The Cokernel of the Johnson Homomorphisms of the Automorphism Group of a Free Metabelian Group

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 4, April 2009, Pages 2085–2107 S 0002-9947(08)04767-3 Article electronically published on November 5, 2008

THE COKERNEL OF THE JOHNSON HOMOMORPHISMS OF THE AUTOMORPHISM GROUP OF A FREE METABELIAN GROUP

TAKAO SATOH

Dedicated to Professor Shigeyuki Morita on the occasion of his 60th birthday

Abstract. In this paper, we determine the cokernel of the k-th Johnson ho- momorphisms of the automorphism group of a free metabelian group for k ≥ 2 and n ≥ 4. As a corollary, we obtain a lower bound on the rank of the graded quotient of the Johnson filtration of the automorphism group of a free group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup product map in the rational second cohomology group of the IA-automorphism group of a free metabelian group, and show that it is iso- morphic to that of the IA-automorphism group of a free group which is already determined by Pettet. Finally, by considering the kernel of the Magnus repre- sentations of the automorphism group of a free group and a free metabelian group, we show that there are non-trivial rational second cohomology classes of the IA-automorphism group of a free metabelian group which are not in the image of the cup product map.

1. Introduction

Let G be a group and ΓG(1) = G,ΓG(2), ... its lower central series. We denote by Aut G the group of automorphisms of G.Foreachk ≥ 0, let AG(k)bethe group of automorphisms of G which induce the identity on the quotient group G/ΓG(k + 1). Then we obtain a descending central filtration

Aut G = AG(0) ⊃AG(1) ⊃AG(2) ⊃··· of Aut G, called the Johnson filtration of Aut G. This filtration was introduced in 1963 with a pioneer work by S. Andreadakis [1]. For each k ≥ 1, set LG(k):= k ab ΓG(k)/ΓG(k +1)andgr (AG)=AG(k)/AG(k +1).LetG be the abelianization of G. Then, for each k ≥ 1, an Aut Gab-equivariant injective homomorphim k ab τk :gr (AG) → HomZ(G , LG(k +1)) is defined. (For a definition, see Subsection 2.1.2.) This is called the k-th Johnson homomorphism of Aut G. Historically, the study of the Johnson homomorphism was begun in 1980 by D. Johnson [17]. He studied the Johnson homomorphism of a mapping class group of a closed oriented surface, and determined the abelianization

Received by the editors May 17, 2007. 2000 Mathematics Subject Classification. Primary 20F28; Secondary 20J06. Key words and phrases. Automorphism group of a free metabelian group, Johnson homomor- phism, second cohomology group, Magnus representation.

c 2008 American Mathematical Society 2085

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of the Torelli group. (See [18].) There is a broad range of remarkable results for the Johnson homomorphisms of a mapping class group. (For example, see [14] and [25].) M Let Fn be a free group of rank n with basis x1,...,xn and Fn the free metabelian M group of rank n.NamelyFn is the quotient group of Fn by the second derived series M [[Fn,Fn], [Fn,Fn]] of Fn. ThenbothabelianizationsofFn and Fn are a free abelian group of rank n, denoted by H. Fixing a basis of H induced from x1,...,xn,wecan ab M identify Aut G with GL(n, Z)forG = Fn and Fn . For simplicity, throughout this L A k A L A paper we write Γn(k), n(k), n(k)andgr ( n)forΓFn (k), Fn (k), Fn (k)and k A M LM AM k AM gr ( Fn ) respectively. Similarly, we write Γn (k), n (k), n (k)andgr( n ) k for Γ M (k), L M (k), A M (k)andgr(A M ) respectively. The first aim of this Fn Fn Fn Fn paper is to determine the GL(n, Z)-module structure of the cokernel of the Johnson M ≥ homomorphisms τk of Aut Fn for n 4 as follows: Theorem 1. For k ≥ 2 and n ≥ 4,

Tr M → k AM −τ→k ∗ ⊗ LM −−−k→ k → 0 gr ( n ) H Z n (k +1) S H 0 is a GL(n, Z)-equivariant exact sequence.

k M Here S H is the symmetric product of H of degree k,andTrk is a certain GL(n, Z)-equivariant homomorphism called the Morita trace introdued by S. Morita [24]. (For definition, see Subsection 3.2.) k From Theorem 1, we can give a lower bound on the rank of gr (An)fork ≥ 2 and n ≥ 4. The study of the Johnson filtration of Aut Fn was begun in the 1960s k by Andreadakis [1] who showed that for each k ≥ 1andn ≥ 2, gr (An)isafree abelian group of finite rank, and that A2(k) coincides with the k-th lower central series of the inner automorphism group Inn F2 of F2. Furthermore, he [1] computed k k rankZ gr (A2) for all k ≥ 1. However, the structure of gr (An) for general k ≥ 2 and n ≥ 3 is much more complicated. Set τk,Q = τk ⊗ idQ, and call it the k-th rational Johnson homomorphism. For any Z-module M,wedenoteM ⊗Z Q by the Q symbol obtained by attaching a subscript Q to M, like MQ and M .Forn ≥ 3, 2 A the GL(n, Z)-module structure of grQ( n) is completely determined by Pettet [31]. 3 A ≥ ≥ In our previous paper [33], we determined those of grQ( n)forn 3. For k 4, k A the GL(n, Z)-module structure of grQ( n) is not determined. Furthermore, even its dimension is also unknown. → M Let νn :AutFn Aut Fn be a natural homomorphism induced from the action M of Aut Fn on Fn . By notable works due to Bachmuth and Mochizuki [5], it is known that νn is surjective for n ≥ 4. They [4] also showed that ν3 is not surjective. In k A → k AM Subsection 3.1, we see that the homomorphism νn,k :gr ( n) gr ( n ) induced from νn is also surjective for n ≥ 4. Hence we have Corollary 1. For k ≥ 2 and n ≥ 4, n + k − 1 n + k − 1 rank (grk(A )) ≥ nk − . Z n k +1 k We should remark that in general, equality does not hold, since for instance 3 4 2 rankZ gr (An)=n(3n − 7n − 8)/12, which is not equal to the right hand side of the inequality above. Next, we consider the second cohomology group of the IA-automorphism group of the free metabelian group. Here the IA-automorphism group IA(G) of a group G

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is defined to be a group which consists of automorphisms of G which trivially act on A M the abelianization of G. By the definition, IA(G)= G(1). We write IAn and IAn M ∗ for IA(Fn)andIA(Fn ) for simplicity. Let H := HomZ(H, Z) be the dual group M ≥ of H. Then we see that the first homology group of IAn for n 4 is isomorphic ∗ ⊗ 2 → M to H Z Λ H in the following way. Let νn,1 :IAn IAn be the restriction of νn to IAn. Bachmuth and Mochizuki [5] showed that νn,1 is surjective for n ≥ 4. This fact sharply contrasts with their previous work [4], which shows there are M infinitely many automorphisms of IA3 which are not contained the image of ν3,1. On the other hand, by independent works of Cohen-Pakianathan [9, 10], Farb [11] ∼ ∗ 2 and Kawazumi [19], H1(IAn, Z) = H ⊗Z Λ H for n ≥ 3. Since the kernel of νn,1 is M M ∼ ∗ ⊗ 2 contained in the commutator subgroup of IAn ,wehaveH1(IAn , Z) = H Z Λ H for n ≥ 4. (See Subsection 2.3.) In general, however, there are few results for M computation of the (co)homology groups of IAn of higher dimensions. In this paper we determine the image of the cup product map in the rational second cohomology M group of IAn , and show that it is isomorphic to that of IAn, using the second 2 1 2 Johnson homomorphism. Namely, let ∪Q :ΛH (IAn, Q) → H (IAn, Q)and ∪M 2 1 M → 2 M Q :Λ H (IAn , Q) H (IAn , Q) be the rational cup product maps of IAn and M IAn respectively. In Subsection 4.2, we show ≥ ∗ ∪M → ∪ Theorem 2. For n 4, νn,1 :Im( Q ) Im( Q) is an isomorphism.

Here we should remark that the GL(n, Z)-module structure of Im(∪Q)iscom- pletely determined by Pettet [31] for any n ≥ 3. M Now, for the study of the second cohomology group of IAn ,itisalsoanimpor- ∪M tant problem to determine whether the cup product map Q is surjective or not. For the case of IAn, it is still not known whether ∪Q is surjective or not. In the ∪M last section, we prove that the rational cup product map Q is not surjective for n ≥ 4. It is easily seen that Kn is an infinite subgroup of IAn,sinceKn contains the second derived series of the inner automorphism group of a free group Fn.The structure of Kn is, however, very complicated. For example (finitely or infinitely many) generators and the abelianization of Kn are still not known. To clarify the structure of Kn, it is also important to study the obstruction for the faithfulness of the Magnus representation of IAn since Kn is equal to the kernel, by a result of Bachmuth [2]. (See Subsection 2.3.) From the cohomological five-term exact sequence of the group extension →K → → M → 1 n IAn IAn 1,

1 K IAn ∪M  it suffices to show the non-triviality of H ( n, Q) to show that Im( Q ) = 2 M K K K ∩A ⊂ 3 A K H (IAn , Q). Set n := n/( n n(4)) gr ( n). Then n naturally has a GL(n, Z)-module structure, and the natural projection Kn → Kn induces an injec- 1 1 IAn tive homomorphism H (Kn, Q) → H (Kn, Q) . In this paper, we determine the GL(n, Z)-module structure of H1(Kn, Q) using the rational third Johnson homo- 1 morphism of Aut Fn. The non-triviality of H (Kn, Q) immediately follows from it. In Subsection 5.1, we show

− ≥ KQ ∼ [2,1] ⊕ ⊗ [3,22,1n 4] Theorem 3. For n 4, τ3,Q( n ) = HQ (D Q HQ ). Here Hλ denotes the Schur-Weyl module of H corresponding to the Young n diagram λ =[λ1,...,λl], and D := Λ H the one-dimensional representation of

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GL(n, Z) given by the determinant map. Since τ3,Q is injective, this shows that

− KQ ∼ [2,1] ⊕ ⊗ [3,22,1n 4] n = HQ (D Q HQ ). As a corollary, we have

Corollary 2. For n ≥ 4, 1 1 rank (H (K , Z)) ≥ n(n2 − 1) + n2(n − 1)(n +2)(n − 3). Z 1 n 3 8 Finally, we obtain

Theorem 4. For n ≥ 4, the rational cup product ∪M 2 1 M → 2 M Q :Λ H (IAn , Q) H (IAn , Q) is not surjective, and 1 dim (H2(IAM , Q)) ≥ n(n − 2)(3n4 +3n3 − 5n2 − 23n − 2). Q n 24 In Section 2, we recall the IA-automorphism group of G and the Johnson homo- morphisms of the automorphism group Aut G of G foragroupG.Inparticular,we concentrate on the case where G is a free group and a free metabelian group. We M also review the definition of the Magnus representations of IAn and IAn . In Section 3, we determine the cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group. In Section 4, we show that the image of the cup ∪M ∪ product map Q is isomorphic to that of Q. Finally, in Section 5, we determine KQ ∪M the GL(n, Z)-module structure of n and show that Q is not surjective.

2. Preliminaries In this section, we recall the definition and some properties of the associated Lie algebra, the IA-automorphism group of G, and the Johnson homomorphisms of the automorphism group Aut G of G for any group G. In Subsections 2.2 and 2.3, we consider the case where G is a free group and a free metabelian group.

2.1. Notation. First of all, throughout this paper we use the following notation and conventions. • For a group G, the abelianization of G is denoted by Gab. • ForagroupG, the 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 xσ. • For a group G and its quotient group G/N,wealsodenotethecosetclass of an element g ∈ G by g ∈ G/N if there is no confusion. • For any Z-module M,wedenoteM ⊗Z Q by the symbol obtained by Q attaching a subscript Q to M,suchasMQ or M . Similarly, for any Z- linear map f : A → B, the induced Q-linear map AQ → BQ is denoted by Q fQ or f . • For elements x and y of a group, the commutator bracket [x, y]ofx and y is defined to be [x, y]:=xyx−1y−1.

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2.1.1. Associated Lie algebra of a group. ForagroupG, we define the lower central series of G by the rule

ΓG(1) := Fn, ΓG(k):=[ΓG(k − 1),G],k≥ 2. L We denote by G(k):=ΓG(k)/ΓG(k + 1) the graded quotient of the lower central L L series of G,andby G := k≥1 G(k) the associated graded sum. The graded sum LG naturally has a graded Lie algebra structure induced from the commutator bracket on G, and is called the associated Lie algebra of G. For any g1,...,gt ∈ G, a commutator of weight k type of ··· ··· ∈{ } [[ [[gi1 ,gi2 ],gi3 ], ],gik ],ij 1,...,t , with all of its brackets to the left of all the elements occuring, is called a simple k-fold commutator among the components g1,...,gt, and we denote it by

[gi1 ,gi2 ,...,gik ] for simplicity. Then we have

Lemma 2.1. If G is generated by g1,...,gt, then each of the graded quotients ΓG(k)/ΓG(k +1) is generated by the simple k-fold commutators ∈{ } [gi1 ,gi2 ,...,gik ],ij 1,...,t . ab Let ρG :AutG → Aut G be the natural homomorphism induced from the abelianization of G.ThekernelIA(G)ofρG is called the IA-automorphism group of G. Then the automorphism group Aut G naturally acts on LG(k)foreachk ≥ 1, and IA(G) acts on it trivially. Hence the action of Aut G/IA(G)onLG(k) is well- defined. 2.1.2. Johnson homomorphisms. For k ≥ 0, the action of Aut G on each nilpotent quotient G/ΓG(k + 1) induces a homomorphism k → ρG :AutG Aut(G/ΓG(k +1)). 0 1 k A The map ρG is trivial, and ρG = ρG. We denote the kernel of ρG by G(k). Then the groups AG(k) define a descending central filtration

Aut G = AG(0) ⊃AG(1) ⊃AG(2) ⊃···

of Aut G,withAG(1) = IA(G). (See [1] for details.) We call it the Johnson filtration of Aut G.Foreachk ≥ 1, the group Aut G acts on AG(k) by conjugation, k A and it naturally induces an action of Aut G/IA(G)ongr( G). The graded sum A k A gr( G):= k≥1 gr ( G) has a graded Lie algebra structure induced from the commutator bracket on IA(G). k To study the Aut G/IA(G)-module structure of each graded quotient gr (AG), we define the Johnson homomorphisms of Aut G in the following way. For each ab k ≥ 1, we consider a map AG(k) → HomZ(G , LG(k + 1)) defined by σ → (g → g−1gσ),x∈ G.

Then the kernel of this homomorphism is just AG(k + 1). Hence it induces an injective homomorphism k ab τk = τG,k :gr (AG) → HomZ(G , LG(k +1)).

The homomorphsim τk is called the k-th Johnson homomorphism of Aut G.It is easily seen that each τk is an Aut G/IA(G)-equivariant homomorphism. Since

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each Johnson homomorphism τk is injective, to determine the cokernel of τk is an k important problem for the study of the structure of gr (AG)asanAutG/IA(G)- module. A Here, we consider another descending filtration of IA(G). Let G(k)bethe ≥ A k-th subgroup of the lower central series of IA(G). Then for each k 1, G(k) is a subgroup of AG(k)sinceAG(k) is a central filtration of IA(G). In general, A A k A it is not known whether or not G(k)coincideswith G(k). Set gr ( G):= A A ≥ A → G(k)/ G(k +1)foreachk 1. The restriction of the homomorphism G(k) ab L A HomZ(G , G(k + 1)) to G(k) induces an Aut G/IA(G)-equivariant homomor- phism   k A → ab L τk = τG,k :gr ( G) HomZ(G , G(k +1)).  In this paper, we also call τk the k-th Johnson homomorphism of Aut G. For any σ ∈AG(k)andτ ∈AG(l), we give an example of computation of −1 σ τk+l([σ, τ ]) using τk(σ)andτl(τ). For σ ∈AG(k)andg ∈ G,setsg(σ):=g g ∈ ΓG(k +1). Then, τk(σ)(g)=sg(σ) ∈LG(k +1). Forany σ ∈AG(k)andτ ∈AG(l), by an easy calculation we have

−1 τ −1 −1 σ−1τ −1 σ−1τ −1 τσ−1τ −1 sg([σ, τ ]) = (sg(τ) ) (sg(σ) ) sg(τ) sg(σ) (1) −1 τ −1 σ −1 ≡ sg(σ) sg(σ) · (sg(τ) sg(τ) ) (mod ΓG(k + l +2)).

Using this fomula, we can easily compute sg([σ, τ ]) from sg(σ)andsg(τ). For example, if sg(σ)andsg(τ)isgivenby

(2) sg(σ)=[g1,g2,...,gk+1] ∈LG(k+1),sg(τ)=[h1,h2,...,hl+1] ∈LG(l+1), then we obtain k+1 l+1 − sg([σ, τ ]) = [g1,...,sgi (τ),...,gk+1] [h1,...,shj (σ),...,hl+1] i=1 j=1

in LG(k + l +1).

2.2. Free groups. In this section we consider the case where G is a free group of finite rank.

2.2.1. Free Lie algebra. For n ≥ 2, let Fn be a free group of rank n with basis ∗ x1,...,xn. We denote the abelianization of Fn by H and its dual group by H := HomZ(H, Z). If we fix the basis of H as a free abelian group induced from the ab basis x1,...,xn of Fn,wecanidentifyAutFn =Aut(H) with the general linear

group GL(n, Z). Furthermore, it is classically well known that the map ρFn : Aut Fn → GL(n, Z) is surjective. (See [21], proposition 4.4.) Hence we also identify Aut(H)/IA(Fn)withGL(n, Z). In this paper, for simplicity, we write Γn(k), Ln(k) L L L and n for ΓFn (k), Fn (k)and Fn respectively. The associated Lie algebra Ln is called the free Lie algebra generated by H.(See [32] for basic material concerning free Lie algebra.) It is classically well known due to Witt [34] that each Ln(k)isaGL(n, Z)-equivariant free abelian group of rank 1 k (3) r (k):= µ(d)n d n k d|k where µ is the M¨obius function.

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Next we consider the GL(n, Z)-module structure of Ln(k). For example, for 1 ≤ k ≤ 3wehave L (1) = H, L (2) = Λ2H, n n 2 Ln(3) = (H ⊗Z Λ H) x ⊗ y ∧ z + y ⊗ z ∧ x + z ⊗ x ∧ y | x, y, z ∈ H. LQ In general, the irreducible decomposition of n (k)asaGL(n, Z)-module is com- pletely determined. For k ≥ 1 and any Young diagram λ =[λ1,...,λl]ofdegree k,letHλ be the Schur-Weyl module of H corresponding to the Young diagram k λ. For example, H[k] = SkH and H[1 ] =ΛkH. (For details, see [12] and [13].) λ LQ λ LQ Let m(HQ, n (k)) be the multiplicity of the Schur-Weyl module HQ in n (k). λ LQ Bakhturin [6] gave a formula for m(HQ, n (k)) using the character of the Specht module of HQ corresponding to the Young diagram λ. However, its character value had remained unknown in general. Then Zhuravlev [35] gave a method of calcula- tion for it. Using these facts, we can give the explicit irreducible decomposition of LQ n (k). For example, LQ ∼ [2,1] LQ ∼ [3,1] ⊕ [2,1,1] (4) n (3) = HQ , n (4) = HQ HQ .

2.2.2. IA-automorphism group of a free group. Now we consider the IA-automor- phism group of Fn. WedenoteIA(Fn)byIAn. It is well known due to Nielsen [27] that IA2 coincides with the inner automorphsim group Inn F2 of F2.Namely, IA2 is a free group of rank 2. However, IAn for n ≥ 3 is much larger than Inn Fn. Indeed, Magnus [22] showed that for any n ≥ 3, the IA-automorphism group IAn is finitely generated by automorphisms −1 xi → xj xixj, Kij : xt → xt (t = i)

for distinct i, j ∈{1, 2,...,n} and −1 −1 xi → xixjxkxj xk , Kijk : xt → xt (t = i)

for distinct i, j, k ∈{1, 2,...,n} such that j

2.2.3. Johnson homomorphisms of Aut Fn. Here, we consider the Johnson homo- morphisms of Aut Fn. Throughout this paper, for simplicity, we write An(k),

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A (k), grk(A )andgrk(A )forA (k), A (k), grk(A )andgrk(A ) respec- n n n Fn Fn Fn Fn tively. Pettet [31] showed 1 (6) rank gr2(A )= n(n + 1)(2n2 − 2n − 3), Z n 6 and in our previous paper [33], we showed 1 rank gr3(A )= n(3n4 − 7n2 − 8). Z n 12 k In general, for any n ≥ 3andk ≥ 4 the rank of gr (An) is still not known. One of k the aims of this paper is to give a lower bound on rankZ gr (An) by studying the Johnson filtration of the automorphism group of a free metabelian group. A A Next, we mention the relation between n(k)and n(k). Since τ1 is the abelian- A A ization of IAn as mentioned above, we have n(2) = n(2). Furthermore, Pettet A A [31] showed that n(3) has at most finite index in n(3). Although it is conjec- A A ≥ tured that n(k)= n(k)fork 3, there are few results for the difference between A A ≥ n(k)and n(k)forn 3. ∗ Let H be the dual group HomZ(H, Z)ofH. For the standard basis x1,...,xn of ∗ ∗ ∗ H induced from the generators of Fn,letx1,...,xn be its dual basis of H .Then ∗ identifying HomZ(H, Ln(k + 1)) with H ⊗Z Ln(k + 1), we obtain the Johnson homomorphism k ∗ τk :gr (An) → H ⊗Z Ln(k +1)

of Aut Fn. Here we give some examples of computation τk(σ)forσ ∈An(k). For the generators K and K of A (1) = IA ,wehave ij ijk n n 1,l= i, 1,l= i, sxl (Kij)= −1 −1 sxl (Kijk)= [xi ,xj ],l= i, [xj,xk],l= i

in Γn(2). Hence ∗ ⊗ ∗ ⊗ (7) τ1(Kij)=xi [xi,xj],τ1(Kijk)=xi [xj,xk] ∗ ⊗ L  in H Z n(2). Then using (1) and (7), we can recursively compute τk(σ)=τk(σ) ∈A for σ n(k). These computations are perhaps easiest explained with examples, so we give two here. For distinct a, b, c and d in {1, 2,...,n},wehave  ∗ ⊗ τ2([Kab,Kbac]) = xa ([sxa (Kbac),xb]+[xa,sxb (Kbac)]) − ∗ ⊗ xb ([sxa (Kab),xc]+[xa,sxc (Kab)]) ∗ ⊗ − ∗ ⊗ = xa [xa, [xa,xc]] xb [[xa,xb],xc] and  τ3([Kab,Kbac,Kad]) ∗ ⊗ = xa ([sxa (Kad), [xa,xc]] + [xa, [sxa (Kad),xc]] + [xa, [xa,sxc (Kad)]]) − ∗ ⊗ xb ([[sxa (Kad),xb],xc]+[[xa,sxb (Kad)],xc]+[[xa,xb],sxc (Kad)]) − ∗ ⊗ xa ([sxa ([Kab,Kbac]),xd]+[xa,sxd ([Kab,Kbac])]) ∗ ⊗ ∗ ⊗ = xa [[xa,xd], [xa,xc]] + xa [xa, [[xa,xd],xc]] − ∗ ⊗ xb [[[xa,xd],xb],xc] − ∗ ⊗ xa [[xa, [xa,xc]],xd]. 2.3. Free metabelian groups. In this section we consider the case where a group G is a free metabelian group of finite rank.

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M  2.3.1. Free metabelian Lie algebra. Let Fn = Fn/Fn be a free metabelian group  of rank n where Fn =[[Fn,Fn], [Fn,Fn]] is the second derived group of Fn.Then M ab M ab we have (Fn ) = H, and hence Aut (Fn ) =Aut(H)=GL(n, Z). Since the M surjective map ρ :AutF → GL(n, Z) factors through Aut F ,amapρ M : Fn n n Fn M → M M Aut Fn GL(n, Z) is also surjective. Hence we identify Aut Fn /IA(Fn )with M M M GL(n, Z). In this paper, for simplicity, we write Γ (k), L (k)andL for Γ M (k), n n n Fn L M (k)andL M respectively. Fn Fn LM The associated Lie algebra n is called the free metabelian algebra generated L LM ≤ ≤ by H.Weseethat n(k)= n (k)for1 k 3. It is also classically well known LM due to Chen [8] that each n (k)isaGL(n, Z)-equivariant free abelian group of rank n + k − 2 (8) rM (k):=(k − 1) . n k

2.3.2. IA-automorphism group of a free metabelian group. Here we consider the IA- M M → automorphism group of FM .LetIAn := IA(Fn ). We denote by νn :AutFn M M Aut Fn the natural homomorphism induced from the action of Aut Fn on Fn .Re- → M stricting νn to IAn, we obtain a homomorphism νn,1 :IAn IAn . Bachmuth and M Mochizuki [4] showed that ν3,1 is not surjective and IA3 is not finitely generated. ≥ M They also showed that in [5], νn,1 is surjective for n 4. Hence IAn is finitely ≥ M generated for n 4. It is, however, not known whether or not IAn is finitely presented for n ≥ 4. From now on, we consider the case where n ≥ 4. Set Kn := Ker(νn). Since Kn ⊂ IAn, we have an exact sequence →K → → M → (9) 1 n IAn IAn 1.

Furthermore, observing Kn ⊂An(2) = [IAn, IAn], we obtain M ab ∼ ab ∼ ∗ ⊗ 2 (10) (IAn ) = IAn = H Z Λ H

M and see that the first Johnson homomorphism τ1 of Aut Fn is an isomorphism.

M 2.3.3. Johnson homomorphisms of Aut Fn . Here we consider the Johnson ho- M k M momorphisms of Aut (F ). We denote A M (k)andgr(A M )byA (k)and n Fn Fn n k M  k  gr (A ) respectively. Furthermore, we also denote A M (k)andgr(A M )by n Fn Fn AM k AM n (k)andgr ( n ) respectively. For each k ≥ 1, restricting νn to An(k), we obtain a homomorphism νn,k : A →AM 1 AM → ∗ ⊗ 2 n(k) n (k). Since τ1 :gr( n ) H Z Λ H is an isomorphism, we AM AM see that n (2) = n (2), and hence νn,2 is surjective. However it is not known whether or not νn,k is surjective for k ≥ 3. Now, the main aim of the paper is to determine the GL(n, Z)-module structure M of the cokernel of the Johnson homomorphisms of Aut Fn .Inthispaper,wegive an answer to this problem for the case where k ≥ 2andn ≥ 4. We remark that by an argument similar to that in Subsection 2.2, we can recursively compute τk(σ)=  ∈AM ∗ ⊗ τk(σ)forσ n (k), using τ1(νn,1(Kij)) = xi [xi,xj]andτ1(νn,1(Kijk)) = ∗ ⊗ xi [xj,xk].

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2.4. Magnus representations. In this subsection we recall the Magnus represen- M ≤ ≤ tation of Aut Fn and Aut Fn . (For details, see [7].) For each 1 i n,let ∂ : Z[Fn] → Z[Fn] ∂xi be the Fox derivation defined by r ∂ 1 − 1 2 (j 1) (w)= j δµ ,ix ···xµ ∈ Z[Fn] ∂x j µ1 j i j=1

1 ··· r ∈ ± → for any reduced word w = xµ1 xµr Fn, j = 1. Let a : Fn H be the abelianization of Fn. Wealsodenotebya the ring homomorphism Z[Fn] → Z[H] ∈ a a ∈ induced from a. For any A =(aij) GL(n, Z[Fn]), let A be the matrix (aij) GL(n, Z[H]). The Magnus representation rep : Aut Fn → GL(n, Z[H]) of Aut Fn is defined by ∂x σ a σ → i ∂xj

for any σ ∈ Aut Fn. This map is not a homomorphism but a crossed homomor- phism. Namely, ∗ rep(στ)=(rep(σ))τ · rep(τ), ∗ where (rep(σ))τ denotes the matrix obtained from rep(σ) by applying the auto- morphism τ ∗ : Z[H] → Z[H] induced from ρ(τ) ∈ Aut(H) on each entry. Hence by restricting rep to IAn, we obtain a homomorphism rep : IAn → GL(n, Z[H]). This is called the Magnus representation of IAn. M M M → Next, we consider the Magnus representation of IAn .Letrep :IAn GL(n, Z[H]) be a map defined by ∂(x σ) a σ → i ∂xj ∈ M σ ∈ M for any σ IAn , where we consider any lift of the element xi Fn to Fn.Then M M we see rep is a homomorphism and rep = rep ◦ νn,1, and call it the Magnus M M representation of IAn . Bachmuth [2] showed that rep is faithful, and determined the image of repM in GL(n, Z[H]). The faithfulness of the Magnus representation M rep shows that the kernel of the Magnus representation rep is equal to Kn.

3. The cokernel of the Johnson homomorphisms

In this section, we determine the cokernel of the Johnson homomorphism τk of M ≥ ≥ Aut Fn for k 2andn 4.

3.1. Upper bound on the rank of cokernel of τk. First we give an upper bound on the rank of the cokernel of τk by reducing its set of generators. By Lemma 2.1, ∗ ⊗ ∗ ⊗ LM we see that elements of type xi [xi1 ,xi2 ,...,xik+1 ] generate H Z n (k +1). First we prepare some lemmas. Let Sl be the symmetric group of degree l.Then we have ≥ ≥ ∈ Lemma 3.1. Let l 2 and n 2. For any element [xi1 ,xi2 ,xj1 ,...,xjl ] LM ∈ n (l +2) and any λ Sl,

[xi1 ,xi2 ,xj1 ,...,xjl ]=[xi1 ,xi2 ,xjλ(1) ,...,xjλ(l) ].

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Proof. Since Sl is generated by transpositions (mm+1)for1 ≤ m ≤ l − 1, it suffices to prove the lemma for each λ =(mm+1).Nowwehave

[[[xi1 ,xi2 ,xj1 ,...,xjm−1 ],xjm ],xjm+1 ] − = [[xjm ,xjm+1 ], [xi1 ,xi2 ,xj1 ,...,xjm−1 ]] − [[xjm+1 , [xi1 ,xi2 ,xj1 ,...,xjm−1 ]],xjm ]

= [[[xi1 ,xi2 ,xj1 ,...,xjm−1 ],xjm+1 ],xjm ] LM in n (m + 3) by Jacobi’s identity. Hence,

[xi1 ,xi2 ,xj1 ,...,xjl ]=[xi1 ,xi2 ,xj1 ,...,xjm−1 ,xjm+1 ,xjm ,...,xjl ]

=[xi1 ,xi2 ,xjλ(1) ...,xjλ(l) ] LM
in n (l +2).

Lemma 3.2. Let k ≥ 1 and n ≥ 4. For any i and i1,i2,...,ik+1 ∈{1, 2 ...,n},if i1,i2 = i, ∗ ⊗ ∈  xi [xi1 ,xi2 ,...,xik+1 ] Im(τk). ∈LM Proof. First, using Lemma 3.1, we remark that [xi1 ,xi2 ,...,xik+1 ] n (k +1)is rewritten as

[xi1 ,xi2 ,xi3 ,...,xil−1 ,xi,xi,...,xi] LM ≤ ≤  in n (k +1)forsomel,3 l k +2.Namely,i1,i2,...,il−1 = i and il = il+1 = ···= ik+1 = i.  We prove this lemma by induction on k.Ifk =1,by(7),wehaveτ1(νn,1(Kii1i2 )) ∗ ⊗ ≥ ≥ ∈ = xi [xi1 ,xi2 ]. Suppose k 2. Since n 4, we can choose an element j M {1, 2,...,n} such that j = i, i1,i2. Define σ ∈A (1) to be n ν (K−1),l≤ k +1, σ = n,1 ij νn,1(Kiik+1 ),l= k +2. Then x∗ ⊗ [x ,x ],l≤ k +1, τ  (σ)= i j i 1 ∗ ⊗ xi [xi,xik+1 ],l= k +2. By the inductive hypothesis, we have an element τ ∈AM (k − 1) such that n x∗ ⊗ [x ,x ,x ,...,x ,x ,x ,...,x ],l≤ k +1, τ  (τ)= j i1 i2 i3 il−1 i i i k−1 ∗ ⊗ xi [xi1 ,xi2 ,xi3 ,...,xik+1 ],l= k +2

where xi appears k − l + 1 times among the components of the commutator above. Then we obtain  ∗ ⊗ τk([σ, τ ]) = xi [xi1 ,xi2 ,xi3 ,...,xil−1 ,xi,xi,...,xi],

where xi appears k − l + 2 times among the components. This completes the proof of Lemma 3.2.

Lemma 3.3. Let k ≥ 1 and n ≥ 4. For any i and i1,i2,...,ik ∈{1, 2,...,n} such that i1,i2 = i, and any transposition λ =(mm+1)∈ Sk, ∗ ⊗ − ∗ ⊗ ∈  xi [xi,xi1 ,...,xik ] xi [xi,xiλ(1) ,...,xiλ(k) ] Im(τk).

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Proof. From Lemma 3.1, we have ∗ ⊗ − ∗ ⊗ xi [xi,xi1 ,...,xik ] xi [xi,xiλ(1) ,...,xiλ(k) ]=0 inthecasewhereλ =(mm+1)for2≤ m ≤ k − 1. Set λ = (1 2). Since − − [[xi,xi1 ],xi2 ] [[xi,xi2 ],xi1 ]= [[xi1 ,xi2 ],xi], by the Jacobi’s identity we have ∗ ⊗ − ∗ ⊗ − ∗ ⊗ xi [xi,xi1 ,xi2 ,...,xik ] xi [xi,xi2 ,xi1 ,...,xik ]= xi [xi1 ,xi2 ,xi,...,xik ]. Therefore Lemma 3.3 follows from Lemma 3.2 immediately.

Lemma 3.4. Let k ≥ 1 and n ≥ 4. For any i2,...,ik+1 ∈{1, 2,...,n}, we have ∗ ⊗ − ∗ ⊗ ∈  xi [xi,xi2 ,...,xik+1 ] xj [xj,xi2 ,...,xik+1 ] Im(τk)

for any i = i2 and j = i2,ik+1. Proof. We may assume j = i.First,set  νn,1(Kijik+1 ), if ik+1 = i, τ = −1 νn,1(Kij ), if ik+1 = i.  ∗ ⊗ ∈AM − Then we have τ1(τ)=xi [xj,xik+1 ]. From Lemma 3.2, there exists a σ n (k 1) such that  ∗ ⊗ τk−1(σ)=xj [xi,xi2 ,...,xik ]. Henceweobtain  ∗ ⊗ − ∗ ⊗ τk([τ,σ]) = xi [xi,xi2 ,...,xik+1 ] xj [xj,xik+1 ,xi2 ,...,xik ]. On the other hand, by Lemma 3.3, we have ∗ ⊗ − ∗ ⊗ ∈  xj [xj,xik+1 ,xi2 ,...,xik ] xj [xj,xi2 ,...,xik+1 ] Im(τk), and hence the lemma.

Using the lemmas above, we can reduce the generators of Coker(τk). We remark  ⊂ that Im(τk) Im(τk). ≥ ≥ n+k−1 Proposition 3.1. For k 2 and n 4, Coker(τk) is generated by k ele- ments. ∗ ⊗ LM Proof. First, as mentioned above, H Z n (k + 1) is generated by { ∗ ⊗ | ≤ ≤ } xi [xi1 ,...,xik+1 ] 1 i, i1,...,ik+1 n . From Lemma 3.2, ∗ ⊗ ≡ xi [xi1 ,...,xik+1 ] 0  ∗ ⊗ ∈ in Coker(τk)ifi1,i2 = i. For any xi [xi,xi2 ,...,xik+1 ], there exists a j {1, 2,...,n} such that j = i2,ik+1 since n ≥ 4. Then ∗ ⊗ ∗ ⊗ ∈ xi [xi,xi2 ,...,xik+1 ]=xj [xj,xi2 ,...,xik+1 ] Coker(τk)

by Lemma 3.4. Hence Coker(τk) is generated by { ∗ ⊗ | ≤ ≤  } E = xi [xi,xi2 ,...,xik+1 ] 1 i, i2,...,ik+1 n, i = i2,ik+1 . ∗ ⊗ ∈ Furthermore, since xi [xi,xi1 ,...,xik ] E does not depend on i by Lemma 3.4, ∗ ⊗ we can set s(i1,...,ik):=xi [xi,xi1 ,...,xik ]. Next, we show s(i1,...,ik)=s(iλ(1),...,iλ(k))

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for any λ ∈ Sk. It suffices to show above for each transposition λ =(mm+1).If 2 ≤ m ≤ k − 2, it is trivial by Lemma 3.3. If m = k − 1, take j ∈{1,...,n} such that j = i1,ik−1,ik.Then ∗ ⊗ ∗ ⊗ s(i1,...,ik−1,ik)=xj [xj,...,xik−1 ,xik ]=xj [xj,...,xik ,xik−1],

= s(i1,...,ik,ik−1)

from Lemma 3.3. Similarly, if m =1,takej ∈{1,...,n} such that j = i1,i2,ik. Then we obtain s(i1,i2,...,ik)=s(i2,i1,...,ik). Hence, we see  E = {s(i1,...,ik) ∈ Coker(τk) | i1 ≤···≤ik}  n+k−1 generates Coker(τk). The order of E is ( k ). This completes the proof of Proposition 3.1.

3.2. Lower bound on the rank of the cokernel of τk. In this subsection we give a lower bound on the rank of Coker(τk) by using the Magnus representation M of Aut Fn . To do this, we use trace maps introduced by Morita [24] with pioneer and remarkable works. Recently, he showed that there is a symmetric product of H of degree k in the cokernel of the Johnson homomorphism of the automorphism group of a free group using trace maps. Here we apply his method to the case M for Aut Fn . In order to define the trace maps, we prepare some notation of the associated algebra of the integral group ring. (For basic materials, see [30], Chapter VIII.) For a group G,letZ[G] be the integral group ring of G over Z. Wedenotethe augmentation map by  : Z[G] → Z.ThekernelIG of  is called the augmentation i ≥ ideal. Then the powers of IG for i 1 provide a descending filtration of Z[G], and the direct sum k k+1 IG := IG/IG k≥1 naturally has a graded algebra structure induced from the multiplication of Z[G]. We call IG the associated algebra of the group ring Z[G].

For G = Fn a free group of rank n,writeIn and In for IFn and IFn respectively. k k+1 It is classically well known due to Magnus [23] that each graded quotient In/In { − − ··· − | ≤ ≤ } is a free abelian group with basis (xi1 1)(xi2 1) (xik 1) 1 ij n ,and k k+1 → ⊗k amapIn/In H defined by − − ··· − → ⊗ ⊗···⊗ (xi1 1)(xi2 1) (xik 1) xi1 xi2 xik induces an isomorphism from I to the tensor algebra n T (H):= H⊗k k≥1 k k+1 ⊗k of H as a graded algebra. We identify In/In with H via this isomorphism. k k+1 ItisalsowellknownthateachgradedquotientIH /IH is a free abelian group { − − ··· − | ≤ ≤ ≤ ··· ≤ ≤ } with basis (xi1 1)(xi2 1) (xik 1) 1 i1 i2 ik n ,andthe associated graded algebra I of H is isomorphic to the symmetric algebra H S(H):= SkH k≥1 of H as a graded algebra. (See [30], Chapter VIII, Proposition 6.7.) We also identify k k+1 k k k+1 → k k+1 IH /IH with S H. Then a homomorphism In/In IH /IH induced from the ⊗k k abelianization a : Fn → H is considered as the natural projection H → S H.

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∈ ∗ ⊗ LM Now, we define trace maps. For any element f H Z n (k +1),set ∂(x f ) a f := i ∈ M(n, SkH) ∂xj where we consider any lift of the element f ∈LM · ∩  xi n (k +1)=Γn(k +1)/(Γn(k +2) Γn(k +1) Fn ) M ∗ ⊗ LM → k to Γn(k + 1). Then we define a map Trk : H Z n (k +1) S H by M Trk (f):=trace( f ). M M It is easily seen that Trk is a GL(n, Z)-equivariant homomorphism. The maps Trk M M ◦ are called the Morita trace maps. We show that Trk is surjective and Trk τk =0 for k ≥ 2andn ≥ 3. By a direct computation, we obtain ∗ ⊗ ∈ ∗ ⊗ LM Lemma 3.5. For f = xi [xi1 ,xi2 ,...,xik+1 ] H Z n (k +1), we have M − k{ ··· − ··· } Trk (f)=( 1) δi1i xi2 xi3 xik+1 δi2i xi1 xi3 xik+1 ,

where δij is the Kronecker delta. ≥ ≥ M Lemma 3.6. For any k 1 and n 2, Trk is surjective. ··· ∈ k ∈{ } Proof. For any generator xi1 xi2 xik S H,wecanchooseanumberi 1,...,n such that i = i1 since n ≥ 2. Then by Lemma 3.5, M ∗ ⊗ − k ··· Trk (xi [xi,xi1 ,...,xik ]) = ( 1) xi1 xi2 xik . M
This shows Trk is surjective. M ◦ Before showing Trk τk = 0, we consider a relation between the Magnus repre- sentation and the Johnson homomorphism. For each k ≥ 1, composing the Magnus M AM → representation rep restricted to n (k) with a homomorphism GL(n, Z[H]) k+1 → k+1 GL(n, Z[H]/IH ) induced from a natural projection Z[H] Z[H]/IH ,weob- M AM → k+1 tain a homomorphism repk : n (k) GL(n, Z[H]/IH ). By the definition of the Magnus representation and the Johnson homomorphism, we obtain M (11) repk (σ)=I + τk(σ) , where I denotes the identity matrix. (See also [24].) ≥ ≥ M Proposition 3.2. For k 2 and n 3, Trk vanishes on the image of τk. ◦ M ∈AM Proof. By Bachmuth [2], we have det rep (σ) = 1 for any σ n (2). This shows that ◦ M M 1=det repk (σ)=1+Trk (τk(σ)) ∈AM M
for any σ n (k). Hence Trk (τk(σ)) = 0. As a corollary, we have Corollary 3.1. For k ≥ 2 and n ≥ 3, n + k − 1 rank (Coker(τ )) ≥ . Z k k

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Combining this corollary with Proposition 3.1, we obtain Theorem 3.1. For k ≥ 2 and n ≥ 4,

Tr M → k AM −τ→k ∗ ⊗ LM −−−k→ k → 0 gr ( n ) H Z n (k +1) S H 0 is a GL(n, Z)-equivariant exact sequence.

M Proof. It suffices to show that Im(τk)=Ker(Trk ). This immediately follows from Proposition 3.1 and Corollary 3.1. This completes the proof of the theorem.

From (8), we obtain Corollary 3.2. For k ≥ 2 and n ≥ 4, n + k − 1 n + k − 1 rank (grk(AM )) = nk − . Z n k +1 k k A → k AM Let νn,k :gr( n) gr ( n ) be the homomorphism induced from νn,k.By theargumentabove,weseethatIm(τk ◦ νn,k)=Im(τk). Since τk is injective, this shows that νn,k is surjective. Hence Corollary 3.3. For k ≥ 2 and n ≥ 4, n + k − 1 n + k − 1 rank (grk(A )) ≥ nk − . Z n k +1 k As mentioned before, in the inequality above, equality does not hold in general, 3 4 2 since rankZ gr (An)=n(3n − 7n − 8)/12, which is not equal to the right hand side of the inequality.

4. Theimageofthecupproduct in the second cohomology group

M In this section, we consider the rational second (co)homology group of IAn .In particular, we determine the image of the cup product map ∪M 2 1 M → 2 M Q :Λ H (IAn , Q) H (IAn , Q). 4.1. A minimal presentation and second cohomology of a group. In this subsection, we consider detecting non-trivial elements of the second cohomology group H2(G, Z)ifG has a minimal presentation. For a group G, a group extension ϕ (12) 1 → R → F −→ G → 1 is called a minimal presentation of G if F is a free group such that ϕ induces an isomorphism

ϕ∗ : H1(F, Z) → H1(G, Z). This shows that R is contained in the commutator subgroup [F, F]ofF .Inthe following, we assume that G has a minimal presentation defined by (12), and fix it. Furthermore we assume that the rank m of F is finite. We remark that considering M M the Magnus generators of IAn and IAn , we see that each of IAn and IAn has such a minimal presentation. From the cohomological five-term exact sequence of (12), we see 2 ∼ 1 G H (G, Z) = H (R, Z) .

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Set LF (k)=ΓF (k)/ΓF (k +1)foreach k ≥ 1. Then LF (k) is a free abelian group of rank rm(k)by(3).Let{Rk}k≥1 be a descending filtration defined by Rk := R ∩ ΓF (k)foreachk ≥ 1. Then Rk = R for k = 1 and 2. For each k ≥ 1, let

ϕk : LF (k) →LG(k) be a homomorphism induced from the natural projection ϕ : F → G. Observing ∼ Rk/Rk+1 = (Rk ΓF (k +1))/ΓF (k + 1), we have an exact sequence

ιk ϕk (13) 0 → Rk/Rk+1 −→LF (k) −−→LG(k) → 0.

ThisshowseachgradedquotientRk/Rk+1 is a free abelian group. Set Rk := R/Rk. The natural projection R → Rk induces an injective homo- morphism k 1 1 ψ : H (Rk, Z) → H (R, Z). Considering the right action of F on R, defined by r · x := x−1rx, r ∈ R, x ∈ F, we see ψk is a G-equivariant homomorphism. Hence it induces an injective homo- morphism, also denoted by ψk:

k 1 G 1 G ψ : H (Rk, Z) → H (R, Z) .

1 G 1 For k =3,H (R3, Z) = H (R3, Z), since G acts on R3 trivially. Here we show that the image of the cup product ∪ :Λ2H1(G, Z) → H2(G, Z) is contained in 1 H (R3, Z). Lemma 4.1. If G has a minimal presentation as above, the image of the cup product ∪ :Λ2H1(G, Z) → H2(G, Z) ∗ 1 L → 1 is isomorphic to the image of ι2 : H ( F (2), Z) H (R3, Z). Proof. First, considering the cohomological five-term exact sequence of →A → → ab → (14) 1 G(2) G G 1, we have → 1 ab → 1 → 1 A G → 2 ab → 2 0 H (G , Z) H (G, Z) H ( G(2), Z) H (G , Z) H (G, Z). 1 ab ∼ 1 1 A G 1 2 A Since H (G , Z) = H (G, Z)andH ( G(2), Z) = H (gr ( G), Z), we obtain an exact sequence → 1 2 A → 2 ab −→ 2 0 H (gr ( G), Z) H (G , Z) H (G, Z). Since H (G, Z) is a free abelian group of finite rank, we have a natural isomorphism 1 ∼ H2(Gab, Z) = Λ2H1(G, Z). Then the map H2(Gab, Z) −→ H2(G, Z) is regarded as the cup product ∪ :Λ2H1(G, Z) → H2(G, Z). On the other hand, we also consider a five-term exact sequence  L → 1 2 A → 1 L → 1 F (2) 0 H (gr ( G), Z) H ( F (2),Z) H (R3, Z) → 2 2 A → 2 L H (gr ( G), Z) H ( F (2), Z)

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1 LF (2) of (13) for k =2.SinceLF (2) acts on R3 trivially, we have H (R3, Z) = 1 H (R3, Z). Then we have a commutative diagram

tg ∪ 0 −−−−→ H1(gr2(A ), Z) −−−−→ H2(Gab, Z) −−−−→ H2(G, Z) G ⏐ ⏐ µ

ϕ∗ ι∗ −−−−→ 1 2 A −−−−2 → 1 L −−−−2 → 1 0 H (gr ( G), Z) H ( F (2), Z) H (R3, Z) where tg is the transgression and µ is a natural isomorphism. Hence we obtain ∪ ∼ ∗
Im( ) = Im(ι2). This completes the proof of Lemma 4.1.

2 A ∪ 1 Here we remark that if gr ( G) is free abelian group, Im( )=H (R3, Z). Fur- 2 1 2 thermore if we consider the rational cup product ∪Q :ΛH (G, Q) → H (G, Q), ∗ 1 L → since Q is a Z-injective module, the induced homomorphism ι2 : H ( F (2), Q) 1 H (R3, Q) is surjective. Hence the image of the rational cup product ∪Q is equal 1 to H (R3, Q).

∪M 4.2. The image of the rational cup product Q . In this subsection, we de- termine the image of the rational cup product ∪M 2 1 M → 2 M Q :Λ H (IAn , Q) H (IAn , Q).

2 1 First, we should remark that the image of the cup product ∪Q :Λ H (IAn, Q) → 2 H (IAn, Q) is completely determined by Pettet [31] who gave the GL(n, Q)-irreduc- ∗ 2 M → ible decomposition of it. Here we show that the restriction of νn,1 : H (IAn , Q) 2 ∪M ∪ H (IAn, Q)toIm( Q ) is an isomorphism onto Im( Q). To do this, we prepare some notation. Let F be a free group on Kij and Kijk which are corresponding to the Magnus generators of IAn.Namely,F is a free group of rank n2(n − 1)/2. Then we have a natural surjective homomorphism ϕ : F → IAn and a minimal presentation ϕ (15) 1 → R → F −→ IAn → 1

of IAn,whereR =Ker(ϕ). From a result of Pettet [31], we have

Lemma 4.2. For n ≥ 3, CR3 is a free abelian group of rank 1 1 α(n):= n2(n − 1)(n3 − n2 − 2) − n(n + 1)(2n2 − 2n − 3). 8 6 M Next, we consider the second cohomology groups of IAn .Fromnowon,we ≥ → M assume n 4. We recall that the natural homomorphism νn,1 :IAn IAn is ab ∼ M ab ∼ ∗ ⊗ 2 surjective, and νn,1 induces an isomorphism IAn = (IAn ) = H Z Λ H for ≥ M ◦ → M n 4. Then we have a surjective homomorphism ϕ := νn,1 ϕ : F IAn and a minimal presentation

M → M → −ϕ−→ M → (16) 1 R F IAn 1

M M of IAn ,whereR =Ker(ϕ). Observe a sequence 2 A → 2 AM → 2 AM grQ( n) grQ( n ) grQ( n )

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A A of surjective homomorphisms. Since n(3)/ n(3) is at most a finite abelian group due to Pettet [31], we see 1 dim (gr2 (A )) = dim (gr2 (A )) = n(n + 1)(2n2 − 2n − 3) Q Q n Q Q n 6 2 AM =dimQ(grQ( n )) 2 AM ∼ 2 AM by (6), and hence grQ( n ) = grQ( n ). Thus,

≥ M Lemma 4.3. For n 4, CR3 is a free abelian group of rank α(n). Proof. Considering the exact sequence of (13) with respect to the minimal presen- tation (16) for k =2andtensoringitwithQ,weobtain

M ϕ → M →LQ −−−2,→Q 2 A M → 0 (R3 )Q F (2) grQ( n ) 0. Hence M M rankZ(R3 )=dimQ((R3 )Q) LQ − 2 AM =dimQ( F (2)) dimQ(grQ( n )) 1 1 = n2(n − 1)(n3 − n2 − 2) − n(n + 1)(2n2 − 2n − 3). 8 6

Therefore, from the functoriality of the spectral sequence, we obtain commuta- tivity of a diagram 0 −−−−→ H1(RM , Q) −−−−→ H2(IAM , Q) ⏐3 ⏐n ∼⏐ ⏐ ∗ = νn,1

1 2 0 −−−−→ H (R3, Q) −−−−→ H (IAn, Q) and ≥ ∗ ∪M → ∪ Theorem 4.1. For n 4, νn,1 :Im( Q ) Im( Q) is an isomorphism. ∪M 2 1 M In Subsection 5.2, we will show the rational cup product Q :ΛH (IAn , Q) → 2 M H (IAn , Q) is not surjective.

5. On the kernel of the Magnus representation of IAn

In this section, we study the kernel Kn of the Magnus representation of IAn for 3 n ≥ 4. Set Kn := Kn/(Kn ∩An(4)) ⊂ gr (An). Since [Kn, Kn] ⊂An(6), we see K K KQ H1( n, Z)= n. Here we determine the GL(n, Z)-module structure of n .Asa ∪M 2 1 M → 2 M corollary, we see that the rational cup product Q :Λ H (IAn , Q) H (IAn , Q) is not surjective.

KQ 5.1. The irreducible decompositon of n . First, we consider the irreducible ∗ ⊗ LQ decomposition of the target HQ Q n (4) of the rational third Johnson homomor-  phism τ3,Q of Aut Fn.LetB and B be subsets of Ln(4) consisting of

[[[xi,xj],xk],xl],i>j≤ k ≤ l,

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and

[[xi,xj], [xk,xl]], i>j,k>l,i>k,

[[xi,xj], [xi,xl]], i>j,i>l,j>l,  respectively. Then B ∪ B forms a basis of Ln(4) due to Hall [15]. Let Gn be the GL(n, Z)-equivariant submodule of Ln(4) generated by elements type of  [[xi,xj], [xk,xl]] for 1 ≤ i, j, k, l ≤ n.ThenB is a basis of Gn, and the quo- L G LM GQ tient module of n(4) by n is isomorphic to n (4). Observing that n is a LQ ∼ [3,1] ⊕ [2,1,1] GQ GL(n, Z)-equivariant submodule of n (4) = HQ HQ , and dimQ( n )= 2 − GQ ∼ [2,1,1] LM ∼ [3,1] n n(n 1)(n +2)/8, we see n = HQ and n,Q(4) = HQ .LetD := Λ H be the one-dimensional representation of GL(n, Z) given by the determinant map. ∗ ∼ ⊗ n−1 Then considering a natural isomorphism HQ = (D Q Λ HQ)asaGL(n, Z)- module, and using Pieri’s formula (see [13]), we obtain Lemma 5.1. For n ≥ 4, 3 2 n−4 ∗ ⊗ GQ ∼ [1 ] ⊕ [2,1] ⊕ ⊗ [3,2 ,1 ] (i) HQ Z n = HQ HQ (D Q HQ ), n−3 ∗ ⊗ LM ∼ [3] ⊕ [2,1] ⊕ ⊗ [4,2,1 ] (ii) HQ Z n,Q(4) = HQ HQ (D Q HQ ).

KQ ⊂ ∗ ⊗ GQ Now it is clear that τ3,Q( n ) HQ Z n . On the other hand, in our previous paper [33], we showed that the cokernel of the rational Johnson homomorphism τ3,Q [3] ⊕ [13] KQ is given by Coker(τ3,Q)=HQ HQ . Hence we see that τ3,Q( n ) is isomorphic to − [2,1] ⊕ ⊗ [3,22,1n 4] KQ ∼ a submodule of HQ (D Q HQ ). In the following, we show τ3,Q( n ) = − [2,1] ⊕ ⊗ [3,22,1n 4] HQ (D Q HQ ). To show this, we prepare some elements of Kn. First, for any distinct p, q, r, s ∈ {1, 2,...,n} such that p>q,rand q>r,set −1 −1 ∈ T (s, p, q, r):=[[Ksp ,Ksr ],Ksqp] IAn. Since T (s, p, q, r)satisfies xs[[xp,xq], [xp,xr]], if t = s, xt → xt, if t = s, ∈K ∗ ⊗ ∈ ∗ ⊗ G T (s, p, q, r) n and τ3(T (s, p, q, r)) = xs [[xp,xq], [xp,xr]] H Z n.Next, for any distinct p, q, r, s ∈{1, 2,...,n} such that p>s,set −1 −1 −1 ∈ E(s, p, q, r):=[[Ksr,Kspq],Krsq](Krs [[Krs,Kspq] ,Krq ]Krs) IAn. Then we have Lemma 5.2. For any n ≥ 4, ∗ ⊗ ∈ ∗ ⊗ G (i) τ3(E(s, p, q, r)) = xs [[xp,xq], [xs,xq]] H Z n. (ii) E(s, p, q, r) ∈Kn. Proof. Part (i) follows from a direct computation. To prove (ii), we show that ∈ M M νn(E(s, p, q, r)) IAn fix each xt of Fn . Here we recall two basic formulae in commutator calculus. For any group G and x, y, z ∈ G, (17) [xy, z]=[x, [y, z]][y, z][x, z], (18) [x, yz]=[x, y][x, z][[z,x],y].

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First, we consider νn([[Ksr,Kspq],Krsq]). Since [Ksr,Kspq] ∈ IAn satisfies −1 xs[[xq,xp],xr ], if t = s, xt → xt, if t = s,

we see that νn([[Ksr,Kspq],Krsq]) fixes xt for t = s, r, and maps xs and xr as follows:

−ν−−−−−−−−−n([Ksr,Kspq→]) −1 −ν−−−−−n(Krsq→) −1 xs xs[[xq,xp],xr ] xs[[xq,xp], [xq,xs]xr ] (18) · −1 · −1 = xs[[xq,xp], [xq,xs]] [[xq,xp],xr ] [[xr , [xq,xp]], [xq,xs]] −1 −1 −1 −ν−−−−−−−−−−n([Ksr,Kspq ]) → −−−−−−−→νn(Krsq) = xs[[xq,xp],xr ] xs xs and

−1 −ν−−−−−−−−−n([Ksr,Kspq→]) −ν−−−−−n(Krsq→) −ν−−−−−−−−−−n([Ksr,Kspq ]) → −1 xr xr xr[xs,xq] xr[xs[xr , [xq,xp]],xq] (17) −1 · −1 · = xr[xs, [[xr , [xq,xp]],xq]] [[xr , [xq,xp]],xq] [xs,xq] · −1 −1 · = xr (xs[[xr , [xq,xp]],xq]xs ) [xs,xq] −1 −−−−−−−→νn(Krsq) · −1 −1 xr (xs[[[xs,xq]xr , [xq,xp]],xq]xs ) (17) · −1 −1 = xr (xs[[xr , [xq,xp]],xq]xs ). ± Next, consider [Krs,Kspq] ∈ IAn. Clearly, these maps fix xt for t = r,andmap xr as follows:

−[−−−−−−Krs,Kspq→] −1 xr xr[xr , [xq,xp]], −1 −−−−−−−−→[Krs,Kspq ] −1 −1 xr xr[[xp,xq], [[xq,xp],xr ]][[xq,xp],xr ]. Observing −1 − νn([Krs,Kspq ] ) 1 ∈ M xr =[[xq,xp],xr ] Fn , we have ν ([[K ,K ]−1,K−1]) −−−−−−−−−−−−−−−→n rs spq rq −1 xr xr[xq, [xr , [xq,xp]]] and ν (K−1[[K ,K ]−1,K−1]K ) −−−−−−−−−−−−−−−−−−−−−n rs rs spq rq rs→ −1 −1 xr xrxs[xq, [xr , [xq,xp]]]xs . Therefore we obtain

−ν−−−−−−−−n(E(s,p,q,r))→ −1 −1 xr xr (xs[xq, [xr , [xq,xp]]]xs ) · −1 −1 −1 −1 (xs[[(xs[[xr , [xq,xp]],xq]xs ) xr , [xq,xp]],xq]xs ) (17) −1 −1 · −1 −1 = xr (xs[xq, [xr , [xq,xp]]]xs ) (xs[[xr , [xq,xp]],xq]xs )=xr. This complets the proof of Lemma 5.2.

− ≥ KQ ∼ [2,1] ⊕ ⊗ [3,22,1n 4] Theorem 5.1. For n 4, τ3,Q( n ) = HQ (D Q HQ ).

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∗ [2,1] Proof. Let Φ : H ⊗Z Ln(4) → H be a map defined by the composition of maps

⊗ f ∗ id ι4 ∗ ⊗4 C ⊗3 [2,1] [2,1] Φ:H ⊗Z Ln(4) −−−→ H ⊗Z H −→ H −−−→ H ⊗4 where ι4 : Ln(4) → H is a natural embedding defined by the rule [X, Y ] → ∗ X ⊗ Y − Y ⊗ X,amapC is a contraction defined by C(x ⊗ x1 ⊗···⊗x4)= ∗ x (x1)x2 ⊗ x3 ⊗ x4,andf[2,1] is a natural projection. Then Φ is a GL(n, Z)- equivariant homomorphism. For the Johnson homomorphism τ3 of Aut,Fn,wedenotebyΦ˜ the restriction of Φtoτ3(Kn). By Lemma 5.2,

Φ˜ Q(E(s, p, q, r)) = −2xq ⊗ xp ∧ xq. { ⊗ ∧ | ≤ } [2,1] ˜ Since xi xj xk j>k i is a basis of HQ ,weseeIm(Φ) is non-trivial. ˜ [2,1] Hence Φ is surjective since HQ is a irreducible GL(n, Z)-module. This shows that KQ [2,1] τ3,Q( n )containsHQ as an irreducible component. On the other hand, since Φ(˜ T (s, p, q, r)) = 0,

− ∈ ˜ ⊗ [3,22,1n 4] T (s, p, q, r) Ker(Φ) = (D Q HQ ). ⊗   KQ KQ Since (id ι4)Q(T (s, p, q, r)) =0,T (s, p, q, r) =0in n . This shows that τ3,Q( n ) − ⊗ [3,22,1n 4] also contains (D Q HQ ) as an irreducible component. Thus we conclude − KQ ∼ [2,1] ⊕ ⊗ [3,22,1n 4]
that τ3,Q( n ) = HQ (D Q HQ ).

Since τ3,Q is injective, this shows that

− KQ ∼ [2,1] ⊕ ⊗ [3,22,1n 4] n = HQ (D Q HQ ) and Corollary 5.1. For n ≥ 4, 1 1 rank (H (K , Z)) ≥ n(n2 − 1) + n2(n − 1)(n +2)(n − 3). Z 1 n 3 8 ∪M 5.2. Non-surjectivity of the cup product Q . In this subsection, we also as- ≥ ∪M 2 1 M → sume n 4. Here we show that the rational cup product Q :Λ H (IAn , Q) 2 M H (IAn , Q) is not surjective. From the rational five-term exact sequence

→ 1 M → 1 → 1 K IAn → 2 M → 2 0 H (IAn , Q) H (IAn, Q) H ( n, Q) H (IAn , Q) H (IAn, Q) of (9), we have an exact sequence

→ 1 K IAn → 2 M → 2 0 H ( n, Q) H (IAn , Q) H (IAn, Q). ∪M By Theorem 4.1, to show the non-surjectivity of the cup product Q it suffices to 1 IAn show the non-triviality of H (Kn, Q) . The natural projection Kn → Kn induces an injective homomorphism

1 1 IAn H (Kn, Q) → H (Kn, Q) . By Theorem 5.1 and the universal coefficients theorem, we see 1 ∼ H (Kn, Q) = HomZ(H1(Kn, Z), Q) =0 .

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Therefore we obtain Theorem 5.2. For n ≥ 4, the rational cup product ∪M 2 1 M → 2 M Q :Λ H (IAn , Q) H (IAn , Q) is not surjective, and 1 dim (H2(IAM , Q)) ≥ n(n − 2)(3n4 +3n3 − 5n2 − 23n − 2). Q n 24 Acknowledgments The author would like to thank Professor Nariya Kawazumi for valuable advice and useful suggestions. He would also like to express his thanks to the referee for helpful comments and correcting typos and grammatical mistakes. This research is supported by JSPS Research Fellowships for Young Scientists.

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Department of Mathematics, Graduate School of Sciences, Osaka University, 1-16 Machikaneyama, Toyonaka-city, Osaka 560-0043, Japan E-mail address: [email protected]

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