On the low dimensional cohomology groups of the IA-automorphism group of a free group of rank three

In this paper we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm{IA}_3$ of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of $\mathrm{IA}_3$, which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm{IA}_3$ has finite index in that of the Andreadakis-Johnson filtration.


Introduction
Let F n be a free group of rank n ≥ 2 with basis x 1 , . . . , x n , and Aut F n the automorphism group of F n . As far as we know, the first contribution to the study of the (co)homology groups of Aut F n was given by Nielsen [36] in 1924, who showed H 1 (Aut F n , Z) = Z/2Z for n ≥ 2 by using a presentation for Aut F n . Now we have a broad range of results for the (co)homology groups of Aut F n due to many authors. In 1984, Gersten [16] showed H 2 (Aut F n , Z) = Z/2Z for n ≥ 5 In 1980s, by introducing the Outer space, Culler and Vogtmann [10] made a breakthrough in the computation of homology groups of the outer automorphism groups Out F n of free groups F n . To put it briefly, the Outer space is an analogue of the Teichmüller space on which the mapping class of a surface naturally acts. By using the geometry of the Outer space, Hatcher and Vogtmann [17] computed H 4 (Aut F 4 , Q) = Q, and On the other hand, by using sophisticated homotopy theory, Galatius [14] showed that the stable integral homology groups of Aut F n are isomorphic to those of the symmetric group S n of degree n. In particular, the stable rational homology groups H q (Aut F n , Q) are trivial for n ≥ 2q + 1. This result is a quite contrast to the case of the mapping class groups of surfaces. Intuitively, we can see this from the fact that the free group has no geometric extra structure like surface groups.
With respect to unstable cohomology groups, Aut F n behave in much different and mysterious way. The unstable cohomology groups of the (outer) automorphism groups of free groups has also been studied by many authors. For unstable case, the Outer space 6. Acknowledgments 23 References 23

Preliminaries
In this section, after fixing some notation and conventions, we recall the IA-automorphism groups, Free Lie algebras, the Andreadakis-Johnson filtration and the representation theory of the general linear group over Q.

Notation and conventions.
Let G be a group.
• The abelianization of G is denoted by G ab .
• The automorphism group Aut G acts on G from the right. For any σ ∈ Aut G and x ∈ G, the action of σ on x is denoted by x σ . • For a normal subgroup N, we often denote the coset class of an element g ∈ G by the same g in the quotient group G/N if there is no confusion. • For elements x and y of G, the commutator bracket [x, y] of x and y is defined to be [x, y] := xyx −1 y −1 . Then for any x, y, z ∈ G, we have For elements g 1 , . . . , g k ∈ G, a simple k-fold commutator [[· · · [[y 1 , y 2 ], y 3 ], · · · ], y k ] is denoted by [y 1 , y 2 , · · · , y k ] for simplicity. For subgroups H and K of G, we denote by [H, K] the commutator subgroup of G generated by [h, k] for h ∈ H and k ∈ K.
• For any Z-module M, we denote M ⊗ Z Q by the symbol obtained by attaching a subscript or a superscript Q to M, like M Q or M Q . Similarly, for any Z-linear map f : A → B, the induced Q-linear map f ⊗ id Q : A Q → B Q is denoted by f Q or f Q .

IA-automorphism groups.
Fix a basis x 1 , . . . , x n of a free group F n . We denote by H the abelianization H 1 (F n , Z) of F n . Let ρ : Aut F n → Aut H be the natural homomorphism induced from the abelianization of F n . We identify Aut H with the general linear group GL(n, Z) by fixing the basis of H induced from the basis x 1 , . . . , x n of F n . The kernel IA n of ρ is called the IA-automorphism group of F n . It is clear that the inner automorphism group Inn F n of F n is contained in IA n . Nielsen [35] showed that IA 2 = Inn F 2 . For n ≥ 3, IA n is much larger than Inn F n . In fact, Magnus [25] showed that IA n is finitely generated by automorphisms for distinct i, j ∈ {1, 2, . . . , n} and for distinct i, j, l ∈ {1, 2, . . . , n} such that j > l. For any 1 ≤ i ≤ n, set ι i := K 1i K 2i · · · K ni . Namely, ι i is the inner automorphism of F n given by x → x −1 i xx i . Cohen-Pakianathan [7,8]CFarb [13] and Kawazumi [21] independently showed as a GL(n, Z)-module where H * := Hom Z (H, Z) is the Z-linear dual group of H. This fact is obtained from the above result of Magnus and the first Johnson homomorphism which is defined below.
2.3. Free Lie algebra generated by H.
Let Γ n (1) ⊃ Γ n (2) ⊃ · · · be the lower central series of a free group F n defined by the rule We denote by L n (k) := Γ n (k)/Γ n (k + 1) the graded quotient of the lower central series of F n , and by L n := k≥1 L n (k) the associated graded sum. See L n (1) = H. Since the group Aut F n naturally acts on L n (k) for each k ≥ 1, and since IA n acts on it trivially, the action of GL(n, Z) on each L n (k) is well-defined. Furthermore, the graded sum L n naturally has a graded Lie algebra structure induced from the commutator bracket on F n , and is isomorphic to the free Lie algebra generated by H. (See [26] and [40] for basic material concerning the free Lie algebra.) It is well known due to Witt [45] that each L n (k) is a GL(n, Z)-equivariant free abelian group of rank where µ is the Möbius function, and d runs over all positive divisors of k. For example, the GL(n, Z)-module structure of L n (k) for 1 ≤ k ≤ 3 is given by Hall [18] constructed an explicit basis of L n (k). More precisely, he introduced basic commutators of F n , and showed that (the coset classes of) basic commutators of weight k form a basis of L n (k). For example, basic commutators of weight less than four are listed below. k basic commutators [19] for details for the basic commutators of the free groups.)

Andreadakis-Johnson filtration.
In this subsection, we recall the Andreadakis-Johnson filtration and Johnson homomorphisms of Aut F n . For each k ≥ 1, the action of Aut F n on the nilpotent quotient group F n /Γ n (k + 1) of F n induces a homomorphism Aut F n → Aut(F n /Γ n (k + 1)).
We denote its kernel by A n (k). Then the groups A n (k) define a descending central filtration of IA n . We call this filtration the Andreadakis-Johnson filtration of Aut F n . Andreadakis showed that (1) For any k, l ≥ 1, σ ∈ A n (k) and x ∈ Γ n (l), x −1 x σ ∈ Γ n (k + l).
(2) For any k and l ≥ 1, For each k ≥ 1, the group Aut F n acts on A n (k) by conjugation, and it naturally induces an action of GL(n, Z) = Aut F n /IA n on the graded quotients gr k (A n ) := A n (k)/A n (k + 1) by Part (2) of Theorem 2.1. The graded sum gr(A n ) := k≥1 gr k (A n ) has a graded Lie algebra structure induced from the commutator bracket on IA n .
The homomorphismsτ k and τ k are called the k-th Johnson homomorphisms of Aut F n . Each τ k is GL(n, Z)-equivariant homomorphism. For the Magnus generators of IA n , their images by τ 1 are given by Hence τ 1 is surjective. From this fact, we see that the first Johnson homomorphism induces the abelianization of IA n . (For details about the Johnson homomorphisms, see [42], [43] and [44] for example.) Let IA n = A ′ n (1) ⊃ A ′ n (2) ⊃ · · · be the lower central series of IA n , and set gr k (A ′ n ) := A ′ n (k)/A ′ n (k +1) for each k. Since the Andreadakis-Johnson filtration is central by Part (2) of Theorem 2.1, we see A ′ n (k) ⊂ A n (k) for any k ≥ 1. Then we have the following conjecture.
In this paper, by abuse of language, we also call τ ′ k the k-th Johnson homomorphism of Aut F n . We can see that each τ ′ k is GL(n, Z)-equivariant by the same way as τ k . Then we have the exact sequence 0 → A n (k + 1)/A ′ n (k + 1) → A n (k)/A ′ n (k + 1) → gr k (A n ) → 0 induced from natural homomorphisms.
Here we briefly review well-known results in representation theory for the general linear group GL(n, Q), including Cartan-Weyl's highest weight theory. The notation we use here is according to our previous paper [12].
For a rational representation V of GL(n, Q), consider the irreducible decomposition V = λ∈P V λ as a T -module where 1 · · · t λn n v for any t ∈ T }. We call this decomposition the weight decomposition of V with respect to T . If V λ = {0}, then we call λ the weight of V . For a weight λ, a non-zero vector v ∈ V λ is call a weight vector of weight λ.
Let U be the subgroup of GL(n, Q) consists of all upper unitriangular matrices in GL(n, Q). For a rational representation V of GL(n, Q), we set We call a non-zero vector Then we have the following. Theorem 2.3 (Cartan-Weyl's highest weight theory).
(1) Any rational representation of V is completely reducible.
(2) Suppose V is an irreducible rational representation of GL(n, Q). Then V U is one-dimensional, and the weight λ of V U = V U λ belongs to P + . We call this λ the highest weight of V , and any non-zero vector v ∈ V U λ is called a highest weight vector of V .
(4) The set of isomorphism classes of irreducible rational representations of G is parameterized by the set P + of dominant integral weights. (5) Let V be a rational representation of GL(n, Q) and χ V the character of V as a T -module. Then for two rational representation V and W , they are isomorphic as G-modules if and only if χ V = χ W .
From the above theorem, we can parameterize the set of isomorphism classes of irreducible rational representations of GL(n, Q) by P + . We can do this with the determinant representations. For any e ∈ Z, let D e : GL(n, Q) → Q × be the e-th power of the determinant representation of GL(n, Q) defined by X → (det X) e . The highest weight of this representation is given by (e, e, · · · , e) ∈ P + . If λ ∈ P + satisfies λ n < 0 then we have then Therefore we can parameterize the set of isomorphism classes of irreducible rational representations of GL(n, Q) by the set {(λ, e)} where λ is a partition such that ℓ(λ) ≤ n and e ∈ Z <0 where ℓ(λ) is the length of λ. Moreover, the set of isomorphism classes of irreducible polynomial representations is parameterized by the set of partitions λ such that ℓ(λ) ≤ n. We remark that the dual representation of L (λ 1 ,λ 2 ,...,λn) is isomorphic to L (−λn,...,−λ 2 ,−λ 1 ) . In the following, for simplicity, for any λ = (λ 1 , . . . , λ n ), we write the irreducible representation L λ as [λ 1 , . . . , λ n ] according to the usual notation in representation theory.
From Pieri's rule, the irreducible decomposition of W Q as a GL(n, Q)-module is given by In Section 4, we consider the irreducible decompositions of several GL(W Q )-modules. In order to find highest weight vectors of them, we have to know the actions of elementary matrices of GL(n, Q) on the Magnus generators of W Q . For any 1 ≤ i = j ≤ n, let E x i x j ∈ Aut F n be the automorphism of F n defined by Let E ij ∈ GL(n, Z) ⊂ GL(n, Q) be the image of E x i x j by the natural homomorphism ρ : Aut F n → GL(n, Z). Then the actions of E ij on the Magnus generators are given as follows: Here 1 ≤ i, j, l, p, q ≤ n are distinct indices.

(Co)homology groups of IA n
In this section, we give combinatorial group theoretic descriptions of low dimensional (co)homology groups of IA n . In particular, by using it, we study the second and the third (co)homology groups of IA 3 .
Since IA 3 is not finitely presentable due to Krstić and McCool [24], it immediately turns out that H 2 (IA 3 , Z) is not finitely generated. This fact is also reproved by Bestvina, Bux and D. Margalit [4]. It is a natural problem to give an explicit generating set of H 2 (IA 3 , Z). However, if we approach this problem, we immediately face the difficulties coming from the complexity of the structure of the group of relators among Magnus generators of IA 3 . In order to make the problem more easy, we consider the rationalization of the problem. After this, we can use the representation theory of the general linear group GL(n, Q), and its highest weight theory. But, it is still too complicated to give a complete answer to the above problem. In the following, we give partial results for this problem.
3.1. A minimal presentation and H 2 (IA n , Z).
In this subsection, we give a combinatorial group theoretical interpretation of the second homology group of IA n . Let F be a free group on the Magnus generators: K ij for any 1 ≤ j < i ≤ n and K ijl for any i = j < l = i. By abuse of the language, we use the same notation K ij and K ijl for the elements in F as the automorphisms K ij and K ijl in IA n . The rank of F is n 2 (n − 1)/2. Then we have a natural surjective homomorphism π : F → IA n , and the group extension of IA n where R = Ker(π). Since the abelianization H 1 (IA n , Z) of IA n is the free abelian group generated by (the coset classes of) the Magnus generators, we verify that π induces the isomorphism This shows that R is contained in the commutator subgroup [F, F ] of F . From the homological five-term exact sequence of (6), we have By considering the right action of F on R, defined by we see ψ k is an F -equivariant homomorphism. Hence it induces the surjective homomorphism Therefore we see that if R/R 3 is a non-trivial module, we can detect non-trivial elements in H 2 (IA n , Z).
In the exact sequence (7) for k = 2, since both of L F (2) is a free abelian group, so is R/R 3 . Since A ′ n (3) has a finite index in A n (3) by Pettet [39], we see gr 2 Q (A ′ n ) = gr 2 Q (A n ). Hence, we have This means H 2 (IA n , Z) contains a free abelian group of this rank.
Next, we consider the kernel of ψ 2 : H 1 (R, Z) F → H 1 (R/R 3 , Z) F . Observe the exact sequences of Z-modules. This shows that if H 2 (IA n , Z) ∼ = R/[F, R] is not isomorphic to R/R 3 , by showing R 3 /(R 4 [F, R]) = 0, we have the potential to detect non-trivial second homology classes which we cannot detect by R/R 3 . In Section 4, we show that (R 3 /(R 4 [F, R]))⊗ Z Q is a non-trivial irreducible representation of GL (3, Q). Consider the homomorphism induced from the commutator bracket of F . This homomorphism induces the homomorphism

3.2.
On the second cohomology group H 2 (IA n , Z).
Here we consider the second cohomology group of IA n , and the image of the cup product ∪ : Λ 2 H 1 (IA n , Z) → H 2 (IA n , Z). From the cohomological five-term exact sequence of (6), we have For any k ≥ 2, the natural projection R → R/R k+1 induces the injective homomorphism In particular, for k = 2, we have H 1 (R/R 3 , Z) F ∼ = H 1 (R/R 3 , Z) since F acts on R/R 3 trivially. Namely we can regard H 1 (R/R 3 , Z) as a Z-submodule of H 2 (IA n , Z). Then we have is isomorphic to the image of induced from the natural inclusion R/R 3 → L F (2).
Proof. First, observing the last three terms of the cohomological five-term exact sequence of , we obtain an exact sequence 0 → H 1 (gr 2 (A ′ n ), Z) → H 2 (IA ab n , Z) − → H 2 (IA n , Z). Since IA ab n is a free abelian group of finite rank, we have the natural isomorphism H 2 (IA ab n , Z) ∼ = Λ 2 H 1 (IA n , Z). Then the map H 2 (IA ab n , Z) − → H 2 (IA n , Z) is regarded as the cup product ∪ : On the other hand, from the cohomological five-term exact sequence where tg is the transgression, µ is the natural isomorphism, and ι * is the homomorphism induced from the inclusion R/R 3 → L F (2). Hence we obtain Im(∪) ∼ = Im(ι * ).
We remark that in the rational coefficients case, we have Im(∪ Q ) ∼ = Im(ι * Q ). Furthermore, if A ′ n (3) = A n (3) then gr 2 (A ′ n ) = gr 2 (A n ) is a free abelian group, and hence ι * is surjective. This implies that Im(∪) ∼ = H 1 (R/R 3 , Z). At the present stage, it is not known if A ′ n (3) coincides with A n (3) for general n ≥ 4. Next, we consider a method to detect non-trivial elements in H 2 (IA n , Z), which do not belong to Im(∪). From the exact sequence Here Hom Z (R/[F, R], Z) ∼ = H 2 (IA n , Z). On the other hand, since R/R 3 is a free abelian group, the split exact sequence Thus by studying Hom Z (R 3 /R 4 [R, F ], Z), we have the potential to detect a non-trivial submodule in H 2 (IA n , Z) which is not contained in Im(∪).

3.3.
On the third cohomology group H 3 (IA n , Z).
Here we consider the third cohomology group of IA n . In particular, we characterize the image of the triple cup product In general, by the reduction theorem, we have the isomorphism This isomorphism is given by the differential homomorphism d 1,1 2 of the Lyndon-Hochshild-Serre spectral sequence of the group extension (6). (See [20] for details.) The IA nequivariant exact sequence In the following, for simplicity, for any Q-vector space V , we write V * Q for the Q-linear dual space Hom Q (V, Q) of V . Then the above exact sequence induces the cohomological long exact sequence of IA n . Since IA n acts on R 3 /R 4 and R 3 /R 4 [F, R] trivially, we have the sequence . Now we consider the restriction of the connecting homomorphism δ to (R 3 /R 4 ) * Q . To begin with, we determine the kernel of δ.
On the other hand, the exact sequence splits since R/R 3 is a free abelian group of finite rank, we have an isomorphism

Conversely, for any
Next we consider the image of α 0 in H 1 (IA n , (R ab ) * Q ). Observe the commutative diagram of IA ab n -modules, containing two exact sequences. This diagram and the natural projection IA n → IA ab n induces the commutative diagram of the first cohomologies of IA n and IA ab n . Here ψ 1 is surjective and ψ 2 is an isomorphism. Remark that through the isomorphism d 1,1 2 , the target of β 1 is isomorphic to the image of the triple cup product  Proof. For any f ∈ Im(β 1 ), from the surjectivity of ψ 2 • ψ 1 , there exists some element By the commutativity of (11), we see β 1 • (ξ 2 • ξ 1 )( f ) = f . With standard arguments and calculations, we can see that the homomorphism ξ 2 • ξ 1 : H 3 (IA ab n , Q) → H 3 (IA n , Q) is equal to the natural homomorphsim induced from that projection IA n → IA ab n . Therefore, we see f ∈ Im(∪ 3 IAn ). On the other hand, since H 1 (IA n , Q) ∼ = H 1 (IA ab n , Q), for any g ∈ Im(∪ 3 IAn ), there exists some g ∈ Im(∪ 3 IA ab n ) such that ξ 2 • ξ 1 ( g) = g. Hence there exists some h ∈ H 1 (IA ab n , (Γ F (2)/Γ F (3)) * Q ) such that β 1 (h) = g and Thus g ∈ Im(β 1 ), and we obtain the required result.

The case where n = 3
In this section, we show our main theorems. In the following, we always assume that n = 3.
To begin with, for W = H 1 (IA 3 , Z), we consider the irreducible decompositions as GL(3, Q)-modules of several submodules of the free Lie algebra L Q F := k≥1 L Q F (k) generated by W Q . Recall that the irreducible decomposition of W Q is given by The list of highest weight vectors of W Q is given as follows: In [39], Pettet gave the irreducible decomposition of L Q The list of linearly independent highest weight vectors are as follows: Consider the injective homomorphism (R/R 3 ) Q → L Q F (2) induced from the natural inclusion R ֒→ Γ F (2). We regard (R/R 3 ) Q as a GL(3, Q)-subrepresentation of L Q W (2). Proof. First, we check v 2 , v 4 ∈ (R/R 3 ) Q . Observe the exact sequence (7) for k = 2, and recall that the second Johnson homomorphism τ ′ 2,Q : gr 2 This shows that (R/R 3 ) Q contains a submodule isomorphic to [1, 1] ⊕ [2, 1, −1]. On the other hand, from the fact that 3 )) = 36 − 18 = 18 and dim Q ([1, 1] ⊕ [2, 1, −1]) = 18, we obtain the required results.
Next we give the irreducible decomposition of L Q F (3). To do this, we prepare some lemmas. In general, for any representations V 1 , V 2 of GL(n, Q), and for any k ≥ 1 we have the formula Then we have the following decompositions. Proof.
(1). By applying (13) to (2). Since n = 3, we have by Pieri's rule. On the other hand, by using Pieri's rule again, we can compute the irreducible decomposition of both of ( Hence by checking the difference between them, we obtain the required results. (1) (1). From (13), Hence, from Lemma 4.2, we obtain the required result.
(2). Consider the GL(3, Q)-equivariant surjective homomorphism On the other hand, we can compute the irreducible decomposition of Λ 2 W Q ⊗ Q W Q by using Pieri's rule and Lemma 4.2 as Thus from Part (1), we obtain the required result. Now, we consider the irreducible decomposition of (R 3 /R 4 ) Q . Consider the GL(n, Q)- The proof of the following proposition requires patient, lengthy and straightforward computation. Thus we give the proof in Section 5 later.
In our previous paper [41], we have determined the cokernel, and hence the image, of the third rational Johnson homomorphism τ 3,Q : gr 3 (A 3 ) → H * Q ⊗ Q L n,Q (4). In particular, we showed that dim Q (gr 3 Q (A 3 )) = 43. From the fact that A ′ 3 (3) = A 3 (3) and A ′ 3 (4) ⊂ A 3 (4), we have the surjective homomorphism gr 3 . This shows that dim Q (gr 3 Q (A ′ 3 )) ≥ 43. Thus, from the exact sequence (7) for k = 3 and dim Q (L Q F (3)) = 240, we see that dim Q ((R 3 /R 4 ) Q ) ≤ 197. On the other hand, by Proposition 4.4, we see that (R 3 /R 4 ) Q has a subspace Im([ , ] Q ) whose dimension is 162, and that dim Namely, the natural surjective homomorphism gr 3 is an isomorphism. Hence A ′ 3 (4) has at most finite index in A 3 (4). Finally, we consider the third cohomology group H 3 (IA 3 , Q). Let IO 3 be the quotient group of IA 3 by the inner automorphism group Inn F 3 . Bestvina, Bux and Margalit [4] showed that IO 3 has 2-dimensional Eilenberg-Maclane space, and that H 2 (IO 3 , Q) is infinitely generated. Hence, by using the spectral sequence of the group extension we see that H 3 (IA 3 , Q) is infinitely generated. The following theorem shows that nontrivial elements in H 3 (IA 3 , Q) cannot be detected by the triple cup product of the first cohomology group of IA 3 .
Theorem 4.7. The image of the triple cup product be the connecting homomorphism defined in Subsection 3.3. By Lemma 3.2, we see that the restriction of δ to On the other hand, since Q )) = 162. Therefore, the homomorphism δ is an isomorphism. Thus β 1 : H 1 (IA 3 , (R/R 3 ) * Q ) → H 3 (IA 3 , Q) is zero map. Then by Proposition 3.3, we obtain the required result.

Proof of Proposition 4.4
In this section, we give the proof of Proposition 4.4. The proof is done by patient lengthy hand calculation. To begin with, we explain how to prove, and show a brief of calculations.
By the natural injective homomorphism R 3 /R 4 ֒→ L F (3), we consider R 3 /R 4 as a submodule of L F (3). We show that the homomorphism given by r ⊗ x → [r, x] is injective. In order to do this, we show that the cokernel L F (3)/Im([ , ]) is a free abelian group of rank From this, we can see that the image Im([ , ]) is a free abelian group of rank 162, and that the bracket map [ , ] is injective.
We give basis of L F (1) and R/R 3 as follows. First, the set E 1 := {K 12 , K 13 , K 21 , K 23 , K 123 , K 213 , K 312 } ⊂ L F (1) forms a basis of L F (1). Consider the relators in R for distinct i, j and k. These elements defines a basis of R/R 3 given as follows. Let E 2 ⊂ R/R 3 be the set of the above elements. Then Im([ , ]) is generated by 162 elements [r, x] for any r ∈ E 2 and x ∈ E 1 . Consider again the Hall basis of L Q F (3) consisting the Magnus generators of F where its order is given by (14). The list of the Hall basis is given in Table 1  (1) [(R1-1), K].
For any K ∈ E 1 , by using [K 23 , K 13 , K] = 0 ∈ L F (3)/Im([ , ]). we remove the generators [K 21 , K, K 31 ] for any K < K 21 , and [K 31 , K 21 , K] for any K 21 ≤ K from E.  For any K ∈ E 1 , consider [K 32 , K 13 , K] − [K 13 , K 12 , K] = 0 ∈ L F (3)/Im([ , ]). We consider the following four cases. Hereafter, we repeat similar computations according to the following order. In each step, the removed generators are attached the step number in Table 1. For example, [K 31 , K 12 , K 12 ] is removed from E at the step (7)-(i). Since all of the computations are straightforward and too lengthy to write down here, we leave it to the high-motivated reader as exercises.
Finally, we see that from the above computations, the cokernel L F (3)/Im([ , ]) is a free abelian group with basis consisting the elements in Table 1 to which no step number attached. This completes the proof of Proposition 4.4. Table 1.

Acknowledgments
The author would like to express his sincere gratitude to Professor Shigeyuki Morita for discussing this research many times after his lecture for undergraduate students at Tokyo University of Science. His insatiable thirst for mathematics and lots of indisputable leading works have been the intense longing and the perfect model of the author as a mathematician. The author has been inspired by his mathematical spirits, and have supported by his unfailing encouragements for more than ten years.
The author would like to thank Andrew Putman for giving me usuful comments about the Andreadakis-Johnson filtration of Aut F n . This work is supported by JSPS KAKENHI Grant Number 16K05155.