefly recall the Johnson filtration and the Johnson-Levine filtration in order to compare them with each other.

Johnson filtration. The lower central series of π is preserved by the Dehn-Nielsen-Baer representation ρ, so for every k ≥ 1 there is a group homomorphism (4.15) ρ k : M -→ Aut(π/Γ k+1 π), defined as the composition

M ρ -→ Aut(π) -→ Aut(π/Γ k+1 π).
Notice that ker(ρ 1 ) is the Torelli group I. The Johnson filtration of M is the descending chain of subgroups (4.16)

M ⊇ I = J 1 M ⊇ J 2 M ⊇ J 3 M ⊇ • • • defined by J k M := ker(ρ k ) for k ≥ 1. Equivalently for k ≥ 1, (4.17) J k M = {h ∈ M | for all x ∈ π : h # (x)x -1 ∈ Γ k+1 π}.
(i) [J k M, J m M] ⊆ J k+m M for all k, m ≥ 1. (ii) k≥1 J k M = {Id Σ }. (iii) For all k ≥ 1 the group J k M is residually nilpotent.
I L = J L 1 M ⊇ J L 2 M ⊇ J L 3 M ⊇ • • • defined by (4.19) J L k M := {h ∈ I L | ι # h # (A) ⊆ Γ k+1 π } for k ≥ 1.
Let H be the subgroup of M consisting of the elements that can be extended to the handlebody V . In Example 3.8 we used these kind of homeomorphisms to give examples of special Lagrangian cobordisms. It is well known that (4.20)

H = {h ∈ M | h # (A) ⊆ A},
(i) For k ≥ 1, J L k M is a subgroup of M. (ii) k≥1 J L k M = H ∩ I L . (iii) J k M ⊆ J L k M for every k ≥ 1. (iv) J L
2 M is generated by simple closed curves which bound in V , equivalently by scc's whose homology class belongs to A.

(v) I L = I • (H ∩ I L ) and J L 2 M = J 2 M • (H ∩ I L ).
We refer to the alternative Johnson filtration, the Johnson filtration and the Johnson-Levine filtration as Johnson-type filtrations.

Comparison between Johnson-type filtrations. Proposition 4.11 gives a first comparison between the three filtrations. Let us give a more general comparison. Lemma 4.12. For every m ≥ 1 there exists a normal subgroup N m of A such that

K m = Γ m π • N m .
Proof. The argument is by strong induction on m. Taking N 1 = {1} and N 2 = A, clearly we have

K 1 = Γ 1 π • N 1 and K 2 = Γ 2 π • N 2 . Suppose m ≥ 3 and let N m-2 , N m-1 be normal subgroups of A such that K m-1 = Γ m-1 π • N m-1 and K m-2 = Γ m-2 π • N m-2 . Thus K m = [K m-1 , K 1 ] • [K m-2 , K 2 ] = [Γ m-1 π • N m-1 , π] • [Γ m-2 π • N m-2 , K 2 ] = [Γ m-1 π, π] • [N m-1 , π] • [Γ m-2 π, Γ 2 π • A] • [N m-2 , K 2 ] = Γ m π • [N m-1 , π] • [Γ m-2 π, Γ 2 π] • [Γ m-2 π, A] • [N m-2 , K 2 ] = Γ m π • N m , where N m = [N m-1 , π] • [Γ m-2 π, A] • [N m-2 , K 2 ] is a normal subgroup of A.
Proposition 4.13. For every m ≥ 1, we have

(i) J a 2m M ⊆ J m M. (ii) J m M ⊆ J a m-1 M. (iii) J a m M ⊆ J L m+1 M.
In particular the Johnson filtration and the alternative Johnson filtration are cofinal.

Proof. Let m ≥ 1. Let h ∈ J a 2m M, then for every x ∈ π we have

h # (x)x -1 ∈ K 2m+1 ⊆ Γ (2m+1)/2 = Γ m+1 π, that is, h ∈ J m M so (i) holds. Let h ∈ J m M, then for every x ∈ π we have h # (x)x -1 ∈ Γ m+1 π ⊆ K m+1 .
In particular, h # (x)x -1 ∈ K m for every x ∈ π and h # (y)y -1 ∈ K m+1 for every y ∈ K 2 . That is, h ∈ J a m-1 M, hence (ii). Finally for (iii) we use Lemma 4.12 to write

K m+2 = Γ m+2 π • N with N a normal subgroup of A. Let h ∈ J a m M. It follows that for every α ∈ A ⊆ K 2 , h # (α)α -1 ∈ K m+2 . Write h # (α)α -1 = xn with x ∈ Γ m+2 π and n ∈ N . Therefore ι # (h # (α)) = ι # (h # (α)α -1 ) = ι # (x)ι # (n) = ι # (x) ∈ Γ m+2 π , whence ι # h # (A) ⊆ Γ m+2 π . Hence h ∈ J L m+1 M. Remark 4.14.
We expect that the subscripts of the relations on Proposition 4.13 are the best possibles.

Remark 4.15. D. Johnson proved in [START_REF] Johnson | The structure of the Torelli group. I. A finite set of generators for I[END_REF] that the Torelli group I g,1 is finitely generated for g ≥ 3. This result together with the short exact sequence

1 -→ I -→ I L σ --→ σ(I L ) -→ 0,
where σ is the symplectic representation, imply that the Lagrangian Torelli group I L g,1

is finitely generated for g ≥ 3. Notice that

σ(I L ) = σ(I a ) = Idg ∆ 0 Idg ∈ Sp(2g, Z)
∆ is symmetric , see Lemma 6.6 and Equation (6.10). Hence σ(I L ) and σ(I a ) are finitely generated. Recently T. Church, M. Ershov and A. Putnam proved in [START_REF] Church | On finite generation of the Johnson filtrations[END_REF] several results concerning the finite generation of the Johnson filtration. In particular they proved [8, Theorem A] that the Johnson kernel K g,1 is finitely generated for g ≥ 4. This result together with the short exact sequences

1 -→ K -→ I ∩ I a τ 1 --→ τ 1 (I ∩ I a ) -→ 0 and 1 -→ I ∩ I a -→ I a σ --→ σ(I a ) -→ 0,
where τ 1 is the first Johnson homomorphism, imply that the alternative Torelli group I a g,1 is finitely generated for g ≥ 4. Besides, it follows from the general result [8, Theorem B] that J a m M g,1 is finitely generated for m ≥ 2 and g ≥ 2m + 1.

Johnson-type homomorphisms

Throughout this section we use the same notations and conventions from Section 4. The aim of this section is the study of a sequence of group homomorphisms {τ a m } m≥0 introduced in [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF], which are defined on each term of the alternative Johnson filtration and taking values in some abelian groups. These abelian groups can be described by means of the first homology group H of the surface Σ, the first homology group B := H of the handlebody V and the subgroup A = ker(ι * ).

Preliminaries. Since [J a

l M, J a m M] ⊆ J a l+m M for all l, m ≥ 0, the quotient group J a m M/J a m+1 M is an abelian group for m ≥ 1 and we can endow

(5.1) Gr(J a • M) = m≥1 J a m M J a m+1 M
with a structure of graded Lie algebra with Lie bracket induced by the commutator operation. K. Habiro and G. Massuyeau show in [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF] that the Lie algebra (5.1) embeds into a Lie algebra of derivations. To achieve this, they define group homomorphisms on each term of the alternative Johnson filtration, even in the 0-th term J a 0 M which is the Lagrangian mapping class group L. We shall call these homomorphisms the alternative Johnson homomorphisms. In order to define them, let us start with some preliminaries.

Free Lie algebra associated to the N -series (K m ) m≥1 . In the definition of the alternative Johnson filtration we use the N -series (K m ) m≥1 defined in Equation (4.10). The graded Lie algebra associated to this N -series is given by

(5.2) Gr(K • ) = m≥1 K m K m+1 = K 1 K 2 ⊕ K 2 K 3 ⊕ • • •
It follows from [23, Proposition 1] that this graded Lie algebra is freely generated in degree 1 and 2, see also [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF]Lemma 10.9]. More precisely, by Hopf's formula we have

(Γ 2 π ∩ A)/[π, A] ∼ = H 2 (π/A) ∼ = H 2 (π ) and H 2 (π ) = 0 because π is a free group. Hence [π, A] = Γ 2 π ∩ A.
Consider the injective homomorphism j : A → K 2 /K 3 given by the composition

(5.3) A ∼ = ←- ab (A • Γ 2 π)/Γ 2 π ∼ = A/(Γ 2 π ∩ A) = A/[π, A] -→ K 2 K 3 .
Identify B = H with K 1 /K 2 . Denote by Lie(B; A) the graded free Lie algebra (over Z) generated by B in degree 1 and A in degree 2:

(5.4)

Lie(B; A) = m≥1 Lie m (B; A) = B ⊕ (Λ 2 B ⊕ A) ⊕ • • •
Therefore we have From the long exact sequence associated to the pair (V, ∂V ) we obtain the short exact sequence

(5.6) 0 -→ H 2 (V, ∂V ; Z) δ * --→ H ι * --→ B -→ 0, whence H 2 (V, ∂V ; Z) ∼ = A.
Besides, by Poincaré-Lefschetz duality there is a canonical isomorphism H 2 (V, ∂V ; Z) ∼ = H 1 (V ; Z) which allows to define the intersection form of the handlebody

(5.7) ω : B × H 2 (V, ∂V ; Z) -→ Z.
Consider the identifications B ∼ = A * given by the isomorphism H/A Using the identification (5.9) we obtain (5.12)

Ψ(d) = g i=1 b * i ⊗ d(b i ) + g i=1 a * i ⊗ d(a i ) = g i=1 a i ⊗ d(b i ) - g i=1 b i ⊗ d(a i ). Hence (5.13) Ξ m Ψ(d) = g i=1 [a i , d(b i )] - g i=1 [b i , d(a i )] = d g i=1 [a i , b i ] = d(Ω ) = 0.

Alternative Johnson homomorphisms.

Let m be a positive integer and consider the space D m (Lie(B; A)) defined in (5.8).

Definition 5.4. The m-th alternative Johnson homomorphism is the group homomorphism

(5.14) τ a m : J a m M -→ D m (Lie(B; A)), that maps h ∈ J a m M to τ a m (h) = (τ a m (h) 1 , τ a m (h) 2 )
, where

τ a m (h) 1 (xK 2 ) = h # (x)x -1 K m+2 and τ a m (h) 2 (a) = h # (y)y -1 K m+3 for all x ∈ π, a ∈ A, here y ∈ A ⊆ K 2 is any lift of a, see (5.3).
We refer to [15, Proposition 6.2] for a proof of the homomorphism property. From the definition of the alternative Johnson homomorphisms it follows that for m ≥ 1 (5.15) ker(τ a m ) = J a m+1 M. Consider the bases as in Proposition 5.3 of H and B and keep the notation {α i , β i } for a free basis of π. Using the identification (5.9) we have that the m-th alternative Johnson homomorphism of h ∈ J a m M is given by

τ a m (h) = g i=1 a i ⊗ (τ a m (h) 1 (β i K 2 )) - g i=1 b i ⊗ (τ a m (h) 2 (a i )) = g i=1 a i ⊗ h # (β i )β -1 i K m+2 - g i=1 b i ⊗ h # (α i )α -1 i K m+3 .
(5.16)

Example 5.5. Consider the Dehn twist h = t α i from Example 4.4, we know that h ∈ J a 1 M = I a . Let us compute its first alternative Johnson homomorphism. We have

h # (α j ) = α j for 1 ≤ j ≤ g, h # (β j ) = β j for 1 ≤ j ≤ g with j = i and h # (β i ) = α -1 i β i . Hence τ a 1 (t α i ) = -a i ⊗ a i .
= α 2 α 1 ∈ K 2 . We have h # (α j ) = α j and h # (β j ) = β j for 3 ≤ j ≤ g and h # (α 1 ) = λ -1 α 1 λ, h # (α 2 ) = λ -1 α 2 λ, h # (β 1 ) = λ -1 β 1 and h # (β 2 ) = λ -1 β 2 . Hence τ a 1 (t α 12 ) = -(a 1 ⊗ a 1 ) -(a 2 ⊗ a 2 ) -(a 1 ⊗ a 2 ) -(a 2 ⊗ a 1
). Similarly for the Dehn twist t α kl from Example 3.11 we have

τ a 1 (t α kl ) = -(a k ⊗ a k ) -(a l ⊗ a l ) -(a k ⊗ a l ) -(a l ⊗ a k ).
= [α 1 , β -1 1 ] ∈ K 3 . We have h # (α j ) = α j and h # (β j ) = β j for 2 ≤ j ≤ g and h # (α 1 ) = λ -1 α 1 λ and h # (β 1 ) = λ -1 β 1 λ. Hence τ a 1 (t δ ) = 0. In particular t δ ∈ J a 2 M. Example 5.8. Let h = t t -1
αg from Example 4.8. The homotopy class of the curve is represented by

λ = β -1 g α -1 g β g [α g-1 , β g-1 ]. We have h # (α i ) = α i and h # (β i ) = β i for 1 ≤ i ≤ g -2, and h # (α g ) = α g , h # (α g-1 ) = λ -1 α g-1 λ, h # (β g-1 ) = λ -1 β g-1 λ and h # (β g ) = α g β g λ. By a direct calculation we obtain τ a 1 (t t -1 αg ) = 0. In particular t t -1
αg ∈ J a 2 M. Notice that in Examples 5.5 and 5.6, we have Ξ 1 τ a 1 (t α i ) = 0 and Ξ 1 τ a 1 (t α 12 ) = 0. This is a more general fact.

Theorem 5.9. Let m ≥ 1. For h ∈ J a m M we have Ξ m τ a m (h) = 0, that is τ a m (h) ∈ D m (B; A) ∼ = Der +,ω
m (Lie(B; A)). In other words, τ a m (h) is a positive symplectic derivation of Lie(B; A).

Proof. The proof is similar to the proof of [START_REF] Morita | Abelian quotients of subgroups of the mapping class group of surfaces[END_REF]Corollary 3.2]. Consider the free basis {α i , β i } of π induced by the system of meridians and parallels in Figure 3.2. Let h ∈ J a m M. Since h preserves the boundary ∂Σ of Σ, then h # fixes the inverse of the homotopy class

[∂Σ] of Σ. So h # ([∂Σ] -1 ) = [∂Σ] -1 , that is, (5.17) h # g i=1 [β -1 i , α i ] = g i=1 [β -1 i , α i ].
For 1 ≤ i ≤ g we have

β -1 i h # (β i ) = δ i ∈ K 1+m and h # (α i )α -1 i = γ i ∈ K 2+m . Whence h # (β -1 i ) = δ -1 i β -1 i . Hence [h # (β -1 i ), h # (α i )] = [δ -1 i β -1 i , γ i α i ] = δ -1 i β -1 i γ i α i β i δ i α -1 i γ -1 i = δ -1 i [β -1 i , γ i ]δ i δ -1 i γ i [β -1 i , α i ]γ -1 i δ i [δ -1 i , γ i ] γ i [δ -1 i , α i ]γ -1 i .
It follows from equation (5.17) that

g i=1 [β -1 i , α i ] = g i=1 δ -1 i [β -1 i , γ i ]δ i δ -1 i γ i [β -1 i , α i ]γ -1 i δ i [δ -1 i , γ i ] γ i [δ -1 i , α i ]γ -1 i .
(5.18)

Now [β -1 i , γ i ] ∈ K 3+m , [δ -1 i , γ i ] ∈ K 3+2m ⊆ K m+4 and [δ -1 i , α i ] ∈ K 3+m .
Therefore, by considering Equation (5.18) modulo K m+4 we obtain

g i=1 [β -1 i , α i ] ≡ g i=1 δ -1 i [β -1 i , γ i ]δ i δ -1 i γ i [β -1 i , α i ]γ -1 i δ i [δ -1 i , γ i ] γ i [δ -1 i , α i ]γ -1 i ≡ g i=1 [β -1 i , α i ] g i=1 [β -1 i , γ i ][δ -1 i , α i ] . Thus (5.19) g i=1 [β -1 i , γ i ][δ -1 i , α i ] ∈ K m+4 .
From (5.19), identification (5.9) and (5.16) we have

0 = Ξ m g i=1 (-b i ) ⊗ (γ i K m+3 ) + g i=1 a i ⊗ (δ i K m+2 ) = g i=1 [a i , h # (β i )β -1 i K m+2 ] - g i=1 [b i , h # (α i )α -1 i K m+3 ] = Ξ m τ a m (h).
(5.20)

In the second equality of (5.20) we use

δ i K m+2 = β i δ i β -1 i K m+2 = h # (β i )β -1 i K m+2 .
Let us briefly recall the Johnson homomorphisms and the Johnson-Levine homomorphisms.

Johnson homomorphisms. To define the Johnson filtration we use the lower central series (Γ m π) m≥1 of π. The associated graded Lie algebra of this filtration is (5.21) Gr(Γ

• π) = m≥1 Γ m π Γ m+1 π ∼ = m≥1 Lie m (H) = Lie(H),
where Lie(H) is the graded free Lie algebra on H. The m-th Johnson homomorphism

(5.22) τ m : J m M -→ Hom(H, Γ m+1 π/Γ m+2 π) ∼ = H * ⊗ Γ m+1 π/Γ m+2 π ∼ = H ⊗ Lie m+1 (H),
sends the isotopy class h ∈ J m M to the map x → h # (x)x -1 Γ m+2 π for all x ∈ H, where x ∈ π is any lift of x. The second isomorphism in (5.22) is given by the identification H ∼ -→ H * that maps x to ω(x, •). These homomorphisms were introduced by D. Johnson in [START_REF] Johnson | An abelian quotient of the mapping class group I g[END_REF][START_REF] Johnson | A survey of the Torelli group[END_REF] and extensively studied by S. Morita in [START_REF] Morita | On the structure of the Torelli group and the Casson invariant[END_REF][START_REF] Morita | Abelian quotients of subgroups of the mapping class group of surfaces[END_REF]. In particular S. Morita proved in [START_REF] Morita | Abelian quotients of subgroups of the mapping class group of surfaces[END_REF]Corollary 3.2] that the m-th Johnson homomorphism takes values in the kernel D m (H) of the Lie bracket [ , ] : H ⊗ Lie m+1 (H) → Lie m+2 (H). Compare this with Theorem 5.9. From the definition it follows that ker(τ m ) = J m+1 M.

Johnson-Levine homomorphisms. J. Levine defined and studied in [START_REF] Levine | Homology cylinders: an enlargement of the mapping class group[END_REF][START_REF] Levine | The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres[END_REF]

τ L m : J L m M → Hom(A, Γ m+1 π /Γ m+2 π ) ∼ = A * ⊗ Γ m+1 π /Γ m+2 π ∼ = H ⊗ Lie m+1 (H ),
is the group homomorphism that sends h ∈ J L m M to the map a ∈ A → ι # h # (α)Γ m+2 π , where α ∈ A is any lift of α. Notice that here we consider the graded free Lie algebra Lie(H ) generated by H . J. Levine showed in [START_REF] Levine | Homology cylinders: an enlargement of the mapping class group[END_REF]Proposition 4.3] that τ L m takes values in the kernel D m (H ) of the Lie bracket [ , ] : H ⊗Lie m+1 (H ) → Lie m+2 (H ). Compare this with Theorem 5.9. From the definition it follows that ker(τ L m ) = J L m+1 M. We refer to the alternative Johnson homomorphisms, the Johnson-Levine homomorphisms and the Johnson homomorphisms as Johnson-type homomorphisms.

Alternative Johnson homomorphisms and Johnson-levine homomorphisms.

In view of Proposition 4.13, for m ≥ 1 we have J a m M ⊆ J L m+1 M. We show that for J a m M the m-th alternative Johnson homomorphism determines the (m + 1)-st Johnson-Levine homomorphism. Recall that B = H . Lemma 5.10. For m ≥ 1, there is a well defined homomomorphism

ι * : D m (B; A) -→ D m+1 (H ).
Proof. It follows from Lemma 4.12 that for m ≥ 1 the map ι # : π → π induces a well-defined homomorphism

ι * : Lie m+2 (B; A) ∼ = K m+2 K m+3 -→ Γ m+2 π Γ m+3 π ∼ = Lie m+2 (H ),
which sends xK m+3 to ι # (x)Γ m+3 π for all x ∈ K m+2 . This map is compatible with the Lie bracket, in particular, the following diagram is commutative

(A ⊗ Lie m+1 (B; A)) ⊕ (B ⊗ Lie m+2 (B; A)) [ , ] / / ι * ⊗ι * Lie m+3 (B; A) ι * H ⊗ Lie m+2 (H ) [ , ]
/ / Lie m+3 (H ).

Whence, we have a well-defined homomorphism ι * : D m (B; A) → D m+1 (H ).

Proposition 5.11. For m ≥ 1, the diagram

J a m M ⊂ / / τ a m J L m+1 M τ L m+1 D m (B; A) ι * / / D m+1 (H )
is commutative. In other words, for J a m M, the homomorphism τ L m+1 is determined by the homomorphism τ a m . Proof. Let h ∈ J a m M. By considering the free basis {α i , β i } of π and the induced symplectic basis {a i , b i } of H, the (m + 1)-st Johnson-Levine homomorphism on h is given by

(5.23) τ L m+1 (h) = - g i=1 ι * (b i ) ⊗ ι # h # (α i )Γ m+3 π .
Applying ι * : D m (B; A) → D m+1 (H ) to Equation (5.16) we obtain exactly the left-hand side of (5.23), that is, ι * τ a m (h) = τ L m+1 (h). Remark 5.12. In general it is not easy to compare the alternative Johnson homomorphisms and the Johnson homomorphisms. In Lemma 6.9 we carry out the comparison between τ a 1 (ψ) and τ 1 (ψ) for ψ ∈ I ∩ I a . 5.3. Alternative Johnson homomorphism on L. In [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF], K. Habiro and G. Massuyeau defined, in a general context, a group homomorphism on L. In this subsection we study in detail this homomorphism, which we shall call here the 0-th alternative Johnson homomorphism.

An automorphism φ of Lie(B; A) is a family φ = (φ i ) i≥1 of group isomorphisms φ i :

Lie i (B; A) → Lie i (B; A), such that φ i+j ([x, y]) = [φ i (x), φ j (y)] for x ∈ Lie i (B; A) and y ∈ Lie j (B; A). Let Aut (Lie(B; A)) denote

the group of automorphisms of Lie(B; A).

Recall that for the N -series (K i ) i≥1 defined in (4.10), we have from Lemma 4.6 that

for h ∈ L, h (K i ) ⊆ K i for i ≥ 1. Here h x = h # (x) for x ∈ K i .
Definition 5.13. The 0-th alternative Johnson homomorphism is the group homomorphism (5.24) τ a 0 : L -→ Aut (Lie(B; A)) which sends h ∈ L to the family τ a 0 (h) = (τ a 0 (h) i ) i≥1 where

τ a 0 (h) i : Lie i (B; A) ∼ = K i K i+1 -→ K i K i+1 ∼ = Lie i (B; A) is defined by τ a 0 (h) i (xK i+1 ) = h # (x)K i+1 for x ∈ K i .
From the definition it follows that ker(τ a 0 ) = J a 1 M = I a . We refer to [15, Proposition 6.1] for a proof of the homomorphism property with the above definition. We will see an equivalent definition of τ a 0 in (5.31) and we prove the homomorphism property with the equivalent definition in Proposition 5. [START_REF] Hirose | The action of the handlebody group on the first homology group of the surface[END_REF].

Let us see how τ a 0 is related to the other alternative Johnson homomorphisms. First, for m ≥ 1 there is an action of L on J a m M by conjugation, that is, for h ∈ L, and

f ∈ J a m M, we set h f = hf h -1 ∈ J a m M.
f ∈ J a m M we have τ a m ( h f ) = τ a 0 (h) τ a m (f ).
Understanding the image of τ a 0 . Recall that Lie(B; A) is the free Lie algebra generated by B in degree 1 and A in degree 2. For h ∈ L the automorphism τ a 0 (h) : Lie(B; A) → Lie(B; A) is completely determined by its parts of degree 1 and degree 2:

(5.26) τ a 0 (h) 1 : Lie 1 (B; A) → Lie 1 (B; A) and τ a 0 (h) 2 : Lie 2 (B; A) → Lie 2 (B; A), but Lie 1 (B; A) = B and Lie 2 (B; A) = Λ 2 B ⊕ A. Consider the subgroup P of Aut(B) × Aut(Lie 2 (B; A)) defined by P = {(u, v) ∈ Aut(B) × Aut(Lie 2 (B; A)) | v ([x, y]) = [u(x), u(y)] ∀x, y ∈ B} .
Besides, consider the set

D = (u, v) ∈ Aut(B) × Hom(A, Lie 2 (B; A)) the map [x,y] + a -→ [u(x),u(y)] + v(a) is an automorphism of [B,B]⊕A = Lie2(B;A) .
For (u, v) ∈ D, define ṽ : Lie 2 (B, A) → Lie 2 (B, A) on the summands by ṽ ([x, y]) = [u(x), u(y)] and ṽ(a) = v(a), for x, y ∈ B and a ∈ A.

Lemma 5.15. There is a bijective correspondence Φ : P → D, which sends (u, v) ∈ P to (u, v |A ). This way, D inherits a group structure from P, so that the the product in D is given by

(5.27) (u 1 , v 1 )(u 2 , v 2 ) = Φ ((u 1 , ṽ1 )(u 2 , ṽ2 )) = Φ(u 1 u 2 , ṽ1 ṽ2 ) = u 1 u 2 , (ṽ 1 ṽ2 ) |A for (u 1 , v 1 ), (u 2 , v 2 ) ∈ D.
Proof. The inverse of Φ is defined by Φ -1 (u, v) = (u, ṽ) for (u, v) ∈ D.

If h ∈ L we have τ a 0 (h) 1 , (τ a 0 (h) 2 ) |A ∈ D.
Hence we can see the 0-th alternative Johnson homomorphism τ a 0 as taking values in D. We can still improve the target of τ a 0 . First, recall that if h ∈ L, the induced map h * : H → H is symplectic, see (4.2). Denote by ĥ * : H/A → H/A the homomorphism induced by h * . Hence τ a 0 (h) 1 = ι * ĥ * ι -1 * : H → H . The symplectic condition on h * implies that there is some information of (τ a 0 (h) 2 ) |A which is already encoded in τ a 0 (h) 1 . More precisely, we have

(5.28) (τ a 0 (h) 2 ) |A = ι * (τ a 0 (h) 2 ) |A + h * |A ∈ Hom(A, Λ 2 B ⊕ A) = Hom(A, Lie 2 (B; A)),
where ι * : Λ 2 B ⊕ A → Λ 2 B denotes the projection on Λ 2 B. For the moment we have that τ a 0 (h) is completely determined by the pair

τ a 0 (h) 1 , ι * (τ a 0 (h) 2 ) |A ∈ Aut(B) × Hom(A, Λ 2 B),
because h * |A and τ a 0 (h) 1 = ι * ĥ * ι -1 * determine each other by the symplectic condition on h * . The set Aut(B) × Hom(A, Λ 2 B) inherits the group structure (5.27) from D, which can be described explicitly as follows. Let h ∈ Aut(B). Using the identification H = B ∼ = A * , described in subsection 5.1, we obtain h ∈ Aut(A * ). Denote by h the automorphism of A such that (h ) * = h . Let µ ∈ Hom(A, Λ 2 B). We consider the following actions of Aut(B) on Hom(A, Λ 2 B)

right action: µ • h := µ • h ∈ Hom(A, Λ 2 B), left action: h • µ := Λ 2 h • µ ∈ Hom(A, Λ 2 B). (5.29)
We have that the product in Aut(B) × Hom(A, Λ 2 B) inherited from (5.27) is given by

(5.30) (h, µ)(f, ν) = (hf, h • ν + µ • f ) for h, f ∈ Aut(B) and µ, ν ∈ Hom(A, Λ 2 B).
Let Aut(B) ×Hom(A, Λ 2 B) denote the set Aut(B) × Hom(A, Λ 2 B) with the product given in (5.30). Hence we can see the 0-th Johnson homomorphism (5.24) as the map (5.31) τ a 0 : L -→ Aut(B) ×Hom(A, Λ 2 B),

which sends h ∈ L to the pair τ a 0 (h) 1 , ι * (τ a 0 (h) 2 ) |A . Moreover, the homomorphism ι * (τ a 0 (h) 2 ) |A takes a ∈ A to ι # h # (α)Γ 3 π
where α ∈ A is any lift of a, and we identify Γ 2 π /Γ 3 π with Λ 2 H = Λ 2 B. In this context we can show the homomorphism property of τ a 0 .

Proposition 5.16. The map τ a 0 : L → Aut(B) ×Hom(A, Λ 2 B) is a group homomorphism and its kernel is the second term J L 2 M of the Johnson-Levine filtration. In particular we have

I a = J a 1 M = J L 2 M. Proof. Let h, f ∈ L. Clearly we have τ a 0 (hf ) 1 = τ a 0 (h) 1 τ a 0 (f ) 1 . Identify Γ 2 π /Γ 3 π with Λ 2 B. Set µ = ι * (τ a 0 (h) 2 ) |A , ν = ι * (τ a 0 (f ) 2 ) |A and κ = ι * (τ a 0 (hf ) 2 ) |A . Let us see that κ = τ a 0 (h) 1 • ν + µ • τ a 0 (f ) 1 .
Let a ∈ A and α ∈ A with ab(α) = a. By Lemma 4.6 we can write f # (α) = βy with β ∈ A and y ∈ Γ 2 π. We have

ab(β) = ab(βy) = ab(f # (α)) = f * (ab(α)) = f * (a). Hence κ = ι # (h # (f # (α)))Γ 3 π = ι # (h # (β))Γ 3 π + ι # (h # (y))Γ 3 π = µ(ab(β)) + ι # (h # (y))Γ 3 π = µ(f * (a)) + Λ 2 h(ν(a)) = (µ • τ a 0 (f ) 1 )(a) + (τ 0 (h) 1 • ν)(a) = (τ 0 (h) 1 • ν)(a) + (µ • τ a 0 (f ) 1 )(a). Whence κ = τ a 0 (h) 1 • ν + µ • τ a 0 (f ) 1 . Thus τ a 0 : L -→ Aut(B) ×Hom(A, Λ 2 B) is a group homomorphism. Now, let h ∈ ker(τ a 0 ), thus τ 0 (h) 1 = Id H . From the symplectic condition we have h * |A = Id A , so h ∈ I L . Let α ∈ A, hence ι # h # (α)Γ 3 π = ι * (τ a 0 (h) 2 ) (ab(α)) = Γ 3 π , that is, ι # h # (α) ∈ Γ 3 π for all α ∈ A, so that h ∈ J L 2 M.
The next proposition follows from the definition (5.31) of τ a 0 and shows that for the elements of I L , the 0-th alternative Johnson homomorphism determines the first Johnson-Levine homomorphism.

Proposition 5.17. Let q : Aut(B) ×Hom(A, Λ 2 B) → Hom(A, Λ 2 B) denote the cartesian projection (which is not a group homomorphism). Then, the diagram

I L ⊂ / / τ L 1 L τ a 0 Hom(A, Λ 2 B) Aut(B) ×Hom(A, Λ 2 B) q o o is commutative.
Remark 5.18. There is a right split short exact sequence

0 Hom(A, Λ 2 B) Aut(B) ×Hom(A, Λ 2 B) Aut(B) 1, j p s
where j(µ) = (Id B , µ), p(h, µ) = h and s(h) = (h, 0) for µ ∈ Hom(A, Λ 2 B) and h ∈ Aut(B). Therefore, the group Aut(B) ×Hom(A, Λ 2 B) is isomorphic to the semidirect product Aut(B) Hom(A, Λ 2 B), where the action of Aut(B) on Hom(A, Λ 2 B) is given by h

* µ = h • µ • h -1 for h ∈ Aut(B) and µ ∈ Hom(A, Λ 2 B).
Here the • means the left and right actions defined in (5.29). The explicit isomorphism

Θ : Aut(B) ×Hom(A, Λ 2 B) -→ Aut(B) Hom(A, Λ 2 B), is given by Θ(h, µ) = (h, µ • h -1 ) for h ∈ Aut(B) and µ ∈ Hom(A, Λ 2 B).
We can do yet another refinement of the target of τ a 0 . Considering the Definition 5.13 we have that τ a 0 (h) 3 (Ω ) = Ω for h ∈ L, where Ω ∈ Lie 3 (B; A) is determined by the intersection form (5.7).

Notice that a pair (h, κ) ∈ Aut(B) × Hom(A, Lie 2 (B; A)) uniquely determines a morphism of Lie algebras (h, κ) : Lie(B; A) → Lie(B; A). Hence Ω is given as in Equation (5.11). If R = ( kj ) is the matrix of h in the basis {ι * (b i )} and P = (λ ij ) is the the matrix of h in the basis {a i }, then

P T R = Id g . Thus (h, h )(Ω ) = g j=1 [h (a j ), h(b j )] = g j=1 g i=1 λ ij a i , g k=1 kj b k = g i=1 g k=1 g j=1 λ ij kj [a i , b k ] = g j=1 [a j , b j ] = Ω . Let (h, µ) ∈ Aut(B) ×Hom(A, Λ 2 B). Let (h i ) i≥1 be the associated automorphism of Lie(B; A). Explicitly we have h 1 = h and h 2 = µ + h where h ∈ Aut(A) is determined by h. Lemma 5.20. The condition h 3 (Ω ) = Ω holds if and only if (h, µ)(Ω ) = 0.
Proof. We use bases as in Lemma 5.19. We have

h 3 (Ω ) = g j=1 [h 2 (a j ), h 1 (b j )] = g j=1 [µ(a j ), h(b j )] + g j=1 [h (a j ), h(b j )] = (h, µ)(Ω ) + (h, h )(Ω ) = (h, µ)(Ω ) + Ω .
Last equality comes from Lemma 5.19. Whence the desired result.

Notice that a pair (h, µ) ∈ Aut(B) ×Hom(A, Λ 2 B), determines an element

µ h ∈ B ⊗ Λ 2 B through the identification Hom(A, Λ 2 B) ∼ = A * ⊗ Λ 2 B ∼ = B ⊗ Λ 2 B h⊗Id Λ 2 B -------→ B ⊗ Λ 2 B, Thus µ h := (h ⊗ Id Λ 2 B )(µ). Set (5.32) G := (h, µ) ∈ Aut(B) ×Hom(A, Λ 2 B) | Ξ 3 (µ h ) = 0 , where Ξ 3 : B ⊗ Lie 2 (B) → Lie 3 (B)
is the Lie bracket. Using bases as in Lemma 5.19 we have

(5.33) G =    (h, µ) ∈ Aut(B) ×Hom(A, Λ 2 B) | Ξ 3 g j=1 h(b j ) ⊗ µ(a j ) = 0    . Proposition 5.21. The set G is a subgroup of Aut(B) ×Hom(A, Λ 2 B).
Proof. The result can be deduced from Lemma 5.20 or from the description of G given in (5.33) as follows. Let (h, µ),

(f, ν) ∈ G. Let us see that (h, µ)(f, ν) = (hf, h • ν + µ • f ) and (h, µ) -1 = (h -1 , -h -1 • µ • h -1 ) belong to G. We have Ξ 3 g j=1 hf (b j ) ⊗ (h • ν + µ • f )(a j ) = g j=1 hf (b j ), Λ 2 h(ν(a j )) + µf (a j ) = Lie 3 (h) g j=1 [f (b j ), ν(a j )] + g j=1 [h(f (b j )), µ(f (a j ))] = g j=1 [h(b j ), µ(a j )] = 0,
where Lie 3 (h) : Lie 3 (B) → Lie 3 (B) is the isomorphism induced by h. The equality

g j=1 [h(f (b j )), µ(f (a j ))] = g j=1 [h(b j ), µ(a j )]
is deduced in a similar way as we did in the proof of Lemma 5.19. Therefore (h, µ)(f, ν) ∈ G. On the other hand

Ξ 3 g j=1 h -1 (b j ) ⊗ (-h -1 • µ • h -1 )(a j ) = g j=1 h -1 (b j ), -Λ 2 h -1 (µ(h -1 ) (a j )) = -Lie 3 (h -1 ) g j=1 [h(h -1 (b j )), µ(h -1 ) (a j )] = -Lie 3 (h -1 ) g j=1 [h(b j ), µ(a j )] = 0. Hence (h, µ) -1 ∈ G.
Lemma 5.22. The 0-th alternative Johnson homomorphism defined in (5.31) takes its values in G.

Proof. Let h ∈ L. Set µ = ι * (τ a 0 (h) 2 ) |A . Hence Ξ 3 g j=1 τ a 0 (h) 1 (b j ) ⊗ µ(a j ) = - g j=1 Ξ 3 ι * h * (-b j ) ⊗ ι # h # (α j )Γ 3 π = - g j=1 Ξ 3 ι # h # (β -1 j )Γ 2 π ⊗ ι # h # (α j )Γ 3 π =   ι # h # g j=1 [β -1 j , α j ] Γ 4 π   -1 = Γ 4 π = 0 ∈ Lie 3 (B).
To sum up, we can write (5.34) τ a 0 : L -→ G. Theorem 5.23. The 0-th alternative Johnson homomorphism τ a 0 : L → G is surjective. Proof. Notice that the diagram

I ⊂ / / τ 1 I L τ L 1 D 1 (H) ι * / / D 1 (H )
is commutative, this can be shown by writing τ 1 and τ L 1 by using a symplectic basis as we did for τ L m+1 in Equation 5.23, see [START_REF] Levine | The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres[END_REF]Section 4] for more details. The map

ι * : D 1 (H) → D 1 (H ) is induced by ι * : H → H . It is easy to show that ι * : D 1 (H) → D 1 (H )
σ(L) = P Q 0 (P T ) -1 ∈ Sp(2g, Z) P -1 Q is symmetric . Let (f, µ) ∈ G. From (5.35), it follows that there is h ∈ L such that τ a 0 (h) 1 = f . Let ν = ι * (τ a 0 (h) 2 ) |A ∈ Hom(A, Λ 2 B). Hence τ a 0 (h) = (f, ν) ∈ G. Consider the element µ = f -1 • (µ -ν) ∈ Hom(A, Λ 2 H ).
Recall that B = H . Let us see that µ ∈ D 1 (H ). Indeed, if Ξ : H ⊗ Lie 2 (H ) → Lie 3 (H ) denotes the Lie bracket, we have

Ξ(µ ) = -Ξ g j=1 b j ⊗ (f -1 • (µ -ν))(a j ) = - g j=1 [b j , Λ 2 f -1 µ(a j )] + g j=1 [b j , Λ 2 f -1 ν(a j )] - g j=1 [f -1 (f (b j )), Λ 2 f -1 µ(a j )] + g j=1 [f -1 (f (b j )), Λ 2 f -1 ν(a j )] = -Lie 3 (f -1 ) j=1 [f (b j ), µ(a j )] + Lie 3 (f -1 ) g j=1 [f (b j ), ν(a j )] = 0.
Whence µ ∈ D 1 (H ). By the surjectivity of τ L 1 and Proposition 5.17, there exists g ∈ I L such that τ a 0 (g) = (Id H , µ ). Therefore

τ a 0 (hg) = τ a 0 (h)τ a 0 (g) = (f, ν)(Id H , µ ) = (f, ν)(Id H , f -1 • (µ -ν)) = (f, f • (f -1 • (µ -ν)) + ν) = (f, µ).
Hence we have the surjectivity of τ a 0 : L → G.

Corollary 5.24. We have the following short exact sequence ---→ Lie m+3 (B; A) .

1 -→ I a ⊂ --→ L τ a 0 ---→ G -→ 1.
The rational versions D m (H) ⊗ Q, D m (H ) ⊗ Q and D m (B; A) ⊗ Q can be interpreted as subspaces of the spaces of connected tree-like Jacobi diagrams A t,c (H), A t,c (H ) and A t,c (B ⊕ A), respectively. See Example 2.3 for the definition. Recall that these spaces are graded by the internal degree. Notice that as spaces A t,c (H) = A t,c (B ⊕ A) but we would like to give a special role to A in the latter space, which will be reflected in a different grading of the space A t,c (B ⊕ A). Let us start by recalling this interpretation.

For a connected tree-like Jacobi diagram

T in A t,c (H) = A t,c (B ⊕ A) or in A t,c (H ), set (5.36) η(T ) = v color(v) ⊗ (T rooted at v),
where the sum ranges over the set of legs (univalent vertices) of T and we interpret a rooted tree as a Lie commutator. Denote by A t,c m (H) the subspace of A t,c (H) generated by diagrams of internal degree m. So if T ∈ A t,c m (H), then T has m + 2 legs and therefore by rooting T at one of its legs we obtain a rooted tree with m + 1 leafs. To sum up, η(T ) ∈ H ⊗ Lie m+1 (H). Moreover if we apply the Lie bracket Ξ : H ⊗ Lie m+1 → Lie m+2 (H) to η(T ), we obtain Ξη(T ) = 0. This way η(T ) ∈ D m (H). The following result is well known. Theorem 5.26. For m ≥ 1 the map (5.37) η : A t,c m (H) -→ D m (H) ⊗ Q, defined as in Equation (5.36), is an isomorphism of Q-vector spaces.

We refer to [START_REF] Levine | Addendum and correction to: "Homology cylinders: an enlargement of the mapping class group[END_REF]Corollary 3.2] or [START_REF] Habegger | Tree level Lie algebra structures of perturbative invariants[END_REF]Theorem 1] for a proof of Theorem 5. [START_REF] Quoc | Representation of the category of tangles by Kontsevich's iterated integral[END_REF].

In particular we have an isomorphism of graded Q-vector spaces (5.38) η :

m≥1 A t,c m (H) -→ m≥1 D m (H) ⊗ Q.
The same statements hold replacing H by H . We define a degree for connected tree-like Jacobi diagrams, which we call alternative degree and denote by a-deg, such that if Here #S denotes the cardinal of the set S.

T ∈ A t,c (B ⊕ A) is such that a-deg(T ) = m then η(T ) ∈ D m (B; A) ⊗ Q. In
In Table 1 we show some examples of tree-like Jacobi diagrams organized by their internal degree in the columns and by the alternative degree in the rows In this case T is a strut whose both legs are colored by elements of A, say a i , a j . Then η(T ) = a i ⊗ a j + a j ⊗ a i , so Ξη(T ) = 0 by the antisymmetry relation. This way for m ≥ 1 we have η T a m (B ⊕ A) ⊆ D m (B; A) ⊗ Q and by (5.38), the map η| T a m (B⊕A) is injective (it is again necessary to consider the case m = 1 separately). For the surjectivity, first consider the case m = 1. The elements in (A ⊗ A) ∩ D 1 (B; A) are linear combinations of elements of the form a i ⊗ a i and a i ⊗ a j + a j ⊗ a i . Now if T is the strut whose both Table 1. legs are colored by a i , then (1/2)η(T ) = a i ⊗ a i and if T is the strut whose legs are colored by a i and a j , then η(T ) = a i ⊗ a j + a j ⊗ a i .

Let m ≥ 1 and consider y ∈ D m (B; A). In the case m = 1, by the previous paragraph, we can suppose that there are no elements of (A ⊗ A) ∩ D 1 (B; A) appearing in y. This way we can see y ∈ m≥1 D m (H) ⊗ Q. By (5.38), there exists T ∈ m≥1 A t,c m (H) such that η(T ) = y. Consider the decomposition of T by the alternative degree

T = T i with T i ∈ T a i (B ⊕ A). Thus η(T ) = η(T i ) = y, but for i = m we know that η(T i ) ∈ D m (B; A) ⊗ Q. Hence η(T i ) = 0 for i = m.
J m M τm ---→ D m (H) ⊗ Q η -1
---→ A t,c m (H) and

(5.42) , where Λ and ∆ are g × g matrices and ∆ is symmetric.

J L m M τ L m ---→ D m (H ) ⊗ Q η -1 ---→ A
We refer to [START_REF] Vera | Johnson-Levine homomorphisms and the tree reduction of the LMO functor[END_REF]Lemma 3.7] or [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF]Lemma 2.12] for a proof. Definition 6.2. The monoid IC a of alternative homology cylinders of Σ is defined as

IC a = {(M, m) ∈ IC L | ∀α ∈ A : ι # ρ 2 (M )(αΓ 3 π) = 1 ∈ π /Γ 3 π }.
Here ι # : π/Γ 3 π → π /Γ 3 π is induced by ι # : π → π .

Notice that the given definition of IC a is motivated by the definition of J L 2 M. There is an equivalent definition motivated by the definition of I a , see Proposition 5.16. Recall that for M a Lagrangian cobordism, Z Y,t (M ) denotes the reduction of the value Z(M ) modulo struts and looped diagrams. Definition 6.4. The alternative tree filtration {F a m C} m≥1 of C is defined by

F a m C = {(M, m) ∈ IC a | Z Y,t (M ) = ∅ + (terms of a-deg ≥ m)}. Let T Y,a
m ( g + g -) denote the subspace of A Y,t ( g + g -) generated by diagrams of a-deg = m. Recall that the square brackets denote an exponential. Let us write

Z Y,t (M • N ) = ∅ + E + (a-deg > m),
where E is a linear combination of connected Jacobi diagrams in A Y,t ( g + ∪ g -) with a-deg ≤ m. We want to show that E only has diagrams with a-deg = m and moreover that E = D M + D N . Since Z Y,t (M • N ) is obtained by considering the reduction of Z(M • N ) modulo struts and looped diagrams, we need to carefully analyse the pairing (6.6). • The diagrams of type do not contribute any connected term to Z Y,t (M •N ). Therefore, the diagrams (which are no struts) with the lowest alternative degree contributed by the diagrams (6.7) after the pairing (6.6) are exactly the diagrams appearing in D N .

• The diagrams of type (6.8) can contribute looped diagrams to Z Y (M • N ) when we consider their pairing with connected diagrams in Z Y,t (M ), so at the end they do not appear in Z Y,t (M • N ). Or they can also contribute connected diagrams to Z Y (M • N ) after their pairing with disconnected diagrams T of Z t (M ), where at least one of the connected components of T has at least one trivalent vertex. In Figure 6.1 we illustrate this situation with three examples of such a T . Let us see that in this case, the obtained connected tree-like Jacobi diagrams are of a-deg > m. Let T be a disconnected diagram in Z t (M ). Then the connected components of T can be struts or diagrams with at least one trivalent vertex. If there are diagrams in T whose all legs are colored by elements of g -, then it is not possible to obtain a connected diagram from T after the pairing (6.6), hence we can suppose that there is not this type of diagrams in T . Also note that if all the legs of all the connected components of T are colored by elements of g + , then after the pairing (6.6) with the struts of type (6.8), we will obtain looped diagrams. This way we can suppose that there is at least one connected component of T which has one leg colored by an element of g -, and moreover that all the struts appearing in T have one leg colored by g + and the other by g -. Let T 1 be a connected component of T with at least one trivalent vertex, then a-deg(T 1 ) ≥ m. Now, T 1 has legs colored by g + and when we do the pairing with the struts of the type (6.8), we connect such legs either with struts which have legs colored by g -or with other trees appearing in T . In either case, we strictly increase the alternative degree of T 1 . To sum up, the connected tree-like Jacobi diagrams which can appear in this case are of a-deg > m.

Therefore, the diagrams (which are not struts) with the lowest alternative degree contributed by the diagrams (6.9) after the pairing (6.6), are exactly the diagrams appearing in D M .

• Let S be a diagram appearing in Z t (N ) and T be a diagram appearing in Z t (M ). If all the legs of every connected component of S are colored by g + , then S does not intervene in the pairing (6.6), except with the empty diagram. Similarly when all the legs of every connected component of T are colored by g -. Hence we can suppose that there is at least a connected component of S (respectively in T ) with at least one trivalent vertex and with at least one g --colored leg (respectively one g + -colored leg). As in the previous case, the connected diagrams without loops obtained from the pairing of S and T strictly increase the alternative degree. The alternative degree only remains stable when we do the pairing with diagrams coming from

In conclusion, the lower alternative degree terms appearing in Z Y,t (M • N ) are exactly D M + D N , that is, E = D M + D N . 6.2. First alternative Johnson homomorphism and the LMO functor. From Definition 5.29, the diagrammatic version of the first alternative Johnson homomorphism of an element in I a is given by a linear combination of the diagrams shown in Figure 6.2. Recall that by a -(respectively by a +) we mean that the color of the leg belongs to A (respectively to B). Besides, the diagrammatic version of the first Johnson homomorphism of an element in I is given by a linear combination of the diagrams shown in Figure 6.3 and the diagrammatic version of the second Johnson-Levine homomorphism is given by a linear combination of diagrams of type (c) in Figure 6.2.

Let us start by identifying the elements in I a whose first alternative Johnson homomorphism only contains diagrams of the type (a) in Figure 6.2. Let N be the subgroup of I a generated by the Dehn twists t α i and t α kl with 1 ≤ i ≤ g and 1 ≤ k < l ≤ g. Here Proof. Let ψ ∈ I ∩ I a . From equation (5.16) we can write (6.15)

τ a 1 (ψ) = g i=1 a i ⊗ ψ # (β i )β -1 i K 3 - g i=1 b i ⊗ ψ # (α i )α -1 i K 4 , besides (6.16) τ 1 (ψ) = g i=1 a i ⊗ ψ # (β i )β -1 i Γ 3 π - g i=1 b i ⊗ ψ # (α i )α -1 i Γ 3 π .
By applying p to (6.15) and q to (6.16) we obtain

pτ a 1 (ψ) = g i=1 a i ⊗ ι # (ψ # (β i )β -1 i )Γ 3 π - g i=1 b i ⊗ ψ # (α i )α -1 i K 4 mod Lie 3 (B) , and
qτ 1 (ψ) = g i=1 a i ⊗ ι # (ψ # (β i )β -1 i )Γ 3 π - g i=1 b i ⊗ ψ # (α i )α -1 i Γ 3 π mod (Λ 2 A + Λ 2 B) .
Thus we need to show that

(ψ(α i )α -1 i K 4 ) mod Lie 3 (B) and (ψ(α i )α -1 i Γ 3 π) mod (Λ 2 A + Λ 2 B) define the same element in A ∧ B. By Lemma 4.12, we can write ψ(α i )α -1 i = y i n i with y i ∈ Γ 3 π ⊆ K 3 and n i ∈ A. Since ψ(α i )α -1 i ∈ Γ 2 π, then n i ∈ Γ 2 π ∩ A = [π, A] ⊆ K 3 . Therefore ψ(α i )α -1 i Γ 3 π = n i Γ 3 π = k<l λ i kl (a k ∧ a l ) + k<l i kl (b k ∧ b l ) + k,l δ i kl (a k ∧ b l ) ∈ Λ 2 H, where λ i kl , i kl , δ i kl ∈ Z. Thus (ψ(α i )α -1 i Γ 3 π) mod (Λ 2 A + Λ 2 B) = k,l δ i kl (a k ∧ b l ) ∈ A ∧ B.
On the other hand ψ(α

i )α -1 i K 4 = y i K 4 + n i K 4 , but y i ∈ Γ 3 π, hence y i K 4 ∈ Lie 3 (B). Therefore we also have (ψ(α i )α -1 i K 4 ) mod Lie 3 (B) = (y i K 4 ) mod Lie 3 (B) + (n i K 4 ) mod Lie 3 (B) = k,l δ i kl (a k ∧ b l ) ∈ A ∧ B.
The first alternative Johnson homomorphism and the LMO functor. In order to relate τ a 1 with the LMO functor we need particular cases of the two theorems that say how the Johnson homomorphisms and the Johnson-Levine homomorphisms are related to the LMO functor. For h ∈ L denote by Z Y,t,+ (c(h)) the element in A Y,t ( g + ) obtained from Z Y (c(h)) by sending all terms with loops or with i --colored legs to 0. Theorem 6.10. [START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF]Corollary 5.11

] Let m ≥ 1. If h ∈ J m M then Z Y,t (c(h)) = ∅ -η -1 τ m (c(h)) |a j →j -,b j →j + + (i-deg > m).
and by Theorem 6.11

(6.19) -η -1 τ L 2 (ψ) b j →j + = Z Y,t,+ (c(ψ)) |i-deg=2 = Z Y,t (c(ψ)) |a-deg=1,i-deg=2 . Therefore Z Y,a 1 (c(f )) = Z Y,a 1 (c(h)) + Z Y,a 1 (c(ψ)) = Z Y,t (c(ψ)) a-deg=1,i-deg=1 + Z Y,t (c(ψ)) |a-deg=1,i-deg=2 = -η -1 q(τ 1 (ψ)) |a j →j -,b j →j + -η -1 τ L 2 (ψ) b j →j + = -η -1 pτ a 1 (ψ) + τ L 2 (ψ) |a j →j -,b j →j + = -η -1 τ a 1 (ψ) |a j →j -,b j →j + = -η -1 τ a 1 (f ) Y |a j →j -,b j →j +
In the first equality we use Theorem 6.5, in the second we use Lemma 6.13, in the third we use (6.18) and (6.19), in the fourth we use Lemma 6.9 and the homomorphism property of η -1 , and in the fifth we use (6.13). Finally in the sixth equality we use (6.17).

Remark 6.15. Theorems 6.10 and 6.11 are valid in the context of homology cobordisms. This suggests that the first alternative Johnson homomorphism could be defined on IC a and that the statement of Theorem 6.14 could be generalized to IC a . It is very likely possible to read the 0-th alternative Johnson homomorphism τ a 0 (h) for h ∈ L in the a-deg = 0 part of Z t (c(h)). 6.3. Higher alternative Johnson homomorphisms and the LMO functor. The aim of this subsection is to prove an analogue of Theorem 6.14 for τ a m with m ≥ 2. That is, we want to prove the following.

Theorem 6.16. Let m ≥ 2. If f ∈ J a m M, then Z Y,t (c(f )) = ∅ -η -1 τ a m (f ) |a j →j -,b j →j + + (a-deg > m). Or equivalently Z Y,a m (c(f )) = -η -1 τ a m (f ) |a j →j -,b j →j + .
An immediate consequence of the above theorem is the following.

Corollary 6.17. For f ∈ J a m M the value Z Y,a m (c(f )) j + →b j ,j -→a j ∈ T a m (B ⊕ A) is independent of the choice of a Drinfeld associator.

One of the key points in the proof of Theorem 6.16 is to show that the LMO functor defines an alternative symplectic expansion of π. We use several results and definitions from [START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF] and [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF].

Alternative symplectic expansions. Let H

Q be the Q-module H 1 (Σ, Q) = H ⊗ Q. Denote by T (H Q ) the free associative Q-algebra generated by H Q in degree 1, that is, T (H Q ) is the tensor algebra of H Q and let T (H Q ) denote its degree completion. Let A Q = A⊗Q and B Q = B⊗Q. Let T (B Q ; A Q ) be the free associative Q-algebra generated of Q[π] indexed by non-negative integers such that F m F n ⊆ F m+n for m, n ≥ 0. Let I be the augmentation ideal of Q[π]. For m ≥ 1 it is well known that I m = (x 1 -1) • • • (x p -1) | p ≥ 1, x i ∈ Γ m i π and m 1 + • • • + m p ≥ m ,
where the angle brackets stand for the generated subspace of Q

[π]. For m ≥ 1, set R m = (x 1 -1) • • • (x p -1) | p ≥ 1, x i ∈ K m i and m 1 + • • • + m p ≥ m .
This way we have the filtrations {I m } m≥0 and {R m } m≥0 of Q[π] where we set

I 0 = R 0 = Q[π]. These filtrations define inverse systems {Q[π]/I m } m and {Q[π]/R m } m .
Consider the I-adic and R-adic completions of Q[π], that is, the inverse limits

Q[Γ + ] = lim ← - m (Q[π]/I m ) and Q[K + ] = lim ← - m (Q[π]/R m ).
Notice that Q[Γ + ] and Q[K + ] are filtered complete Hopf algebras, with filtrations { I m } m≥0 and { R m } m≥0 defined by

I m = lim ← - l (I m /I l ) and R m = lim ← - l (R m /R l ),
for m ≥ 0. From now on, let θ : π → T (B Q ; A Q ) be an alternative expansion of π relative to A and denote by θ : π → T (H Q ) the associated group-like expansion of π. The Quillen's description [START_REF] Daniel | On the associated graded ring of a group ring[END_REF] of the associated graded of the filtered ring Q[π] with respect to {I m } m , can be generalized [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF]Theorem 11.2] to describe the associated graded of Q[π] with respect to {R m } m . Moreover, we have the following. U / / ρ (h)

Q[K + ] ρ(h) Q[Γ + ] U / / Q[K + ]
is commutative. Besides, using Proposition 6.24, we define = ρ θ (h).

In the second equality we use (6.20), and in the third equality we use Proposition 6.25.

From Proposition 4.11 we have J a 2 M ⊆ J 1 M = I. Hence, by considering the restriction of ρ θ to I and the restriction of ρ θ to J a 2 M and using Proposition 6.27 we obtain the following.

where D m (Lie(B Q , A Q )) is defined by considering the rational version of Equation (5.8). The subscripts m in the right-hand side of the above equations denote the terms of degree m in θ (h) |B⊕A and θ (f ) |H , respectively. Remark 6.30. Notice that in the left-hand sides of (6.22) and (6.23) we are actually considering the rationalization of the Johnson-type homomorphims, but there is no loss of information by doing this. Furthermore, if θ is symplectic, then the maps θ and θ take values in the completions Der

Lie(B

Q ; A Q ) ∼ = m D m (B Q ; A Q ) ∼ = T a (B ⊕ A),
where the first isomorphism is given by Proposition 5.

Lie(H

Q ) ∼ = m D m (H Q ) ∼ = A t,c (H),
where the second isomorphism is given by Theorem 5.26. This way, with the hypothesis that θ is symplectic, Theorem 6.29 can be restated by saying that for m ≥ 1 the diagrams T a (B ⊕ A) A t,c (H) is commutative.

The LMO functor defines a symplectic alternative expansion. We want to apply diagrams (6.24) and (6.25) with a particular symplectic alternative expansion of π relative to A. In [32, Section 5], G. Massuyeau constructed a symplectic expansion of π from the LMO functor. It turns out that this expansion is a symplectic alternative expansion of π relative to A. Let us recall this construction. Fix two points p, q ∈ int(Σ). A bottom knot in Σ × [-1, 1] is the isotopy class (relative to the boundary) of a connected framed oriented tangle starting at q × {-1} and ending at p × {-1}, see Figure 6 Two bottom knots K, K ∈ B are said to be homotopic, denoted K K , if K can be obtained from K by framing changes and a finite number of crossing changes. This relation is compatible with the monoid structure, that is, if K, K , L, L ∈ B are such that K K and L L , then K • L K • L . There is a canonical monoid morphism Now, a bottom knot K ∈ B gives rise to an element in LCob(g, g + 1) by "digging" along K, more precisely, let (M K , m) be the cobordism obtained from Σ × [-1, 1] by removing a tubular neighborhood of K and define the parametrization m on the first handle of the bottom surface of M K by using the framing of K and as the identity elsewhere. We continue to denote the cobordism (M K , m) by K and we endow its top and bottom with the right-handed non-associative words as in Convention 3.13. This way we have K ∈ LCob q (g, g + 1) and we can apply the LMO functor to it, to obtain Z(K) ∈ ts A(g, g + 1) ⊆ A( g + g + 1 -). Change the colors in Z(K) as follows 1 -→ r and i -→ (i -1) -, ∀i = 2, . . . , g + 1; so that, the variable r refers to the bottom knot. Thus Z(K) ∈ A( g + g -{r}).

Example 6.31. In Example 3.12 we considered a cobordism N i ∈ s LCob(g, g + 1), for i = 1, . . . , g, with bottom-top tangle presentation shown in Figure 3.8. Analyzing carefully Figure 3.8 we see that the cobordism N i corresponds to the bottom knot, also denoted N i , in Σ × [-1, 1] such that (N i ) is the homotopy class of the meridian α i in Σ.

Recall the space H(r) defined in Example 2.4. Hence, Z(K) mod H(r) is an exponential series of tree-like Jacobi diagrams with at most one r-colored leg and which only depends on the homotopy class of K, see [START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF]Lemma 5.5]. Moreover, the series consisting of the terms without r-colored legs in Z(K) mod H(r) is exactly the identity morphism in ts A(g, g). To sum up is a series of connected tree-like Jacobi diagrams with legs colored by g + g - {r} and with exactly one r-colored leg. Hence, we can see S Z (K) as an element in Lie(B Q , A Q ) after the replacement of colors i + → b i and i -→ a i for i = 1, . . . , g. The map LCob q (g, g)

-log Z Y,t

x x A t,c (H) is commutative. Where c denotes the cylinder map and for h ∈ I we endow the top and bottom of c(h) with the right-handed non-associative words as in Convention 3.13. In order to see log Z(c(h)) as an element of A t,c (H) we consider the change of colors i + → b i and i -→ a i for i = 1, . . . , g. We obtain the desired result by putting together the commutative diagrams (6.24), (6.25) and (6.28).

Remark 6.33. In fact [START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF]Theorem 5.13] is more general than the commutativity of diagram (6.28); it says that for every homology cylinder M ∈ IC we have η -1 θ (M ) = -log Z Y,t (M ) . Besides, Theorem 6.5 is proved in the setting of homology cobordisms. This suggests that our results could be generalized to the setting of homology cobordisms. More precisely, we expect that the alternative Johnson filtration and the alternative Johnson homomorphisms extend to homology cobordisms and that the diagrammatic version of such alternative Johnson homomorphisms can be read in the tree reduction of the LMO functor.

  The m-th Johnson-Levine homomorphism τ L m : J L m M → D m (B) is defined on J L m M and it takes values in the kernel D m (B) of the Lie bracket [ , ] : B ⊗ Lie m+1 (B) → Lie m+2 (B).

  an orientation-preserving homeomorphism such that the top and bottom restrictions m ± | Σ×{±1} : Σ × {±1} → M of m induce isomorphisms in homology. Denote by C the monoid of homology cobordims of Σ (or C g,1 where g is the genus of Σ). In particular, if h ∈ M, we can consider the homology cobordism c(h) := (Σ × [-1, 1], m h ) where m h is such that m h + = h and m h -= Id Σ . Moreover, h ∈ L if and only if the cobordism c(h) is a Lagrangian cobordism. Thus c(h) belongs to the source category of the LMO functor and therefore we can compute Z(c(h)). The alternative Johnson homomorphisms motivate the definition of the alternative degree, denoted a-deg, for connected tree-like Jacobi diagrams. If T is a tree-like Jacobi diagram colored by B ⊕ A, then a-deg(T ) = 2|T A | + |T B | -3, where |T A | (respectively |T B |) denotes the number of univalent vertices of T colored by A (respectively by B). See Figure 1.1 (a) and (b) for some examples.

Figure 1 . 1 .

 11 Figure 1.1. Tree-like Jacobi diagrams of a-deg = 3 in (a) and of a-deg = 1 in (b). (c) Looped Jacobi diagram. Here a, a ∈ A and b, b , b 1 , . . . , b 4 ∈ B.

Theorem D .

 . The alternative degree induces a filtration {F a m C} m≥1 of C by submonoids. Consider the map Z Y,a m :F a m C -→ T Y,a m (B ⊕ A), where Z Y,am (M ) is defined as the Jacobi diagrams with at least one trivalent vertex and of

Figure 2 . 1 .

 21 Figure 2.1. Jacobi diagrams with X = in (a), X = ↓ ↓ in (b) and X = ∅ in (c). Here C = {1, 2, 3}.

  AS, IHX, Q-multilinearity , where the relations STU, AS, IHX are local and the multilinearity relation applies to the colored legs. See Figure 2.2.

Figure 2 . 2 .

 22 Figure 2.2. Relations on Jacobi diagrams.

  denoted i-deg(D), e-deg(D) and deg(D) respectively, as i-deg(D) := number of trivalent vertices of D, e-deg(D) := number of legs of D, deg(D) := 1 2 (i-deg(D) + e-deg(D)).

2. 2 .

 2 Operations on Jacobi diagrams. Let us recall some operations on the spaces of Jacobi diagrams. Hopf algebra structure. There is a product in A(C) given by disjoint union, with unit the empty diagram, and a coproduct defined by ∆(D) = D ⊗ D where the sum ranges over pairs of subdiagrams D , D of D such that D D = D. For instance:

Figure 2 . 3 .

 23 Figure 2.3. (a) Strut, (b) Jacobi diagram in A( 4 + 3 -), (c) Treelike Jacobi diagram. Here a, b, c, d ∈ C where C is any finite set.

Figure 2 . 4 .

 24 Figure 2.4. Definition of the doubling map and box notation.

Figure 2 . 5 .

 25 Figure 2.5. Definition of orientation-reversal map. Here we suppose that there are k legs attached to the chosen interval.

Theorem 2 . 5 .

 25 [START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF] Theorem 8] Let n ≥ 1 and let D be a Jacobi diagram on (X ↑ s , C) with at most l ≤ n legs attached to ↑ s . Denote by D the Jacobi diagram on (X, C {s}) obtained from D by erasing ↑ s and coloring with s all the legs that were attached to ↑ s . Set σ 1 (D) = D and for n > 1

Figure ( 3 . 1 )

 31 (b) and (c) for two different parenthesizations of the same framed tangle.

Figure 3 . 1 .

 31 Figure 3.1. A framed tangle and two different q-tangles obtained from it.

3 . 3 . 4 .

 334 Using Examples 3.1, 3.2 and Equations (3.1) we have Example Recall the space H(↑ r ) defined in Example 2.4. We have 3.2.

  an orientation-preserving homeomorphism, such that the bottom and top inclusions m ± (•) := m(•, ±1) : Σ g,1 → M induce isomorphisms in homology. Two pairs (M, m) and (M , m ) are equivalent if there exists an orientationpreserving homeomorphism ϕ : M → M such that ϕ • m = m . The composition (M, m)•(M , m ) of two homology cobordisms (M, m) and (M , m ) of Σ g,1 is the equivalence class of the pair ( M , m -∪m + ), where M is obtained by gluing the two 3-manifolds

Example 3 . 6 .

 36 In Figure 3.3 we illustrate the procedure to obtain the bottom-top tangle presentation of the trivial cobordism Σ g,1 × [-1, 1].

Figure 3 . 3 .

 33 Figure 3.3. Obtaining the bottom-top tangle presentation of the trivial cobordism Σ g,1 × [-1, 1].

  Figure 3.5 (a) shows the bottom-top tangle presentation of the trivial cobordism Σ g,1 × [-1, 1] (in thin line) together with a (-1)-framed knot (in thick line) such that the surgery along this knot gives the bottom-top tangle presentation of c(t α i ) showed in Figure 3.5 (b). Notice that going from Figure 3.5 (a) to Figure 3.5 (b) is exactly a Fenn-Rourke move.

Figure 3 . 5 .

 35 Figure 3.5. Bottom-top tangle presentation of c(t α i ).

Figure 3 .

 3 7 (b) shows the bottom-top tangle presentation of c(t α kl ).

Figure 3 . 6 .

 36 Figure 3.6. (a) Curve α 12 and (c) bottom-top tangle presentation of c(t α 12 ).

Figure 3 . 7 .

 37 Figure 3.7. (a) Curve α kl and (b) bottom-top tangle presentation of c(t α kl ).

Figure 3 . 8 .

 38 Figure 3.8. Bottom-top tangle presentation of N i ∈ s LCob(g, g + 1).

Figure 3 . 9 .

 39 Figure 3.9. (a) Composition in ts A and (b) identity morphism in ts A(g, g).

Figure 3 . 10 .

 310 Figure 3.10. Bottom-top tangle presentation of M .

Figure 3 . 11 .

 311 Figure 3.11. Value of Z(M ) in terms of Z(T ).

  3. Now, by a straightforward but long computation we obtainExample 3.18. Example 3.17 can be generalized to the cobordism c(t α kl ) from Example 3.11. In this case we obtain Example 3.19. Consider the special Lagrangian cobordism N 1 from Example 3.12, equipped with non-associative words as in Convention 3.13. Denote by w the righthanded non-associative word in • of length g -1. Denote by P •,•,w the q-cobordism ((••)w) → (•(•w)) whose underlying cobordism is the identity LCob(g + 1, g + 1). Thus we can decompose N 1 as N 1 = P •,•,w • (T ⊗ Id w ), where T is the special Lagrangian cobordism whose bottom-top tangle presentation is shown in Figure3.12.

Figure 3 . 12 .

 312 Figure 3.12. Bottom top-tangle presentation of T .

4. 2 .

 2 Alternative Torelli group. Let V (or V g ) be a handlebody of genus g. Consider a disk D on ∂V such that ∂V = Σ∪D, where D and Σ are glued along their boundaries. Let ι : Σ → V be the inclusion of Σ into ∂V ⊆ V , see Figure4.1.

Figure 4 . 1 .

 41 Figure 4.1. The inclusion Σ ι -→ V .

Figure 4 .

 4 Figure 4.1 also shows the fixed system of meridians and parallels of Σ used in subsection 3.2. Moreover we suppose that the images ι(α i ) of the meridians α i , under the embedding ι, bound pairwise disjoint disks in V . Set H = H 1 (V ; Z) and π = π 1 (V, ι( * )) and denote by ab : π → H the abelianization map. Consider the following subgroups of π and H that arise when looking at the induced maps by ι in homotopy and in homology:

Example 4 . 4 .

 44 The Dehn twists t α i and t α kl from Examples 3.9 and 3.11 are elements of the alternative Torelli group which do not belong to the Torelli group. Example 4.5. Consider the parallel β 1 and the curve γ as shown in Figure 4.2. These curves form a bounding pair. Consider the Dehn twists t β 1 and t γ along these curves. It can be shown that the homeomorphism t γ t -1 β 1 belongs to I ∩ I a .

Figure 4 . 2 .

 42 Figure 4.2. Curves β 1 and γ.

4. 3 .

 3 Alternative Johnson filtration. This subsection is devoted to the study of a filtration of the alternative Torelli group introduced in[START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF] which we shall call here the alternative Johnson filtration. We compare this filtration with the Johnson filtration and the Johnson-Levine filtration. Let us start by recalling some terminology.An N -series (G m ) m≥1 of a group G is a decreasing sequence

Definition 4 . 7 .

 47 The alternative Johnson filtration of M is the descending chain of subgroups {J a m M} m≥0 . Example 4.8. Consider the curves δ, and the meridian α g as show in Figure 4.3. It can be show that t δ and t t -1

Figure 4 . 3 .Proposition 4 . 9 .

 4349 Figure 4.3. Curves δ, and α g .

Proposition 4 .

 4 10. [34, Corollary 3.3] The Johnson filtration satisfies the following properties.

Johnson-

  Levine filtration. J. Levine introduced in [29, 31] a different filtration of the mapping class group by means of the embedding ι : Σ → V , see Figure 4.1, and the lower central series of π = π 1 (V, ι( * )). The Johnson-Levine filtration of M is the descending chain of subgroups (4.18)

(5. 5 )

 5 Gr(K • ) ∼ = Lie(B; A). Positive symplectic derivations of Lie(B; A). Recall that a derivation of Lie(B; A) is a linear map d : Lie(B; A) → Lie(B; A) such that d([x, y]) = [d(x), y] + [x, d(y)] for every x, y ∈ Lie(B; A). The set Der(Lie(B, A)) of derivations of Lie(B; A) is a Lie algebra with Lie bracket [d, d ] = dd -d d.

Definition 5 . 1 .Proposition 5 . 2 . 5

 51525 B and by sending x + A ∈ H/A to ω(x, •) ∈ A * ; and A ∼ = B * given by the isomorphism H 2 (V ; ∂V ; Z) ∼ = A and by sending a ∈ A to ω (•, a) ∈ B * . This way, the intersection form ω determines an element Ω ∈ Lie 3 (B; A). The intersection form ω : H ⊗ H → Z determines an element Ω ∈ Lie 2 (H) ⊆ Lie(H), where Lie(H) is the graded Lie algebra freely generated by H in degree 1. The relation between the intersection form ω of Σ and ω of V is given by the commutativity of the diagram:B × H 2 (V, ∂V ) Let d be a derivation of Lie(B; A). (i) We say d is a positive derivation if d(B) ⊆ Lie ≥2 (B; A) and d(A) ⊆ Lie ≥3 (B; A). (ii) Let m ≥ 1. We say that d is a derivation of degree m if d(B) ⊆ Lie m+1 (B; A) and d(A) ⊆ Lie m+2 (B; A). (iii) We say that d is a symplectic derivation if d(Ω ) = 0.Denote by Der +,ω (Lie(B; A)) the set of positive symplectic derivations of Lie(B; A). This set is a Lie subalgebra of Der(Lie(B; A)). Let m ≥ 1, denote by Der m (Lie(B; A)) the subgroup of of derivations of Lie(B; A) of degree m. Notice that a derivation d of Lie(B; A) of degree m is a family d = (d i ) i≥1 of group homomorphisms d i : Lie i (B; A) -→ Lie i+m (B; A), satisfying d i+j [x, y] = [d i (x), y] + [x, d j (y)] for x ∈ Lie i (B; A) and y ∈ Lie j (B; A). Set (5.8) D m (Lie(B; A)) = Hom Z (B, Lie m+1 (B; A)) ⊕ Hom Z (A, Lie m+2 (B; A)). The following is a classical result, see for instance [40, Lemma 0.7]. For every m ≥ 1, there is a bijection Der m (Lie(B; A)) Ψ --→ D m (Lie(B; A)), defined by Ψ(d) = d |B + d |A for d ∈ Der m (Lie(B; A)). By using the identifications B ∼ = A * and A ∼ = B * , we have D m (Lie(B; A)) ∼ = (B * ⊗ Lie m+1 (B; A)) ⊕ (A * ⊗ Lie m+2 (B; A)) ∼ = (A ⊗ Lie m+1 (B; A)) ⊕ (B ⊗ Lie m+2 (B; A)). (5.9) Hence we can see the map Ψ from Proposition 5.2 as taking values in the space on the left-hand side of Equation (5.9). For m ≥ 1, consider the Lie bracket map (5.10) Ξ m : (A ⊗ Lie m+1 (B; A)) ⊕ (B ⊗ Lie m+2 (B; A)) -→ Lie m+3 (B; A). Set D m (B; A) := ker(Ξ m ). Denote by Der +,ω m (Lie(B; A)) the subgroup of positive symplectic derivations of degree m. Proposition 5.3. Let d ∈ Der + m (Lie(B; A)). Then ΞΨ(d) = 0 if and only if d(Ω ) = 0. That is Der +,ω m (Lie(B; A)) ∼ = D m (B; A). Proof. Consider the symplectic basis {a i , b i } induced by the systems of meridians and parallels {α i , β i } on Σ shown in Figure 3.2 and identify ι * (b i ) ∈ B with b i ∈ H. Then, in this basis, the element Ω is given by (, b i ] ∈ Lie 3 (B; A).

Example 5 . 6 .

 56 Consider the Dehn twist h = t α 12 from Examples 4.4 and 3.10, which is an element of I a . The homotopy class of the curve α 12 is represented by λ

Example 5 . 7 .

 57 Consider the Dehn twist h = t δ , from Example 4.8. We have that t δ ∈ I a . The homotopy class of the curve δ is represented by the commutator λ

  a version of the Johnson homomorphisms for the Johnson-Levine filtration. Identify H/A with A * by sending x + A ∈ H/A to ω(x, •) ∈ A * and H/A with H via the isomorphism ι * . The m-th Johnson-Levine homomorphism

1 .

 1 On the other hand, there is an action of Aut(Lie(B; A)) on the group Der m (Lie(B; A)) of derivations of degree m of Lie(B; A). Let φ ∈ Aut(Lie(B; A)) and d ∈ Der m (Lie(B; A)), set (5.25) φ d = φdφ -More precisely, ( φ d) i (x) = φ m+i d i φ -1 i (x) for x ∈ Lie i (B; A) and i ≥ 1. The following is an instance of a part of [15, Theorem 6.4]. Proposition 5.14. Let m ≥ 1 and h ∈ L. The m-th alternative Johnson homomorphism τ a m : J a m M → Der m (B; A) satisfies the following equivariant property: for every

Lemma 5 . 19 .

 519 Let h ∈ Aut(B) and let h : A → A ⊆ Lie 2 (B; A) be the automorphism of A determined by h. Then (h, h )(Ω ) = Ω . Proof. Consider the bases of H and B as in Proposition 5.3 and identify ι * (b i ) with b i .

5. 4 .

 4 Diagrammatic versions of the Johnson-type homomorphisms. In subsection 5.2 we have seen that for m ≥ 1, the m-th Johnson homomorphism, the m-th Johnson-Levine homomorphism and the m-th alternative Johnson homomorphism take values in the abelian groups D m (H), D m (H ) = D m (B) and D m (B; A), respectively. These spaces were defined as D m (H) = ker H ⊗ Lie m+1 (H) [ , ] ---→ Lie m+2 (H) , D m (H ) = ker H ⊗ Lie m+1 (H ) [ , ] ---→ Lie m+2 (H ) and D m (B; A) = ker (A ⊗ Lie m+1 (B; A)) ⊕ (B ⊗ Lie m+2 (B; A)) [ , ]

Example 5 . 25 .

 525 Consider the tree where a, a ∈ A and b, b ∈ B. Hence, We have that η(T 0 ) ∈ H ⊗ Lie 3 (H) and η(T 0 ) ∈ A ⊗ Lie 4 (B; A) ⊕ B ⊗ Lie 5 (B; A) . Moreover, by the Jacobi identity, if we apply the Lie bracket Ξ to η(T 0 ) we obtain Ξη(T 0 ) = 0. Therefore η(T 0 ) ∈ D 2 (H) and η(T 0 ) ∈ D 3 (B; A).

  Example 5.25, η(T 0 ) ∈ D 3 (B; A), so we want a-deg(T 0 ) = 3. Definition 5.27. Let T be a connected tree-like Jacobi diagram with legs colored by B ⊕ A. The alternative degree of T , denoted a-deg(T ), is defined as a-deg(T ) = 2#{A-colored legs of T } + #{B-colored legs of T } -3.

  . The legs colored by + (respectively by -) in the diagrams represent legs colored by elements of B (respectively of A). Notice that a strut diagram D whose both legs are colored by elements of B is such that a-deg(D) = -1.For m ≥ 1, let T a m (B ⊕ A) denote the subspace of A t,c (B ⊕ A) generated by diagrams of alternative degree m. Proposition 5.28. For m ≥ 1 the map η defined in (5.36) induces an isomorphism(5.39) η : T a m (B ⊕ A) -→ D m (B; A) ⊗ Q of Q-vector spaces.Proof. Let T be a (B ⊕ A)-colored connected tree-like Jacobi diagram with a-deg(T ) = m. Let v be a leg of T and denote by T v the Lie commutator obtained from T rooted at v. If v is colored by an element of B, then deg(T v ) = (a-deg(T ) + 3) -1 = m + 2. Hence color(v) ⊗ T v ∈ B ⊗ Lie m+2 (B; A). On the other hand, if v is colored by an element of A then deg(T v ) = (a-deg(T )+3)-2 = m + 1. Therefore color(v) ⊗ T v ∈ A ⊗ Lie m+1 (B; A).To sum up η(T ) ∈ (A ⊗ Lie m+1 (B; A)) ⊕ (B ⊗ Lie m+2 (B; A). The argument [30, Lemma 3.1] used in the proof of Theorem 5.26 to show that Ξη(T ) = 0 is still valid. The only caveat is when a-deg(T ) = 1 and i-deg(T ) = 0.

1 -

 1 By the injectivity of η, we obtain T i = 0 for i = m. Therefore T = T m ∈ T a m (B ⊕ A) and η(T ) = y. Theorem 5.26 and Proposition 5.28 allow to define diagrammatic versions of the Johnson-type homomorphisms. Definition 5.29. Let m ≥ 1. The diagrammatic version of the m-th alternative Johnson homomorphism is defined as the composition(→ D m (B; A) ⊗ Q η ---→ T a m (B ⊕ A).Similarly, the diagrammatic versions of the m-th Johnson homomorphism and of the m-th Johnson-Levine homomorphism are defined as the compositions(5.41)

Example 6 . 3 .

 63 If c : M → C is the mapping cylinder monoid homomorphism, then c(L) ⊆ LC, c(I) ⊆ IC, c(I L ) ⊆ IC L and c(I a ) ⊆ IC a .

Theorem 6 . 5 .

 65 For m ≥ 1, the set F a m C is a submonoid of C. Consider the map Z Y,a m : F a m C -→ T Y,a m ( g + g -),where Z Y,a m (M ) is defined as the terms of a-deg= m in Z Y,t (M ) for M ∈ F a m C. Then Z Y,a m is a monoid homomorphism. Proof. Let M, N ∈ F a m C and write Z Y,t (M ) = ∅ + D M + (a-deg > m) and Z Y,t (N ) = ∅ + D N + (a-deg > m), where D M and D N are linear combinations of connected Jacobi diagrams in A Y,t ( g + g -) of a-deg = m. We have to show that M • N ∈ F a m C and that (6.4) Z Y,t (M • N ) = ∅ + (D M + D N ) + (a-deg > m).where Λ = (m ij ) and ∆ = (n ij ) are symmetric g × g matrices. We have(6.5) Z(M • N ) = Z(M ) • Z(N ) = Z(M ) |j + →j * , Z(N ) |j -→j * g * . By Lemma 3.15 we can write (6.6)

•

  It is possible for Z Y (M ) or Z Y (N ) to have diagrams with loops. A diagram in Z(M • N ) coming from the pairing of a diagram with loops, in Z Y (M ) or in Z Y (N ), with any other diagram will still have loops. Hence the diagrams with loops in Z Y (M ) or in Z Y (N ) do no contribute any term to Z Y,t (M • N ).

Figure 6

 6 Figure 6.1.

Proposition 6 . 24 .T

 624 [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF] Proposition 12.2][32, Proposition 2.10] The maps θ and θ extend uniquely to complete Hopf algebra isomorphismsθ : Q[K + ] -→ T (B Q ; A Q ) and θ : Q[Γ + ] -→ T (H Q ),which are the identity at the graded level.Since Γ m π ⊆ K m ⊆ Γ m/2 π for m ≥ 1, see (4.10), then I m ⊆ R m and R m ⊆ I m/2. Hence, the identity automorphism of π induces a morphism of inverse systems{u m : Q[π]/I m -→ Q[π]/R m } which induces an isomorphism U : Q[Γ + ] -→ Q[K + ]of complete Hopf algebras. The following two results are straightforward. Proposition 6.25. The diagramQ[Γ + ] (H) T (B; A) is commutative.For a complete Hopf algebra F denote by Aut(F ) the group of automorphisms of F . Corollary 6.26. The diagramAut( Q[Γ + ])Recall that we can restrict the Dehn-Nielsen representation (4.1) to the Lagrangian mapping class group L.Now, if h ∈ L, we have ρ(h)(Γ m π) = h # (Γ m π) ⊆ Γ m π and ρ(h)(K m ) = h # (K m ) ⊆ K m ,see Lemma 4.6. This way, we obtain representations ρ : L -→ Aut( Q[Γ + ]) and ρ : L -→ Aut( Q[K + ]). Notice that for h ∈ L, the diagram (6.20) Q[Γ + ]

θ 1 =

 1 ρ θ : L -→ Aut( T (H Q )) and ρ θ : L -→ Aut( T (B Q ; A Q )), by ρ θ (h) = θ ρ (h) θ -1 and ρ θ(h) = θ ρ(h) θ-1 for h ∈ L. Aut T (H Q ) Aut T (B Q ; A Q ) is commutative. Proof. Let h ∈ L, then ρ θ(h) = θ ρ(h) θ-θU ρ (h)U -1 θ-1 = θ ρ (h) θ -1

  Lie(BQ ; A Q ) and Der +,ω Lie(H Q ) of positive symplectic derivations of Lie(B Q ; A Q ) and Lie(H Q ), respectively. Now Der +,ω

  2 and the second by Proposition 5.28. Similarly Der +,ω

  c (H) are commutative. In the same way, if θ is symplectic, Corollary 6.28 can be restated by saying that

  .4 (a) for an example. Let B denote the set of bottom knots in Σ × [-1, 1]. There is a monoid structure in B. If K, L ∈ B, then K • L is the bottom knot obtained from K and L by joining K and L as shown in Figure 6.4 (b).

Figure 6 . 4 .

 64 Figure 6.4. (a) Bottom knot in Σ × [-1, 1] and (b) monoid structure in B.

( 6

 6 .26) : B/ -→ π, which assigns to the homotopy class of K ∈ B the based loop in Σ × [-1, 1] as shown in Figure 6.5. Then identify π with π 1 (Σ × [-1, 1], * ), see [32, Lemma 5.3].

Figure 6 . 5 .

 65 Figure 6.5. The based loop (K).

( 6 .

 6 27)θ Z : π -→ T (B Q ; A Q ),defined by θ Z ( (K)) = exp ⊗ S Z (K) is a symplectic expansion of π, see[START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF] Proposition 5.6]. Proposition 6.32. The symplectic expansion θZ : π → T (B Q ; A Q ) satisfies θ Z (α) = 1 + {α} + (a-deg > 2)for all α ∈ A. Therefore θ Z is a symplectic alternative expansion of π relative to A.Proof. By[START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF] Proposition 5.6] we know that for every α ∈ A,θ Z (α) = 1 + {α} + (deg > 1).First, notice that a tree of i-deg > 1 with one leg colored by r (the root) and the other legs colored by elements of B ⊕ A gives rise to a Lie commutator in Lie(B Q , A Q ) of a-deg > 2. Therefore, we only need to calculate the terms of i-deg = 1 in S Z (K) for bottom knots K such that (K) ∈ A.Now A is the normal closure of the subgroup α i | i = 1, . . . , g of π generated by the homotopy classes of the meridians and by Example 6.31, the cobordism N i is such that (N i ) is the homotopy class of α i . Therefore, by the homomorphism property of Z i-deg =1 , it is enough to calculate the terms of i-deg = 1 in Z(N i ) mod H(r) and see whether they give rise to Lie commutators in Lie(B Q , A Q ) of a-deg = 1. In Example 3.19 we see that each of the terms with i-deg = 1 in Z(N i ) mod H(r) has one r-colored leg and one i --colored leg, thus the associated commutator has a-deg > 2 which completes the proof.Proof of Theorem 6.16. Let θ denote the symplectic alternative expansion of π relative to A defined by the LMO functor and denote by θ the associated symplectic expansion of π. In [32, Theorem 5.13] G. Massuyeau proved that the diagram (

  Alternative Johnson homomorphism on L 5.4. Diagrammatic versions of the Johnson-type homomorphisms 6. Alternative Johnson homomorphisms and the LMO functor 6.1. The filtration on Lagrangian cobordisms induced by the alternative degree 6.2. First alternative Johnson homomorphism and the LMO functor 6.3. Higher alternative Johnson homomorphisms and the LMO functor References 1. Introduction

	1. Introduction
	2. Spaces of Jacobi diagrams and their operations
	2.1. Generalities
	2.2. Operations on Jacobi diagrams
	3. The Kontsevich integral and the LMO functor
	3.1. Kontsevich Integral
	3.2. The LMO functor
	4. Johnson-type filtrations
	4.1. Preliminaries
	4.2. Alternative Torelli group
	4.3. Alternative Johnson filtration
	5. Johnson-type homomorphisms
	5.1. Preliminaries
	5.2. Alternative Johnson homomorphisms
	5.3.

  see[START_REF] Griffiths | Automorphisms of a 3-dimensional handlebody[END_REF] Theorem 10.1]. The group H is called the handlebody group because it is isomorphic to the mapping class group of V .

	Proposition 4.11. (Levine [29, 31]) The Johnson-Levine filtration satisfies the follow-
	ing properties.

  is surjective and it is well known that the first Johnson homomorphism τ 1 is surjective [17, Theorem 1]. Hence τ L 1 is surjective. Using the symplectic basis {a i , b i } to identify Sp(H) with Sp(2g, Z) we have that the image of the L under the symplectic representation (4.2) is (5.35)

  M belongs to IC L g,1 if and only if B is a homology cube and Lk(M ) = (iii) M belongs to IC g,1 if and only if B is a homology cube and Lk(M ) =

	(ii) 0 Idg Idg ∆ ,
	0 Idg
	Idg 0
	t,c m (H ),

Institut de Recherche Mathématique Avancée, Université de Strasbourg 7 rue René Descartes, 67084 Strasbourg, France E-mail address: vera@math.unistra.fr

respectively.

Example 5.30. In Example 5.5 we calculated τ a 1 (t α i ) = -a i ⊗ a i , for the Dehn twist t α i from Example 4.4. Therefore Example 5.31. In Example 5.6 we calculated τ a 1 (t α kl ) = -(a k ⊗ a k ) -(a l ⊗ a l ) -(a k ⊗ a l ) -(a l ⊗ a k ), for the Dehn twist t α kl from Example 4.4. Hence Comparing these results with the low degree values of the LMO functor on the cobordisms c(t α i ) and c(t α kl ) computed in Examples 3.16, 3.17 and 3.18, we can see that, for these examples, the diagrammatic version of the first alternative Johnson homomorphism appears in the LMO functor with an opposite sign. This is a more general fact which we develop in next section.

Alternative Johnson homomorphisms and the LMO functor

In this section we establish the relation between the LMO functor and the alternative Johnson homomorphisms. From Proposition 4.13, we know that for m ≥ 2, J a m M ⊆ I. Hence, we can use some known results involving the Torelli group. Therefore we carry this out in two stages separately. First we establish this relation for the alternative Johnson homomorphism τ a 1 . Then we consider τ a m for m ≥ 2. First of all let us start by defining the filtration on cobordisms induced by the alternative degree. The monoid IC of homology cylinders of Σ is defined as IC = ker(ρ 1 ). The monoid LC of Lagrangian homology cobordisms of Σ is defined as

and the monoid IC L of strongly Lagrangian homology cobordisms of Σ is defined as

The monoids LC, IC L and IC are characterized in terms of the linking matrix as follows.

Lemma 6.1. Let M ∈ C g,1 and let (B, γ) be its bottom-top tangle presentation. Then α i denotes the i-th meridional curve as in Figure 3.2; and α kl is as shown in Figure 3.7 (a). Using the symplectic basis {a i , b i } to identify Sp(H) with Sp(2g, Z) we have that the image of N under the symplectic representation (4.2) is (6.10)

Equality (6.10) is precisely [START_REF] Levine | The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres[END_REF]Lemma 6.3]. It also follows from the computations made in Examples 5.5 and 5.6. Notice that N is contained in the handlebody group H defined in (4.20).

Lemma 6.6. We have the equality

Proof. Let h ∈ I a . In the symplectic basis {a i , b i }, the matrix of

with ∆ a symmetric matrix. From (6.10) there exists f ∈ N such that the matrix of

We have already computed in Examples 5.5 and 5.6 the first alternative Johnson homomorphism for the generators of N . These computations imply the following. Proposition 6.7. For h ∈ N the first alternative Johnson homomorphism τ a 1 (h) can be computed from the action of h in homology and reciprocally. More precisely, if the matrix of σ

Proof. By Examples 5.5 and 5.6, the result holds for the generators t α i and t α kl of N . The general result follows from the homomorphism property of τ a 1 and the equality

, for all matrices ∆ and ∆ of size g × g. Whence we have the desired result from the homomorphism property of τ a 1 and η -1 . (Equality (6.11) can also be deduced from the description of the composition of cobordisms in terms of their bottom-top tangle presentations).

At this point, we understand the first alternative Johnson homomorphism for the elements of N and moreover we know that the diagrammatic version only contains diagrams of type (a) in Figure 6.2. By Lemma 6.6, in order to understand τ a 1 for all I a we need to understand it for the elements of I ∩ I a . Recall that B = H .

Consider the projection (6.12) p : ). Thus, we can apply p to τ a 1 (ψ). Diagrammatically, when we apply p we kill the diagrams in η -1 (τ a 1 (ψ)) of type (c) in Figure 6.2. Besides, by Proposition 5.11, we have that ι * (τ a 1 (ψ)) = τ L 2 (ψ). Here the map ι * is the map appearing in Proposition 5.11. Diagrammatically we are killing all the diagrams in η -1 (τ a 1 (ψ)) with at least one leg colored by an element of A. This way, we have (6.13)

On the other hand we can also consider the first Johnson homomorphism τ 1 (ψ) of ψ. Hence, the diagrammatic version of τ 1 (ψ) is a linear of the diagrams shown in Figure 6.3. We want to compare τ 1 (ψ) and τ a 1 (ψ). Thus, we need to kill the diagrams of type (a), (c) and (d) in Figure 6.3 from η -1 τ 1 (ψ). For this, consider the projection (6.14)

which in the first direct summand is given by the Id B tensored with the projection, and in the second direct summand is given by Id A ⊗ Λ 2 ι * , where Λ 2 ι * : Λ 2 H → Λ 2 H is induced by ι * : H → H . Lemma 6.9. Via the canonical isomorphism

we have p(τ a 1 (ψ)) = q(τ 1 (ψ)) for every ψ ∈ I ∩ I a . Theorem 6.11. [START_REF] Vera | Johnson-Levine homomorphisms and the tree reduction of the LMO functor[END_REF]Theorem 5.4

Remark 6.12. We state Theorems 6.10 and 6.11 in the context of the mapping class group, but the original versions are stated in the context of homology cobordisms.

1 (c(h)) = 0. Proof. The diagrams with a-deg = 1 have i-deg between 0 and 2 included. In Examples 3.16, 3.17 and 3.18 we have seen that for the generators h ∈ N there are no diagrams of a-deg = 1 and i-deg = 1 in Z Y,t (c(h)). Now, the diagrams of a-deg = 1 and i-deg = 2 are of type (c) in Figure 6.2. Since N is included in the handlebody group, this kind of diagrams do not appear in Z Y,t , see [START_REF] Cheptea | A functorial LMO invariant for Lagrangian cobordisms[END_REF]Corollary 5.4]. Therefore we have the stated result for the generators of N and the general statement follows by Theorem 6.5.

. By Lemma 6.6 we can write f = hψ with h ∈ N and ψ ∈ I ∩ I a . From (6.13) and Lemma 6.9 we have

. By Lemma 3.15 and Proposition 6.8 we have where η -1 τ a 1 (f ) s is the reduction of η -1 τ a 1 (f ) modulo diagrams with i-deg ≥ 1. On the other hand, by Theorem 6.10

by B Q in degree 1 and A Q in degree 2. We call the induced degree in T (B Q , A Q ) the alternative degree. Hence

Remark 6.20. Definition 6.19 implies that for i ≥ 1 and

Hence an alternative expansion of π relative to A is an expansion of the N -series (K i ) i≥1 in the sense of [START_REF] Habiro | Generalized Johnson homomorphisms for extended N-series[END_REF]Section 12].

Notice that T (H Q ) = T (B Q ; A Q ) as Q-algebras as soon as we have chosen a section of ι * : H → H with isotropic image and identified B = H to the image of this section. Thus if θ is an alternative expansion of π relative to A, then, in particular, θ is a group-like expansion of π. We denote this group-like expansion by θ . In [START_REF] Massuyeau | Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant[END_REF]Lemma 2.16] Massuyeau shows that symplectic expansions exist by "deforming" the expansion of Example 6.21. It can be verified that the constructed symplectic expansion in that lemma is actually an alternative symplectic expansion of π relative to A. We will see that the LMO functor also gives an example of an alternative symplectic expansion of π relative to A.

Completions of the group algebra Q[π]. Let Q[π] be the group Q-algebra of π. We have considered two N -series of π; the lower central series Γ + = (Γ m π) m≥1 and the N -series K + = (K m ) m≥1 defined in (4.10). Each one of these defines a filtration of Q[π], that is, a decreasing sequence