数理解析研究所講究録
第 1660 巻 2009 年 95-113

95

THE COKERNEL OF THE JOHNSON HOMOMORPHISMS OF THE
AUTOMORPHISM GROUP OF A FREE METABELIAN GROUPTAKAO SATOH
Graduate School of Sciences, Department of Mathematics, Osaka University
1-16 Machikaneyama, Toyonaka-city, Osaka 560-0043, Japan
ABSTRACT. In this paper, we determine the cokernel of the k-th Johnson homomorphisms of the automorphism group of afree metabelian group for $k\geq 2$ and $n\geq 4$ .
As acorollary, we obtain alower bound on the rank of the graded quotient of theJohnson filtration of the automorphism group of afree group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup productmap in the rational second cohomoloy group of the IA-automorphism group of afree
metabelian group, and show that it is isomorphic to that of the IA-automorphismgroup of afree group which is already determined by Pettet. Finally, by considering
the kernel of the Magnus representations of the automorphism group of afree group
and afree metabelian group, we show that there are non-trivial rational second cohomology claeses of the $IAarrow automorphism$ group of ametabelian group which are
not in the image of the cup product map.
$h\infty$

1. INTRODUCTIONLet
be a group and $\Gamma_{G}(1)=G,$
its lower central series. We denote
by Aut the group of automorphisms of . For each $k\geq 0$ , let
be the groupof automorphisms of which induce the identity on the quotient group $G/\Gamma_{G}(k+1)$ .
Then we obtain a descending central filtration$G$$\Gamma_{G}(2),$$G$$\ldots$$G$$\mathcal{A}_{G}(k)$

$G$

Aut

$G=\mathcal{A}_{G}(0)\supset \mathcal{A}_{G}(1)\supset \mathcal{A}_{G}(2.)\supset\cdots$of Aut , called the Johnson filtration of Aut . This filtration was introduced in 1963$:=\Gamma_{G}(k)/\Gamma_{G}(k+1)$
with a pioneer work by S. Andreadakis [1]. For each $k\geq 1$ , set
and
. Let
be the abelianization of . Then, for each$k\geq 1$ , an Aut-equivariant injective homomorphim
$G$

$G$

$\mathcal{L}_{G}(k)$$gr^{k}(\mathcal{A}_{G})=\mathcal{A}_{G}(k)/\mathcal{A}_{G}(k+1)$$G^{ab}$$G$$G^{ab}$$\tau_{k}$:$gr^{k}(\mathcal{A}_{G})arrow Hom_{Z}(G^{ab}, \mathcal{L}_{G}(k+1))$

is defined. (For definition, see Subsection 2.1.2.) This is called the k-th Johnson homomorphism of Aut . 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 of theTorelli 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].)$G$

2000 Mathematics Subject Classification. 20F28(Primary), 20J06(Secondly).
Key words and phrases. automorphism group of a free metabelian group, Johnson homomorphism,
second cohomology group, Magnus representation.

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, and
Let
be a free group of rank with basis
the free metabeliangroup of rank . Namelyby the second derived series
is the quotient group ofof . Then both abelianizations ofandare a free abeliangroup of rank , denoted by $H$ . Fixing a basis of $H$ induced from
, we
$(n,
Z)$
can identify Aut
for $G=F_{n}$ andwith GL. For simplicity, throughout this paper, we write
andforrespectively. Similarly, we write
and
and
for
respectively. The first aim of the paper isandto determine the GL $(n, Z)$ -module structure of the cokemel of the Johnson homomorphismsof Autfor $n\geq 4$ as follows:$F_{n}$$n$$x_{1},$$\ldots,$$n$$[[F_{n}, F_{n}], [F_{n}, F_{n}]]$$F_{n}^{M}$$x_{n}$$F_{n}^{M}$$F_{n}$$F_{n}$$F_{n}$$F_{n}^{M}$$n$$x_{1},$$G^{ab}$$\Gamma_{n}(k),$$\mathcal{L}_{n}(k),$$gr^{k}(\mathcal{A}_{n})$$\mathcal{A}_{n}(k)$$gr^{k}(\mathcal{A}_{F_{n}})$$\Gamma_{F_{n}^{M}}(k),$$\Gamma_{n}^{M}(k),$$\mathcal{L}_{F_{n}^{M}}(k),$$x_{n}$$\ldots,$$F_{n}^{M}$$\mathcal{A}_{F_{n}^{M}}(k)$$\Gamma_{F_{n}}(k),$$\mathcal{L}_{n}^{M}(k),$$\mathcal{A}_{n}^{M}(k)$$\mathcal{L}_{F_{n}}(k),$$\mathcal{A}_{F_{n}}(k)$$gr^{k}(\mathcal{A}_{n}^{M})$$gr^{k}(\mathcal{A}_{F_{n}^{M}})$$F_{n}^{M}$$\tau_{k}$Theorem 1. For $k\geq 2$ and $n\geq 4$ ,$0arrow gr^{k}(\mathcal{A}_{n}^{M})arrow H^{*}\tau_{k}\otimes_{Z}\mathcal{L}_{n}^{M}(k+1)\prime S^{k}H\underline{Tr_{h_{\iota}}^{M}}arrow 0$

is

$a$

GL $(n, Z)$ -equivariant exact sequence.

Here $S^{k}H$ is the symmetric product of $H$ of degree , and Tr is a certain GL $(n, Z)-$
equivariant homomorphism called the Morita trace introdued by S. Morita [24]. (Fordefinition, see Subsection 3.2.)
From Theorem 1, we can give a lower bound on the rank offor $k\geq 2$$n\geq4$
and. The study of the Johnson filtration of Aut was begun in $1960$ ’s byAndreadakis [1] who showed that for each $k\geq 1$ and $n\geq 2,$
is a free abeliangroup of finite rank, and thatcoincides with the k-th lower central series of theinner automorphism group Innof . FMrthermore, he [1] computed
for all $k\geq 1$ . However, the structure offor general $k\geq 2$ and $n\geq 3$ is, and call it the k-th rational Johnson
much more complicated. Set$M$, we denote $M\otimes_{Z}Q$ by the symbol obtained
homomorphism. For any Z-module
by attaching a subscript. For $n\geq 3$ , the GL $(n, Z)$ -module
to $M$ , likeandstructure ofis completely determined by Pettet [31]. In our previous paper [33],
we determined those of
for $n\geq 3$ . For $k\geq 4$ , the GL $(n, Z)$ -module structureofis not determined. Furthermore, even its dimension is also unknown.
$k$

$kM$

$gr^{k}(\mathcal{A}_{n})$$F_{n}$$gr^{k}(\mathcal{A}_{n})$$\mathcal{A}_{2}(k)$$F_{2}$$rank_{Z}gr^{k}(\mathcal{A}_{2})$$F_{2}$$gr^{k}(\mathcal{A}_{n})$$\tau_{k,Q}=\tau_{k}\otimes id_{Q}$$Q$$M^{Q}$$M_{Q}$$gr_{Q}^{2}(\mathcal{A}_{n})$$gr_{Q}^{3}(\mathcal{A}_{n})$$gr_{Q}^{k}(\mathcal{A}_{n})$

: Aut
Let
Aut
be a natural homomorphism induced from the action ofAuton. By notable works due to Bachmuth and Mochizuki [5], it is known thatis surjective for $n\geq 4$ . They [4] also showed that is not surjective. In Subsection
3.1, we see that the homomorphism
induced fromis alsosurjective for $n\geq 4$ . Hence we have
Corollary 1. For $k\geq 2$ and $n\geq 4$ ,$F_{n}arrow$$\nu_{n}$$F_{n}$$F_{n}^{M}$$F_{n}^{M}$$\nu_{n}$$\nu_{3}$$\overline{\nu}_{n_{i}k}:gr^{k}(\mathcal{A}_{n})arrow gr^{k}(\mathcal{A}_{n}^{M})$$rank_{Z}(gr^{k}(\mathcal{A}_{n}))\geq nk(\begin{array}{ll}n+k -1k +1\end{array})-(\begin{array}{ll}n+k -1k \end{array})$$\nu_{n}$.

We should remark that in general, equality does not hold, since for instance $rank_{Z}$, 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 groupisdefined to be a group which consists of automorphisms ofwhich trivially act on$gr^{3}(\mathcal{A}_{n})=n(3n^{4}-7n^{2}-8)/12$

$G$

$G$

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the abelianization of . By the definition, IA $(G)=\mathcal{A}_{G}(1)$ . We write
andfor IAand IA
for simplicity. Let$:=Hom_{Z}(H, Z)$ be the dual group of$H$ .
Then we see that the first homology group of
for $n\geq 4$ is isomorphic to$IA_{n}arrow
IA_{n}^{M}$in the following way. Let:
be the restriction ofto. Bachmuth and Mochizuki [5] showed thatis surjective for $n\geq 4$ . This fact
sharply contrasts with their previous work [4] which shows there are infinitely many
automorphisms ofwhich are not contained the image of. On the other hand,by an independent works of Cohen-Pakianathan [9, 10], Farb [11] and Kawazumi [19],$H_{1}(IA_{n}, Z)\cong H^{*}\otimes_{Z}\Lambda^{2}H$for $n\geq 3$ . Since the kernel ofis contained in thecommutator subgroup of IA , we havefor $n\geq 4$ . (See
Subsection 2.3.) In general, however, there are few results for computation of the(co)homology groups ofof higher dimensions. In this paper we determine theimage of the cup product map in the rational second cohomology group of IA , andshow that it is isomorphic to that of, using the second Johnson homomorphism.
Namely, let:
IA $n’ Q)arrow H^{2}(IA_{n}, Q)$ and: $\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow H^{2}(IA_{n}^{M}, Q)$
be the rational cup product maps of IA andrespectively. In Subsection 4.2, we
show
$G$

$(F_{n})$$IA_{n}$$(F_{n}^{M})$$IA_{n}^{M}$$H^{*}$$IA_{n}^{M}$$H^{*}\otimes_{Z}\Lambda^{2}H$$\nu_{n,1}$$\nu_{n}$$IA_{n}$$\nu_{n,1}$$IA_{3}^{M}$$\nu_{3,1}$$\nu_{n,1}$$H_{1}(IA_{n}^{M}, Z)\cong H^{*}\otimes_{Z}\Lambda^{2}H$$nM$$IA_{n}^{M}$$nM$$IA_{n}$$\Lambda^{2}H^{1}($$U_{Q}$$\bigcup_{Q}^{M}$$IA_{n}^{M}$$n$

Theorem 2. For$n\geq 4_{j}\nu_{n,1}^{*}$:${\rm Im}( \bigcup_{Q}^{M})arrow{\rm Im}(\bigcup_{Q})$is an isomo rphism.

Here we should remark that the GL $(n, Z)$ -module structure ofis completely
determined by Pettet [31] for any $n\geq 3$ .
Now, for the study of the second cohomology group of IA , it is also an important
problem to determine whether the cup product map
is surjective or not. For the, it is still not known whether
case ofis surjective or not. In the last section, we
prove that the rational cup product map
is not surjective for $n\geq 4$ . by studying thekemelof the homomorphism. It is easily seen that is an infinite subgroupof IA sincecontains the second derived series of the inner automorphism groupof a free group. The structure ofis, however, much complicated. For example,(finitely or infinitely many) generators and the abelianization ofare still not known.
To clarify the structure of, it is also important to study the obstruction for thefaithfulness of the Magnus representation ofis equal to the kernel, by asince
result of Bachmuth [2]. (See Subsection 2.3.)
From the cohomological five-term exact sequence of the group extension,
it suffices to show the non-triviality of.
to show
Set. Thennaturally has a GL $(n, Z)$ -module
structure, and the natural projection
induces an injective homomorphism. In this paper, we determine the GL $(n, Z)$ -module structure ofusing the rational third Johnson homomorphism of Aut . The
non-triviality of
immediately follows from it. In Subsection 5.1, we show${\rm Im}( \bigcup_{Q})$$nM$$\bigcup_{Q}^{M}$$IA_{n}$$\bigcup_{Q}$$\bigcup_{Q}^{M}$$\mathcal{K}_{n}$$\mathcal{K}_{n}$$\nu_{n,1}$$\mathcal{K}_{n}$$n$$F_{n}$$\mathcal{K}_{n}$$\mathcal{K}_{n}$$\mathcal{K}_{n}$$IA_{n}$$\mathcal{K}_{n}$$1arrow \mathcal{K}_{n}arrow IA_{n}arrow IA_{n}^{M}arrow 1$$H^{1}(\mathcal{K}_{n}, Q)^{IA_{n}}$$\overline{\mathcal{K}}_{n}$$:=\mathcal{K}_{n}/(\mathcal{K}_{n}\cap \mathcal{A}_{n}(4))\subset gr^{3}(\mathcal{A}_{n})$${\rm Im}(U_{Q}^{M})\neq H^{2}(IA_{n}^{M}, Q)$$\overline{\mathcal{K}}_{n}$$\mathcal{K}_{n}arrow\overline{\mathcal{K}}_{n}$$H^{1}(\overline{\mathcal{K}}_{n}, Q)arrow H^{1}(\mathcal{K}_{n}, Q)^{IA_{n}}$$H_{1}(\overline{\mathcal{K}}_{n}, Q)$$F_{n}$$H^{1}(\overline{\mathcal{K}}_{n}, Q)$

Theorem 3. For $n\geq 4,$

Here$H^{\lambda}$$\tau_{3_{2}Q}(\overline{\mathcal{K}}_{n}^{Q})\cong H_{Q}^{[2,1]}\oplus(D\otimes_{Q}H_{Q}^{[3,2_{2}^{2}1^{n-4}]})$.

denotes the Schur-Weyl module of $H$ corresponding to the Young diagram, and $D$
the one-dimensional representation of GL $(n, Z)$ given$\lambda=[\lambda_{1}, \ldots, \lambda_{l}]$$:=\Lambda^{n}H$3

98
by the determinant map. Since$\tau_{3,Q}$

is injective, this shows that$\overline{\mathcal{K}}_{n}^{Q}\cong H_{Q}^{[2,1]}\oplus(D\otimes_{Q}H_{Q}^{[3,2^{2},1^{narrow 4}]})$.

As a corollary, we have
Corollary 2. For$n\geq 4_{f}$$rank_{Z}(H_{1}(\mathcal{K}_{n}, Z))\geq\frac{1}{3}n(n^{2}-1)+\frac{1}{8}n^{2}(n-1)(n+2)(n-3)$.

Finally, we obtain
Theorem 4. For $n\geq 4$ , the rational cup product$\bigcup_{Q}^{M}:\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow H^{2}(IA_{n}^{M}, Q)$

is not surjective, and$\dim_{Q}(H^{2}(IA_{n}^{M}, Q))\geq\frac{1}{24}n(n-2)(3n^{4}+3n^{3}-5n^{2}-23n-2)$.

In Section 2, we recall the IA-automorphism group of
and the Johnson homomorphisms of the automorphism group Aut offor a group . In particular, weconcentrate on the case where is a free group and a free metabelian group. We also
review the definition of the Magnus representations ofand. In Section 3, we
determine the cokernel of the Johnson homomorphisms of the automorphism group of afree metabelian group. In Section 4, we show that the image of the cup product map
is isomorphic to that of. Finally, in Section 5, we determine the GL $(n, Z)$ -module, and show thatstructure ofis not surjective.
$G$

$G$

$G$

$G$

$G$

$IA_{n}$$IA_{n}^{M}$$\bigcup_{Q}^{M}$$\bigcup_{Q}$$\overline{\mathcal{K}}_{n}^{Q}$$\bigcup_{Q}^{M}$

CONTENTS
1. Introduction
2. Preliminaries
2.1. Notation
2.2. Free groups
2.3. Free metabelian groups
2.4. Magnus representations
3. The cokemel of the Johnson homomorphisms
3.1. Upper bound on the rank of cokemel of
3.2. Lower bound on the rank of the cokemel of
4. The image of the cup product in the second cohomology group
4.1. A minimal presentation and second cohomology of a group
4.2. The image of the rational cup product5. On the kemel of the Magnus representation of IA
5.1. The irreducible decompositon of
5.2. Non surjectivity of the cup product6. Acknowledgments
References$\tau_{k}$$\tau_{k}$$\bigcup_{Q}^{M}$$n$$\overline{\mathcal{K}}_{n}^{Q}$$U_{Q}^{M}$

4

1
5
5
7
9
10
11
11
12
14
14
15
16

16

18
18
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2. PRELIMINARIESIn this section, we recall the definition and some properties of the associated Liealgebra, the IA-automorphism group of , and the Johnson homomorphisms of theautomorphism group Aut offor any group . In Subsections 2.2 and 2.3, weconsider the case where is a free group and a free metabelian group.
$G$

$G$

$G$

$G$

$G$

2.1. Notation.

First of all, throughout this paper we use the following notation and conventions.
For a group , the abelianization of is denoted by.
For a group , the group Aut acts on from the right. For any
Aut
and $x\in G$ , the action of on is denoted by .
For a group , and its quotient group $G/N$ , we also denote the coset class ofan element $g\in G$ by $g\in G/N$ if there is no confusion.
For any Z-module $M$ , we denoteby the symbol obtained by attaching a
subscript $Q$ to $M$ , likeor
. Similarly, for any Z-linear map : $Aarrow B$ ,the induced Q-linear map $A_{Q}arrow B_{Q}$ is denoted byor .
For elements and of a group, the commutator bracket $[x, y]$ of and isdefined to be $[x, y]:=xyx^{-1}y^{-1}$ .
$G$$\bullet$$G^{ab}$$G$$\bullet$$G$$\bullet$

$G$

$G$

$G$

$x^{\sigma}$$x$$\sigma$$G$$\sigma\in$$M\otimes_{Z}Q$$\bullet$$M^{Q}$$M_{Q}$$f$$f_{Q}$$x$$\bullet$

$x$

$y$

2.1.1. Associated Lie algebra of a group.
For a group , we define the lower central series of$G$$\Gamma_{G}(1)$$f^{Q}$$:=F_{n}$,$\Gamma_{G}(k)$

$G$

$y$

by the rule$:=[\Gamma_{G}(k-1), G]$,$k\geq 2$.

$:=\Gamma_{G}(k)/\Gamma_{G}(k+1)$ the graded quotient of the lower central series
We denote byof , and by
the associated graded sum. The graded sum
naturally has a graded Lie algebra structure induced from the commutator bracket on, and called the accosiated Lie algebra of .
$g_{t}\in G$ , a commutator of weight
For any
type of$\mathcal{L}_{G}(k)$$G$$\mathcal{L}_{G}$$:=\oplus_{k>1}\mathcal{L}_{G}(k)$$\mathcal{L}_{G}$

$G$

$G$

$g_{1},$$k$$\ldots,$$[[\cdots[[g_{i_{1}}, g_{i_{2}}], g_{i_{3}}], \cdots], g_{i_{k}}]$,$i_{j}\in\{1, \ldots, t\}$with all of its brackets to the left of all the elements occuring is called a simple k-foldcommutator among the components, and we denote it by$g_{1},$$\ldots,$$g_{t}$$[g_{i_{1}}, g_{i_{2}}, \cdots, g_{i_{k}}]$for simplicity. Then we have

Lemma 2.1. If is generated by, then each of the graded quotients
1 is generated by the simple k-fold commutators$G$$g_{1},$$\ldots,$$g_{t}$$\Gamma_{G}(k)/\Gamma_{G}(k+$$)$$[g_{i_{1}}, g_{i_{2}}, \ldots, g_{i_{h}}]$,$i_{j}\in\{1, \ldots, t\}$.

Let
: Aut $Garrow$ Aut
be the natural homomorphism induced from the abelian$G$$(G)$ization of . The kemel IAofis called the IA-automorphism group of . Thenthe automorphism group Aut naturally acts onfor each $k\geq 1$ , and IA $(G)$ actson it trivially. Hence the action of Aut $G/$ IA $(G)$ on
is well-defined.
$\rho_{G}$$G^{ab}$$G$$\rho c$$G$$\mathcal{L}_{G}(k)$$\mathcal{L}_{G}(k)$5100
2.1.2. Johnson homomorphisms.
For $k\geq 0$ , the action of Aut $G$ on each nilpotent quotient $G/\Gamma_{G}(k+1)$ induces ahomomorphism: Aut $Garrow$ Aut $(G/\Gamma_{G}(k+1))$ .
The map
is trivial, and. We denote the kemel of by. Then thegroupsdefine a descending central filtrationAutof Aut , with
IA $(G)$ . (See [1] for details.) We call it the Johnson filtration$k\geq
1$ , the group Aut
of Aut . For eachacts onby conjugation, and it
naturally induces an action of Aut $G/IA(G)$ on. The graded sum gr$:=$
has a graded Lie algebra structure induced from the commutator bracket
on IA $(G)$ .
To study the Aut $G/$ IA $(G)$ -module structure of each graded quotient, wedefine the Johnson homomorphisms of Aut in the following way, For each $k\geq 1$ , weconsider a map
defined by, $x\in G$ .
Then the kernel of this homomorphism is just. Hence it induces an injective$\rho_{G}^{k}$$\rho_{G}^{1}=\rho_{G}$$\rho_{G}^{0}$$\rho_{G}^{k}$$\mathcal{A}_{G}(k)$$\mathcal{A}_{f}(k)$$G=\mathcal{A}_{G}(0)\supset \mathcal{A}_{G}(1)\supset \mathcal{A}_{G}(2)\supset\cdots$$G$$\mathcal{A}_{G}(1)=$

$G$

$G$

$\mathcal{A}_{G}(k)$$gr^{k}(\mathcal{A}_{G})$$(\mathcal{A}_{G})$$\oplus_{k>1}gr^{k}(\mathcal{A}_{G})$$gr^{k}(\mathcal{A}_{G})$$G$$\mathcal{A}_{G}(k)arrow Hom_{Z}(G^{ab}, \mathcal{L}_{G}(k+1))$$\sigma\mapsto(g\mapsto g^{-1}g^{\sigma})$$\mathcal{A}_{C}(k+1)$

homomorphism$\tau_{k}=\tau_{G_{1}k}:gr^{k}(\mathcal{A}_{G})\mapsto Hom_{Z}(G^{ab}, \mathcal{L}_{G}(k+1))$.

The homomorphsim is called the k-th Johnson homomorphism of Aut . It is easily
seen that each is an Aut $G/IA(G)$ -equivariant homomorphism. Since each Johnson
homomorphism is injective, to determine the cokernel of is an important problem
for the study of the structure ofas an Aut $G/IA(G)$ -module.
Here, we consider another descending filtration of IA $(G)$ . Let
be the k-th
subgroup of the lower central series of IA $(G)$ . Then for each $k\geq 1,$
is a subgroupofsince
is a central filtration of IA $(G)$ . In general, it is not known whether
coincides withor not. Set
for each $k\geq 1$ .
The restriction of the homomorphism.
toinducesan Aut $G/$ IA $(G)$ -equivariant homomorphism$G$$\tau_{k}$$\tau_{k}$$\tau_{k}$$\tau_{k}$$gr^{k}(\mathcal{A}_{G})$$\mathcal{A}_{G}’(k)$$\mathcal{A}_{G}’(k)$$\mathcal{A}_{G}(k)$$\mathcal{A}_{G}(k)$$\mathcal{A}_{G}’(k)$$gr^{k}(\mathcal{A}_{G}’)$$\mathcal{A}_{C}(k)$$:=\mathcal{A}_{G}’(k)/\mathcal{A}_{G}’(k+1)$$\mathcal{A}_{G}(k)arrow Hom_{Z}(G^{ab}, \mathcal{L}_{G}(k+1))$$\tau_{k}’=\tau_{G,k}’:gr^{k}(\mathcal{A}_{G}’)arrow Hom_{Z}(G^{ab}, \mathcal{L}_{G}(k+1))$

In this paper, we also call$\tau_{k}’$$\mathcal{A}_{G}’(k)$.

the k-th Johnson homomorphism of Aut $G$ .

For any, we give an example of computation of
and
using
. For
and
and $g\in G$ , set $s_{g}(\sigma);=g^{-1}g^{\sigma}\in\Gamma_{G}(k+1)$ .
$\tau_{k}(\sigma)(g)=s_{g}(\sigma)\in\mathcal{L}_{G}(k+1)$
Then,. For any
, by an easy
andcalculation, we have$\sigma\in \mathcal{A}_{G}(k)$$\tau_{k}(\sigma)$$\tau_{l}(\tau)$$\tau\in \mathcal{A}_{G}(l)$$\tau_{k+l}([\sigma, \tau])$$\sigma\in \mathcal{A}_{G}(k)$$\sigma\in \mathcal{A}_{G}(k)$$\tau\in \mathcal{A}_{G}(l)$$s_{g}([\sigma, \tau])=(s_{g}(\tau)^{-1})^{\tau^{-1}}(s_{g}(\sigma)^{-1})^{\sigma^{arrow 1}\tau^{arrow 1}}s_{g}(\tau)^{\sigma^{arrow 1}\tau^{-1}}s_{g}(\sigma)^{\tau\sigma^{-1}\tau^{-1}}$(1)$\equiv s_{g}(\sigma)^{-1}s_{g}(\sigma)^{\tau}\cdot(s_{g}(\tau)^{-1}s_{g}(\tau)^{\sigma})^{-1}$

Using this fomula, we can easily computeif
andis given by$s_{g}(\sigma)$(2)$s_{g}([\sigma, \tau])$

from

,

$(mod \Gamma_{G}(k+l+2))$$s_{g}(\sigma)$and$s_{g}(\tau)$.

. For example,$s_{g}(\tau)$$s_{g}(\sigma)=[g_{1},g_{2}, \ldots, g_{k+1}]\in \mathcal{L}_{G}(k+1)$,6$s_{g}(\tau)=[h_{1}, h_{2}, \ldots, h_{l+1}]\in \mathcal{L}_{G}(l+1)$,

101then we obtain$s_{g}([ \sigma, \tau])=(\sum_{i=1}^{k+1}[g_{1}, \ldots, s_{g_{i}}(\tau), \ldots, g_{k+1}])-(\sum_{j=1}^{l+1}[h_{1}, \ldots, s_{h_{j}}(\sigma), \ldots, h_{l+1}])$

in $\mathcal{L}_{G}(k+l+1)$ .
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 $n\geq 2$ , letbe a free group of rank with basis. We denote the$H$abelianization of
by , and its dual group by$:=Hom_{Z}(H, Z)$ . If we fix thebasis of $H$ as a free abelian group induced from the basisof , we can$(H)$identify Aut
Aut
with the general linear group GL $(n, Z)$ . Furthermore, it isclassically well known that the map: Aut
GL $(n, Z)$ is surjective. (See [21],$identi\phi$
proposition 4.4.) Hence we also
Aut $(H)/IA(F_{n})$ with GL $(n, Z)$ . In this paper,for simplicity, we write
andforrespectively.
andThe associated Lie algebrais called the free Lie algebra generated by H. (See[32] for basic material conceming free Lie algebra.) It is classically well known due to
Witt [34] that eachis a GL $(n, Z)$ -equivariant free abelian group of rank
$F_{n}$$n$$x_{1},$$\ldots,$$x_{n}$$H^{*}$$F_{n}$$x_{1},$$\ldots,$$x_{n}$$F_{n}$$F_{n}^{ab}=$$F_{n}arrow$$\rho_{F_{n}}$$\Gamma_{n}(k),$$\mathcal{L}_{n}(k)$$\Gamma_{F}.(k),$$\mathcal{L}_{n}$$\mathcal{L}_{F}.(k)$$\mathcal{L}_{F_{n}}$$\mathcal{L}_{n}$$\mathcal{L}_{n}(k)$(3)$r_{n}(k):= \frac{1}{k}\sum_{d|k}\mu(d)n^{k}z$

where

$\mu$

is the M\"obius function.

Next we consider the GL $(n, Z)$ -module structure of$\mathcal{L}_{n}(k)$we have$\mathcal{L}_{n}(1)=H$,$\mathcal{L}_{n}(2)=\Lambda^{2}H$. For example, for$1\leq k\leq 3$,$\mathcal{L}_{n}(3)=(H\otimes_{Z}\Lambda^{2}H)/\langle x\otimes y\wedge z+y\otimes z\wedge x+z\otimes x\wedge y|x,$$y,$ $z\in H\rangle$.

In general, the irreducible decomposition of
as a GL $(n, Z)$ -module is completely$k\geq1$
determined. For
and any Young diagramof degree , let
be the Schur-Weyl module of $H$ corresponding to the Young diagram . For example,$H^{[k]}=S^{k}H$ and. (For details, see [12] and [13].) Letbethe multiplicity of the Schur-Weyl module. Bakhturin [6] gave a formulainforusing the character of the Specht module ofcorresponding to
the Young diagram . However, its character value had remained unknown in general.
Then Zhuravlev [35] gave a method of calculation for it. Using these facts, we can givethe explicit irreducible decomposition of. For example,$\mathcal{L}_{n}^{Q}(k)$$k$$\lambda=[\lambda_{1}, \ldots, \lambda_{l}]$$\lambda$$H^{[1^{k}]}=\Lambda^{k}H$$m(H_{Q}^{\lambda}, \mathcal{L}_{n}^{Q}(k))$$H_{Q}^{\lambda}$$\mathcal{L}_{n}^{Q}(k)$$m(H_{Q}^{\lambda}, \mathcal{L}_{n}^{Q}(k))$$H_{Q}$$\lambda$$\mathcal{L}_{n}^{Q}(k)$(4)$\mathcal{L}_{n}^{Q}(3)\cong H_{Q}^{[2,1]}$,$\mathcal{L}_{n}^{Q}(4)\cong H_{Q}^{[3,1]}\oplus H_{Q}^{[2_{I}1,1]}$

7.

$H^{\lambda}$

102
2.2.2. IA-automorphism group of a free group.
Now we consider the IA-automorphism group of. We denote IAby.
It is well known due to Nielsen [27] thatcoincides with the inner automorphsim
group Innof . Namely,is a free group of rank 2. However,for $n\geq 3$
is much larger than Inn . Indeed, Magnus [22] showed that for any $n\geq 3$ , theIA-automorphism groupis finitely generated by automorphisms$(F_{n})$$F_{n}$$IA_{n}$$IA_{2}$$F_{2}$$IA_{2}$$F_{2}$$IA_{n}$$F_{n}$$IA_{n}$$K_{ij}:\{\begin{array}{ll}x_{i} \mapsto x_{j}^{-1}x_{i}x_{j}, x_{t} \mapsto x_{t}, (t\neq i)\end{array}$for distinct$j\in\{1,2, \ldots, n\}$

$i,$

and

$K_{ijk}:\{\begin{array}{l}x_{i} \mapsto x_{i}x_{j}x_{k}x_{J^{-1}}x_{k}^{-1},x_{t} \mapsto x_{t}, (t\neq i)\end{array}$for distinct

$j,$

$i,$

$k\in\{1,2, \ldots, n\}$such that $j<k$ .

For any $n\geq 3$ , although a generating set ofis well known as above, any presentation foris still not known. For $n=3$ , Krsti\v{c} and McCool [20] showed thatis not finitely presentable. For $n\geq 4$ , it is also not known whether
is finitely
presentable or not.
$IA_{n}$$IA_{n}$$IA_{3}$$IA_{n}$Andreadakis [1] showed that the first Johnson homomorphismof Autis surjective by computing the image of the generators ofabove. Furthermore, recently,
Cohen-Pakianathan [9, 10], Farb [11] and Kawazumi [19] inedepedently showed thatinduces the abelianization of IA . Namely, for any $n\geq 3$ , we have$F_{n}$$\tau_{1}$$IA_{n}$$\tau_{1}$

$n$

(5)

$IA_{n}^{ab}\cong H^{*}\otimes_{Z}\Lambda^{2}H$

as a GL $(n, Z)$ -module.
2.2.3. Johnson homomorphisms of Aut .
Here, we consider the Johnson homomorphisms of Aut . Throughout this paper, forsimplicity, we writeforand$F_{n}$$F_{n}$$\mathcal{A}_{m}(k),$and$\mathcal{A}_{n}’(k),$$gr^{k}(\mathcal{A}_{n})$$gr^{k}(\mathcal{A}_{n}’)$$\mathcal{A}_{F_{n}}(k),$$\mathcal{A}_{F_{n}}’(k),$$gr^{k}(\mathcal{A}_{F_{n}})$respectively. Pettet [31] showed$gr^{k}(\mathcal{A}_{F_{n}}’)$(6)$rank_{Z}gr^{2}(\mathcal{A}_{n})=\frac{1}{6}n(n+1)(2n^{2}-2n-3)$,

and in our previous paper [33], we showed$rank_{Z}$gr3$( \mathcal{A}_{n})=\frac{1}{12}n(3n^{4}-7n^{2}-8)$.

In general, for any $n\geq 3$ and $k\geq 4$ the rank ofis still not known. One of theaims of this paper is to give a lower bound on
by studying the Johnson
filtration of the automorphism group of a free metabelian group.
Next, we mention the relation between. Since is the abelianandization of IA as mentioned above, we have
. FMrthermore, Pettet [31]showed that
. Although it is conjectured thathas at most finite index infor $k\geq 3$ , there are few results for the difference between
andfor $n\geq 3$ .
$gr^{k}(\mathcal{A}_{n})$$rank_{Z}gr^{k}(\mathcal{A}_{n})$$\mathcal{A}_{n}’(k)$$\mathcal{A}_{n}(k)$$\tau_{1}$$\mathcal{A}_{n}’(2)=\mathcal{A}_{n}(2)$$n$$\mathcal{A}_{m}(3)$$\mathcal{A}_{n}’(3)$$\mathcal{A}_{n}’(k)=\mathcal{A}_{n}(k)$$\mathcal{A}_{n}’(k)$$\mathcal{A}_{n}(k)$

8

103
Let
be the dual group $Hom_{Z}(H, Z)$ of $H$ . For the standard basis
of $H$
induced from the generators of , let
be its dual basis of. Then identifying $Hom_{Z}(H, \mathcal{L}_{n}(k+1))$ with, we obtain the Johnson homomorphism$H^{*}$$x_{1},$$F_{n}$$x_{1}^{*},$$\ldots,$$\ldots,$$x_{n}$$H^{*}$$x_{n}^{*}$$H^{*}\otimes_{Z}\mathcal{L}_{n}(k+1)$$\tau_{k}:gr^{k}(\mathcal{A}_{n})\mapsto H^{*}\otimes_{Z}\mathcal{L}_{n}(k+1)$

of Aut . Here we give some examples of computation
generators
and, we haveof$F_{n}$$K_{ijk}$$K_{ij}$$\Gamma_{n}(2)$$\sigma\in \mathcal{A}_{n}(k)$. For the$s_{x_{l}}(K_{ijk})=\{\begin{array}{ll}1, l\neq i,[x_{j}, x_{k}], l=i\end{array}$. Hence

(7)

in

for

$\mathcal{A}_{m}(1)=IA_{n}$$s_{x_{1}}(K_{ij})=\{\begin{array}{ll}1, l\neq i,[x_{1}^{-1}, x_{j}^{-1}], l=i,\end{array}$in$\tau_{k}(\sigma)$$\tau_{1}(K_{ij})=x_{i}^{*}\otimes[x_{i},x_{j}]$,$\tau_{1}(K_{ijk})=x_{i}^{*}\otimes[x_{j}, x_{k}]$. Then using (1) and (7), we can recursively computefor. These computations are perhaps easiest explained with examples, so we$H^{*}\otimes_{Z}\mathcal{L}_{n}(2)$$\sigma\in \mathcal{A}_{n}^{l}(k)$$\tau_{k}(\sigma)=\tau_{k}’(\sigma)$

give two here. For distinct$a,$ $b,$ $c$

andin 1, 2,

$d$

$\{$

$\ldots,$$n\}$, we have$\tau_{2}’([K_{ab}, K_{bac}])=x_{a}^{*}\otimes([s_{x_{O}}(K_{bac}), x_{b}]+[x_{a}, s_{x_{b}}(K_{bac})])$$-x_{b}^{*}\otimes([s_{x_{a}}(K_{ab}), x_{c}]+[x_{a}, s_{x_{e}}(K_{ab})])$,$=x_{a}^{*}\otimes[x_{a}, [x_{a}, x_{c}]]-x_{b}^{*}\otimes[[x_{a}, x_{b}], x_{c}]$and$\tau_{3}’([K_{ab},K_{bac}, K_{ad}])$,$=x_{a}^{*}\otimes([s_{x_{a}}(K_{ad}), [x_{a}, x_{c}]]+[x_{a}, [s_{x_{a}}(K_{ad}),x_{c}]]+[x_{a}, [x_{a}, s_{x_{c}}(K_{ad})]])$$-x_{b}^{*}\otimes([[s_{x_{a}}(K_{ad}), x_{b}], x_{c}]+[[x_{a}, s_{x_{b}}(K_{ad})],x_{c}]+[[x_{a}, x_{b}], s_{x_{c}}(K_{ad})])$$-x_{a}^{*}\otimes([s_{x_{a}}([K_{ab}, K_{bac}]), x_{d}]+[x_{a}, s_{x_{d}}([K_{ab}, K_{bac}])])$,$=x_{a}^{*}\otimes[[x_{a}, x_{d}], [x_{a},x_{c}]]+x_{a}^{*}\otimes[x_{a}, [[x_{a}, x_{d}], x_{c}]]$$-x_{b}^{*}\otimes[[[x_{a}, x_{d}], x_{b}], x_{c}]$$-x_{a}^{*}\otimes[[x_{a}, [x_{a},x_{c}]],x_{d}]$.

2.3. Free metabelian groups.
In this section we consider the case where a groupfinite rank.

$G$

is a free metabelian group of

2.3.1. Free metabelian Lie algebra.
Letbe a free metabelian group of rank where $F_{n}’’=[[F_{n}, F_{n}], [F_{n}, F_{n}]]$is the second derived group of . Then we have $(F_{n}^{M})^{ab}=H$ , and hence Aut
Aut $(H)=$ GL $(n, Z)$ . Since the surjective map: Aut
GL $(n, Z)$ factors through$(n,Z)$, a map
Aut
: Aut
GL
is also surjective. Hence we identify
Autwith GL $(n, Z)$ . In this paper, for simplicity, we write$F_{n}^{M}=F_{n}/F_{n}’’$$n$$(F_{n}^{M})^{ab}=$$F_{n}$$\rho_{F_{n}}$$F_{n}^{M}$$\rho_{F_{n}^{M}}$$F_{n}^{M}/IA(F_{n}^{M})$and$\mathcal{L}_{n}^{M}$

for$\Gamma_{F_{n}^{M}}(k),$$F_{n}arrow$$F_{n}^{M}arrow$$\Gamma_{n}^{M}(k),$$\mathcal{L}_{F_{n}^{M}}(k)$and$\mathcal{L}_{F_{n}^{M}}$$\mathcal{L}_{n}^{M}(k)$respectively.

The associated Lie algebrais called the free metabelian algebra generated by $H$ .
$1\leqk\leq 3$ . It is also classically well known due to Chen
We see thatfor$\mathcal{L}_{n}^{M}$$\mathcal{L}_{n}(k)=\mathcal{L}_{n}^{M}(k)$9104[8] that each$\mathcal{L}_{n}^{M}(k)$is a GL $(n, Z)$ -equivariant free abelian group of rank(8)$r_{n}^{M}(k):=(k-1)(\begin{array}{ll}n+k -2k \end{array})$.

2.3.2. IA-automorphism group of a free metabelian group.
Here we consider the IA-automorphism group of. Let $IA_{n}^{M};=$ IA. We
denote by: Aut
Aut
the natural homomorphism induced from the actionof Auton. Restricting to, we obtain a homomorphism: IA.
Bachmuth and Mochizuki [4] showed thatis not surjective andis not finitely
generated. They also showed that in [5],is surjective for $n\geq 4$ . Hence IA
is finitely generated for $n\geq 4$ . It is, however, not known whether IA is finitely
presented or not for $n\geq 4$ .
$:=Ker(\nu_{n})$ . Since
From now on, we consider the case where $n\geq 4.$ . Set,
we have an exact sequence$F_{M}$$F_{n}arrow$$\nu_{n}$$F_{n}^{M}$$F_{n}$$(F_{n}^{M})$$F_{n}^{M}$$IA_{n}$$\nu_{n}$$narrow IA_{n}^{M}$$\nu_{n,1}$$IA_{3}^{M}$$\nu_{3,1}$$nM$$\nu_{n,1}$$nM$$\mathcal{K}_{n}\subset IA_{n}$$\mathcal{K}_{n}$(9)$1arrow \mathcal{K}_{n}arrow IA_{n}arrow IA_{n}^{M}arrow 1$

Furthermore, observing$\mathcal{K}_{n}\subset \mathcal{A}_{n}(2)=[IA_{n}, IA_{n}]$

(10)

.

, we obtain,$(IA_{n}^{M})^{ab}\cong IA_{n}^{ab}\cong H^{*}\otimes_{Z}\Lambda^{2}H$and see that the first Johnson homomorphism$\tau_{1}$of Aut2.3.3. Johnson homomorphisms of Aut.
Here we consider the Johnson homomorphisms of Aut$F_{n}^{M}$

is an isomorphism.

$F_{n}^{M}$$gr^{k}(\mathcal{A}_{F_{n}^{M}})$

and

by

$\mathcal{A}_{n}^{M}(k)$$gr^{k}(\mathcal{A}_{F_{n}^{M}}’)$

by

and

$\mathcal{A}_{n}^{M}’(k)$$gr^{k}(\mathcal{A}_{n}^{M})$and$(F_{n}^{M})$. We denote$\mathcal{A}_{F_{n}^{M}}(k)$respectively. FMrthermore, we also denoterespectively.

and$\mathcal{A}_{F_{n}^{M}}’(k)$’$gr^{k}(\mathcal{A}_{n}^{M})$, we obtain a homomorphism
For each $k\geq 1$ , restrictingto:. Since :is an isomorphism, we see that, and hence
is surjective. However it is not known whether
is surjective$k\geq3$
or not for.
Now, the main aim of the paper is to determine the GL $(n, Z)$ -module structure of
the cokernel of the Johnson homomorphisms of Aut. In this paper, we give ananswer to this problem for the case where $k\geq 2$ and $n\geq 4$ . We remark that by anargument similar to that in Subsection 2.2, we can recursively compute, using
forand.
$\nu_{n}$$\mathcal{A}_{n}^{M}(k)$$\tau_{1}$$\mathcal{A}_{n}(k)$$\nu_{n_{2}k}$$gr^{1}(\mathcal{A}_{n}^{M})’arrow H^{*}\otimes_{Z}\Lambda^{2}H$$\mathcal{A}_{m}(k)arrow$$\mathcal{A}_{n}^{M}(2)=$$\mathcal{A}_{n}^{M}(2)/$$\nu_{n,k}$$\nu_{n_{2}2}$$F_{n}^{M}$$\tau_{k}(\sigma)=\tau_{k}’(\sigma)$$\sigma\in \mathcal{A}_{n^{M}}^{l}(k)$$\tau_{1}(\nu_{n,1}(K_{1j}))=x;\otimes[x_{i}, x_{j}]$$\tau_{1}(\nu_{n,1}(K_{ijk}))=x_{i}^{*}\otimes[x_{j}, x_{k}]$2.4. Magnus representations.

In this subsection we recall the Magnus representation of Aut
details, see [7]. For each $1\leq i\leq n$ , let$)$$\frac{\partial}{\partial x_{i}}:Z[F_{n}]arrow Z[F_{n}]$10$F_{n}$and Aut$F_{n}^{M}$. (For105
be the Fox derivation defined by$\frac{\partial}{\partial x_{i}}(w)=\sum_{j_{arrow}^{-1}}^{r}\epsilon_{j}\delta_{\mu_{j},i}x_{\mu_{1}}^{\epsilon 1\ldots B^{(\epsilon_{j}-1)}}x_{\mu_{j}}^{1}\in Z[F_{n}]$for any reduced word
. Let $a:F_{n}arrow H$ be the abelianization of . We also denote by the ring homomorphism $Z[F_{n}]arrow Z[H]$ induced from
. For any $A=(a_{1j})\in$ GL $(n, Z[F_{n}])$ , let
be the matrix
GL $(n, Z[H])$ . The
Magnus representation rep: Aut
GL $(n, Z[H])$ of Aut is defined by$w=x_{\mu_{1}}^{\epsilon_{1}}\cdots x_{\mu_{\Gamma}^{r}}^{\epsilon}\in F_{n},$$F_{n}$$\epsilon_{j}=\pm 1$$a$$A^{a}$$a$$(a_{ij}^{a})\in$$F_{n}arrow$$F_{n}$$\sigma\mapsto(\frac{\partial x_{i^{\sigma}}}{\partial x_{j}})^{a}$for any
Namely,$\sigma\in$Aut$F_{n}$. This map is not a homomorphism but a crossed homomorphism,$\overline{rep}(\sigma\tau)=(\overline{rep}(\sigma))^{\tau^{*}}\cdot\overline{rep}(\tau)$

where

by applying the automordenotes the matrix obtained fromphism
: $Z[H]arrow Z[H]$ induced from
Aut $(H)$ on each entry. Hence by
restricting rep to IA ’ we obtain a homomorphism rep: $IA_{n}arrow$ GL $(n, Z[H])$ . This iscalled the Magnus representation of IA .
$(\overline{rep}(\sigma))^{\tau^{*}}$$\overline{rep}(a)$$\rho(\tau)\in$$\tau^{*}$

$n$

$n$

Next, we consider the Magnus representation of
be a map defined by$IA_{n}^{M}$. Let rep:$M$

$IA_{n}^{M}arrow$

GL $(n, Z[H])$$\sigma\mapsto(\frac{\partial(x_{1}^{\sigma})}{\partial x_{j}})^{a}$for any
where we consider any lift of the element
to . Then we see
rep is a homomorphism and rep rep, and call it the Magnus representation. Bachmuth [2] showed that rep is faithful, and determined the image of repof$(n,Z[H])$ . The faithfulness of the Magnus representation rep
in GLshows that thekemel of the Magnus representation rep is equal to.
$\sigma\in IA_{n}^{M}$$x_{i^{\sigma}}\in F_{n}^{M}$

$M$

$=$

$F_{n}$$M_{O\nu_{n,1}}$$M$$IA_{n}^{M}$

$M$

$M$

$\mathcal{K}_{n}$3. THE COKERNEL OF THE JOHNSON HOMOMORPHISMS
In this section, we determine the cokemel of the Johnson homomorphism
Aut
for $k\geq 2$ and $n\geq 4$ .

$\tau_{k}$of$F_{n}^{M}$

3.1. Upper bound on the rank of cokernel of .
First we give an upper bound on the rank of the cokemel ofof generators. By Lemma 2.1, we see that elements type ofgenerate. First we prepare some lemmas. Let
group of degree . Then we have$\tau_{k}$$\tau_{k}$by reducing its set$x_{i}^{*}\otimes[x_{i_{1}},x_{i_{2}}, .

$H^{*}\otimes_{Z}\mathcal{L}_{n}^{M}(k+1)$$\mathfrak{S}_{1}$.

, , x_{i_{k+1}}]$be the symmetric$l$Lemma 3.1. Let $l\geq 2$ and $n\geq 2$ . For any element,
and any$[x_{i_{1}}, x_{i_{2}}, x_{j_{1}}, \ldots, x_{j_{l}}]\in \mathcal{L}_{n}^{M}(l+2)$$\lambda\in \mathfrak{S}_{l}$.

$[x_{i_{1}}, x_{i_{2}}, x_{j_{1}}, \ldots, x_{j_{l}}]=[x_{i_{1}},x_{i_{2}},x_{j_{\lambda(1)}}\ldots, x_{j_{\lambda(l)}}]$

Lemma 3.2. Let $k\geq 1$ and $n\geq 4$ . For any$i_{1},$$i_{2}\neq i$

,

$i$

and

$i_{1},$$i_{2},$$x_{i}^{*}\otimes[x_{i_{1}}, x_{i_{2}}, \ldots, x_{i_{k+1}}]\in{\rm Im}(\tau_{k}’)$11.

$\ldots,$$i_{k+1}\in\{1,2\ldots, n\}$

, if

106
Lemma 3.3. Let $k\geq 1$ and $n\geq 4$ . For any,, and any$i_{1},$$i_{2}\neq i$

$i$

and

$i_{1},$$i_{2},$$\ldots,$$i_{k}\in\{1,2\ldots, n\}$such that$\lambda\in 6_{k}$$x_{i}^{*}\otimes[x_{i}, x_{i_{1}}, \ldots,x_{i_{k}}]-X_{1}^{*}\otimes[x_{i}, x_{i_{\lambda(1)}}, \ldots, x_{i_{\lambda(k)}}]\in{\rm Im}(\tau_{k}^{!})$

Lemma 3.4. Let $k\geq 1$ and $n\geq 4$ . For any$i_{2},$$i_{k+1}\in\{1,2, \ldots, n\}$$\ldots,$.
, we have$x_{i}^{*}\otimes[x_{i},x_{i_{2}}, \ldots, x_{i_{k+1}}]-x_{j}^{*}\otimes[x_{j}, x_{i_{2}}, \ldots, x_{i_{k+1}}]\in{\rm Im}(\tau_{k}’)$for any$i\neq i_{2}$and$j\neq i_{2},$ $i_{k+1}$.

Using the lemmas above, we can reduce the generators of Coker${\rm Im}(\tau_{k}’)\subset{\rm Im}(\tau_{k})$$(\tau_{k})$.

Proposition 3.1. For $k\geq 2$ and $n\geq 4$ , Coker

is generated by$(\tau_{k})$. We remark that$(\begin{array}{l}n+k-1k\end{array})$

elements.

3.2. Lower bound on the rank of the cokernel of .
In this subsection we give a lower bound on the rank of Coker
by using the Magnus
representation of Aut. To do this, we use trace maps introduced by Morita [24] withpioneer and remarkable works. Recently, he showed that there is a symmetric productof $H$ of degree in the cokemel of the Johnson homomorphism of the automorphismgroup of a free group using trace maps. Here we apply his method to the case forAut. In order to define the trace maps, we prepare some notation of the associatedalgebra of the integral group ring. (For basic materials, see [30], Chapter VIII.)For a group , let $Z[G]$ be the integral group ring ofover Z. We denote the$Z[G]arrow
Z$. The kemelaugmentation map by :of is called the augmentation
ideal. Then the powers offor $i\geq 1$ provide a descending filtration of $Z[G]$ , and the
direct sum$\tau_{k}$$(\tau_{k})$$F_{n}^{M}$$k$$F_{n}^{M}$

$G$

$G$

$I_{G}$$\epsilon$$\epsilon$$I_{G}^{i}$$?_{G}:= \bigoplus_{k>1,arrow}I_{G}^{k}/I_{G}^{k+1}$

naturally has a graded algebra structure induced from the multiplication of $Z[G]$ . We
callthe associated algebra of the group ring $Z[G]$ .
$\sigma_{G}$For $G=F_{n}$ a free group of rank , write
andrespectively.
forand
It is classically well known due to Magnus [23] that each graded quotient
is afree abelian group with basis $\{(x_{i_{1}}-1)(x_{i_{2}}-1)\cdots(x_{i_{k}}-1)|1\leq i_{j}\leq n\}$ , and a map
defined by$n$$2_{n}$$I_{n}$$I_{F_{n}}$$x_{F_{n}}$$I_{n}^{k}/I_{n}^{k+1}$$I_{n}^{k}/I_{n}^{k+1}arrow H^{\otimes k}$$(x_{i_{1}}-1)(x_{t_{2}}-1)\cdots(x_{i_{k}}-1)\mapsto x_{i_{1}}\otimes x_{i_{2}}\otimes\cdots\otimes x_{i_{k}}$induces an isomorphism from$r_{n}$

to the tensor algebra$T(H):= \bigoplus_{k\geq 1}H^{\Phi k}$of $H$ as a graded algebra. We identify$I_{n}^{k}/I_{n}^{k+1}$

with$H^{\emptyset k}$

via this isomorphism.

It is also well known that each graded quotientis a free abelian group withbasis $\{(x_{i_{1}}-1)(x_{i_{2}}-1)\cdots(x_{i_{k}}-1)|1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq n\}$ , and the associatedgraded algebraof $H$ is isomorphic to the symmetric algebra$I_{H}^{k}/I_{H}^{k+1}$$X_{H}$$S(H):= \bigoplus_{k\geq 1}S^{k}H$

12

107
of $H$ as a graded algebra. (See [30], Chapter VIII, Proposition 6.7.) We also identify
with $S^{k}H$ . Then a homomorphism
induced from the$a:F_{n}arrowH$abelianizationis considered as the natural projection.
$I_{H}^{k}/I_{H}^{k+1}$$I_{n}^{k}/I_{n}^{k+1}arrow I_{H}^{k}/I_{H}^{k+1}$$H^{\otimes k}arrow S^{k}H$Now, we define trace maps. For any element$f\in H^{*}\otimes_{Z}\mathcal{L}_{n}^{M}(k+1)$, set$\Vert f\Vert:=(\frac{\partial(x_{i^{f}})}{\partial x_{j}})^{u}\in M(n, S^{k}H)$

where we consider any lift of the element$x_{i^{f}}\in \mathcal{L}_{n}^{M}(k+1)=\Gamma_{n}(k+1)/(\Gamma_{n}(k+2)\cdot\Gamma_{n}(k+1)\cap F_{n}^{li})$

to $\Gamma_{n}(k+1)$ . Then we define a map$Tr_{k}^{M}:H^{*}\otimes_{Z}\mathcal{L}_{n}^{M}(k+1)arrow S^{k}H$

TJ $kM(f)$ $:=$ trace$(\Vert f\Vert)$by.

It is easily seen that Tr is a GL $(n, Z)$ -equivariant homomorphism. The maps Tr
called the Morita trace maps. We show that Tr is surjective and Tr$k\geq 2$ and $n\geq 3$ . By a direct computation, we obtain

$kM$

are

$kM_{O\mathcal{T}_{k}}=0$

for

$kM$

$kM$

Lemma 3.5. For$f=x_{i}^{*}\otimes[x_{i_{1}},x_{i_{2}}, \ldots,x_{i_{k+1}}]\in H^{*}\otimes_{Z}\mathcal{L}_{n}^{M}(k+1)$, we have$Tr_{k}^{M}(f)=(-1)^{k}\{\delta_{i_{1}i}x_{i_{2}}x_{i_{3}}\cdots x_{i_{k+1}}-\delta_{i_{2}i}x_{i_{1}}x_{i_{\theta}}\cdots x_{i_{k+1}}\}$

where$\delta_{ij}$is the Kronecker delta.

Lemma 3.6. For any $k\geq 1$ and$n\geq 2_{f}Tr_{k}^{M}$

is surjective.

Before showing TJ $kM_{o\tau_{k}}=0$ , we consider a relation between the Magnus representation and the Johnson homomorphism. For each $k\geq 1$ , composing the Magnus representation rep restricted towith a homomorphism GL $(n, Z[H])arrow$$(n,
Z[H]/I_{H}^{k+1})$GLinduced from a natural projection $Z[H]arrow Z[H]/I_{H}^{k+1}$ , we obtain ahomomorphism:
GL $(n, Z[H]/I_{H}^{k+1})$ . By the definition of the Magnus
representation and the Johnson homomorphism, we obtain
(11)
rep
$M$

$\mathcal{A}_{n}^{M}(k)$$rep_{k}^{M}$$\mathcal{A}_{n}^{M}(k)arrow$$kM(\sigma)=I+\Vert\tau_{k}(\sigma)\Vert$

where

$I$

denotes the identity matrix. (See also [24].)

Proposition 3.2. For $k\geq 2$ and $n\geq 3,$$Tr_{k}^{M}$

vanishes on the image ofAs a corollary, we have
Corollary 3.1. For $k\geq 2$ and$n\geq 3_{f}$$rank_{Z}$(Coker$(\tau_{k})$)$\geq(\begin{array}{ll}n+k -1k \end{array})$.

Combining this corollary with Proposition 3.1, we obtain

Theorem 3.1. For $k\geq 2$ and $n\geq 4$ ,$0 arrow gr^{k}(\mathcal{A}_{n}^{M})arrow H^{*}\tau_{k}\otimes_{Z}\mathcal{L}_{n}^{M}(k+1)\frac{Tr_{k_{t}}^{M}}{r}S^{k}Harrow 0$

is

$a$

GL $(n, Z)$ -equivariant exact sequence.

From (8), we obtain13$\tau_{k}$.

108Corollary 3.2. For $k\geq 2$ and $n\geq 4$ ,$rank_{Z}(gr^{k}(\mathcal{A}_{n}^{M}))=nk(\begin{array}{ll}n+k -lk +1\end{array})-(\begin{array}{ll}n+k -1k \end{array})$Let:
argument above, we see thatis surjective. Hencethat$\overline{\nu}_{n_{2}k}$$gr^{k}(\mathcal{A}_{n})arrow gr^{k}(A_{n}^{M})$.

be the homomorphism induced from
. By the. Since is injective, this shows$\nu_{n,k}$${\rm Im}(\tau_{k}0\overline{\nu}_{n_{1}k})={\rm Im}(\tau_{k})$$\tau_{k}$$\overline{\nu}_{n,k}$Corollary 3.3. For $k\geq 2$ and $n\geq 4$ ,$rank_{Z}(gr^{k}(\mathcal{A}_{n}))\geq nk(\begin{array}{ll}n+k -1k +1\end{array})-(\begin{array}{ll}n+k -1k \end{array})$.

As mentioned before, in the inequality above, equality does not hold in general. Since, which is not equal to the right hand side of the$rank_{Z}gr^{3}(\mathcal{A}_{n})=n(3n^{4}-7n^{2}-8)/12$

inequality.

4. THE IMAGE OF THE CUP PRODUCT IN THE SECOND COHOMOLOGY GROUP

In this section, we consider the rational second (co)homology group of IA . In
particular, we determine the image of the cup product map: $\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow H^{2}(IA_{n}^{M}, Q)$ .
$nM$$\bigcup_{Q}^{M}$

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 $H^{2}(G, Z)$ if has a minimal presentation. For a group , a group extension
$G$

$G$

(12)

$1arrow Rarrow Farrow^{\varphi}Garrow 1$

is called a minimal presentation of
isomorphism$G$

if $F$ is a free group such that$\varphi_{*}:H_{1}(F, Z)arrow H_{1}(G, Z)$$\varphi$induces an.

This shows thatis contained in the commutator subgroup $[F, F]$ of $F$ . In the following, we assume thathas a minimal presentation defined by (12), and fix it.
Furthermore we assume that the rank $m$ of $F$ is finite. We remark that considering, we see that each of
the Magnus generators ofandand
has a suchminimal presentation. From the cohomological five-term exact sequence of (12), we see
$R$

$G$

$IA_{n}^{M}$$IA_{n}$$IA_{n}$$IA_{n}^{M}$$H^{2}(G, Z)\cong H^{1}(R, Z)^{G}$for each $k\geq 1$ . Then
is a free abelian group of$:=R\cap\Gamma_{F}(k)$rank $r_{m}(k)$ by (3). Let
be a descending filtration defined byfor each $k\geq 1$ . Then $R_{k}=R$ for $k=1$ , and 2. For each $k\geq 1$ , let

Set$\mathcal{L}_{F}(k)=\Gamma_{F}(k)/\Gamma_{F}(k+1)$$\mathcal{L}_{F}(k)$$\{R_{k}\}_{k>1}$$R_{k}$$\varphi_{k}:\mathcal{L}_{F}(k)arrow \mathcal{L}_{G}(k)$: $Farrow G$ . Observing
be a homomorphism induced from the natural projection$R_{k}/R_{k+1}\cong(R_{k}\Gamma_{F}(k+1))/\Gamma_{F}(k+1)$ , we have an exact sequence$\varphi$(13)$0arrow R_{k}/R_{k+1}arrow\iota_{k}\mathcal{L}_{F}(k)arrow^{\varphi_{h}}\mathcal{L}_{G}(k)arrow 0$.

This shows each graded quotient $R_{k}/R_{k+1}$ is a free abelian group.

14

109
Set

$\overline{R}_{k}$$:=R/R_{k}$. The natural projection$\psi^{k}$we seeinduces an injective homomorphism: $H^{1}(\overline{R}_{k}, Z)arrow H^{1}(R, Z)$ .

Considering the right action of $F$ on$r\cdot x$$Rarrow\overline{R}_{k}$$R$, defined by$:=x^{-1}rx$,,$x\in F$$r\in R,$is an G-equivariant homomorphism. Hence it induces an injective homomorphism, also denoted by,$\psi^{k}$$\psi^{k}$$\psi^{k}$:$H^{1}(\overline{R}_{k}, Z)^{G}arrow H^{1}(R, Z)^{G}$.

For $k=3,$
since $G$ acts ontrivially. Here we show that$\cup:\Lambda^{2}H^{1}(G,Z)arrowH^{2}(G,Z)$
the image of the cup productis contained in.
$H^{1}(\overline{R}_{3}, Z)^{G}=H^{1}(\overline{R}_{3}, Z)$$\overline{R}_{3}$$H^{1}(\overline{R}_{3}, Z)$

Lemma 4.1. If$G$

has a minimal presentation as above, the image of the cup product$\cup:\Lambda^{2}H^{1}(G, Z)arrow H^{2}(G, Z)$

is isomorphic to the image of$\iota_{2}^{*}:$$H^{1}(\mathcal{L}_{F}(2), Z)arrow H^{1}(\overline{R}_{3}, Z)$.

Here we remark that if
is free abelian group,. Furthermore if we consider the rational cup product: $\Lambda^{2}H^{1}(G, Q)arrow H^{2}(G, Q)$ , Since $Q$
is a Z-injective module, the induced homomorphismissurjective. Hence the image of the rational cup productis equal to.
${\rm Im}(\cup)=H^{1}(\overline{R}_{3}, Z)$$gr^{2}(\mathcal{A}_{G}’)$$\bigcup_{Q}$$\iota_{2}^{*}:H^{1}(\mathcal{L}_{F}(2), Q)arrow H^{1}(\overline{R}_{3}, Q)$$H^{1}(\overline{R}_{3}, Q)$$\bigcup_{Q}$4.2. The image of the rational cup product$\bigcup_{Q}^{M}$.

In this subsection, we determine the image of the rational cup product: $\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow H^{2}(IA_{n}^{M}, Q)$ .
First, we should remark that the image of the cup product:$H^{2}(IA_{n}, Q)$ is completely determined by Pettet [31] who gave the GL $(n, Q)$ -irreducible
decomposition of it. Here we show that the restriction of: $H^{2}(IA_{n}^{M}, Q)arrow$$H^{2}(IA_{n}, Q)$ to.
is an isomorphism onto
To do this, we prepare some notation. Let $F$ be a free group on
and
which
are corresponding to the Magnus generators of. Namely, $F$ is a free group of rank$n^{2}(n-1)/2$ . Then we have a natural surjective homomorphism: $Farrow$ IA ’ and aminimal presentation$\bigcup_{Q}^{M}$$\Lambda^{2}H^{1}(IA_{n}, Q)arrow$$\bigcup_{Q}$$\nu_{n,1}^{*}$${\rm Im}(U_{Q}^{M})$${\rm Im}( \bigcup_{Q})$$K_{ijk}$$K_{ij}$$IA_{n}$$\varphi$

(14)

of

$IA_{n}$$n$$1arrow Rarrow Farrow^{\varphi}IA_{n}arrow 1$

where $R=Ker(\varphi)$ . From a result of Pettet [31], we have

Lemma 4.2. For $n\geq 3,$$\alpha(n)$$\overline{R}_{3}$is a free abelian group of rank$:= \frac{1}{8}n^{2}(n-1)(n^{3}-n^{2}-2)-n(n\vec{6}1+1)(2n^{2}-2n-3)$.

Next, we consider the second cohomology groups of IA . From now on, we assume$n\geq 4$ . We recall that the natural homomorphism:
is surjective, andinduces an isomorphism
for $n\geq 4$ . Then we have asurjective homomorphism, and a minimal presentation$M$$\nu_{n,1}$$\nu_{n,1}$$IA_{n}arrow IA_{n}^{M}$$IA_{n}^{ab}\cong(IA_{n}^{M})^{ab}\cong H^{*}\otimes_{Z}\Lambda^{2}H$$\varphi^{M}:=\nu_{n_{1}1}0\varphi:Farrow IA_{n}^{M}$(15)$1 arrow R^{M}arrow F\frac{\varphi_{\iota}^{M}}{r}IA_{n}^{M}arrow 1$

16

110
of $IA^{M}$ where $R^{M}=Ker(\varphi)$ . Observe a sequence$gr_{Q}^{2}(\mathcal{A}_{n}’)arrow gr_{Q}^{2}(\mathcal{A}_{n}^{M})/arrow gr_{Q}^{2}(\mathcal{A}_{\mathfrak{n}}^{M})$of surjective homomorphisms. Since
to Pettet [31], we see$\mathcal{A}_{m}(3)/\mathcal{A}_{n}’(3)$is at most finite abelian group due$\dim_{Q}(gr_{Q}^{2}(\mathcal{A}_{n}’))=\dim_{Q}(gr_{Q}^{2}(\mathcal{A}_{n}))=\frac{1}{6}n(n+1)(2n^{2}-2n-3)$$=\dim_{Q}(gr_{Q}^{2}(\mathcal{A}_{n}^{M}))$by (6), and hence$gr_{Q}^{2}(\mathcal{A}_{n}^{M}’)\cong gP_{Q}(\mathcal{A}_{n}^{M})$

Lemma 4.3. For $n\geq 4,$$\overline{R_{3}^{M}}$. Thus,is a free abelian group of rank$\alpha(n)$.

Therefore, from the functoriality of the spectral sequence, we obtain commutativityof a diagram$0arrow H^{1}(\overline{R_{3}^{M}}, Q)arrow H^{2}(IA_{n}^{M}, Q)$$\simeq\downarrow$$\downarrow\nu_{\mathfrak{n},1}^{*}$$0arrow H^{1}(\overline{R_{3}}, Q)arrow H^{2}(IA_{n}, Q)$

andTheorem 4.1. For $n\geq 4,$$\nu_{n,1}^{*}$:${\rm Im}(U_{Q}^{M})arrow{\rm Im}(U_{Q})$is an isomorphism.

In the subsection 5.2, we will show that the rational cup product$H^{2}(IA_{n}^{M}, Q)$ is not surjective.

$\bigcup_{Q}^{M}$:$\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow$

5. ON THE KERNEL OF THE MAGNUS REPRESENTATION OF$IA_{n}$In this section, we study the kemelof the Magnus representation of
for $n\geq 4$ .
Set
. Since, we see.
Here we determine the GL $(n, Z)$ -module structure of. As a corollary, we see thatthe rational cup productis not surjective.
$IA_{n}$$\mathcal{K}_{n}$$\overline{\mathcal{K}}_{n}$$:=\mathcal{K}_{n}/(\mathcal{K}_{n}\cap \mathcal{A}_{n}(4))\subset gr^{3}(\mathcal{A}_{n})$$[\mathcal{K}_{n}, \mathcal{K}_{n}]\subset \mathcal{A}_{n}(6)$$H_{1}(\overline{\mathcal{K}}_{n}, Z)=\overline{\mathcal{K}}_{n}$$\overline{\mathcal{K}}_{n}^{Q}$$\bigcup_{Q}^{M}:\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow H^{2}(IA_{n}^{M}, Q)$

5.1. The irreducible decompositon of.
First, we consider the irreducible decomposition of the targetof the$B$$B’$rational third Johnson homomorphismof Aut . Let andbe subsets ofconsisting of$\overline{\mathcal{K}}_{n}^{Q}$$H_{Q}^{*}\otimes_{Q}\mathcal{L}_{n}^{Q}(4)$$F_{n}$$\tau_{3,Q}$$\mathcal{L}_{n}(4)$$i>j\leq k\leq l$$[[[x_{i},x_{j}], x_{k}], x_{l}],$and$[[x_{i}, x_{j}], [x_{k}, x_{l}]],$$i>j,$ $k>l,$ $i>k$ ,$[[x_{i}, x_{j}], [x_{i},x_{l}]],$$i>j,$ $i>l,$ $j>l$

respectively. Then $B\cup B’$ forms a basis ofdue to Hall [15]. Let
be theGL $(n, Z)$ -equivariant submodule ofgenerated by elements type offor $1\leq i,j,$ $k,$ $l\leq n$ . Then
is a basis ofand the quotient module ofbyis isomorphic to. Observing thatis a GL $(n, Z)$ -equivariant submodule of$\mathcal{L}_{n}(4)$$[[x_{i}, x_{j}], [x_{k}, x_{l}]]$$\mathcal{L}_{n}(4)$$B^{t}$$\mathcal{G}_{n}$$\mathcal{L}_{n}^{M}(4)$$\mathcal{G}_{n}$$\mathcal{G}_{n}$$\mathcal{G}_{n}^{Q}$

16$\mathcal{L}_{n}(4)$111$\mathcal{L}_{n}^{Q}(4)\cong H_{Q}^{[3_{2}1]}\oplus H_{Q}^{[2,1,1]}$. Let, and $\dim_{Q}(\mathcal{G}_{n}^{Q})=n(n^{2}-1)(n+2)/S$ , we see$\mathcal{G}_{n}^{Q}\cong H_{Q}^{[2_{1}1,1]}$andbe the one-dimensional representation of GL $(n, Z)$
given by the determinant map. Then considering a natural isomorphism
as a GL $(n, Z)$ -module, and using Pieri’s formula (See [13].), we obtain$\mathcal{L}_{n,Q}^{M}(4)\cong H_{Q}^{[3,1]}$$D$$:=\Lambda^{n}H$$H_{Q}^{*}\cong(D\otimes_{Q}$$\Lambda^{n-1}H_{Q})$

Lemma 5.1. For(i)$n\geq 4_{f}$$H_{Q}^{*}\otimes z\mathcal{G}_{n}^{Q}\cong H_{Q}^{[1^{3}]}\oplus H_{Q}^{[2,1]}\oplus(D\otimes_{Q}H_{Q}^{[3,2^{2},1^{n-4}]})$

(ii)

,

$H_{Q}^{*}\otimes_{Z}\mathcal{L}_{n,Q}^{M}(4)\cong H_{Q}^{[3]}\oplus H_{Q}^{[2,1]}\oplus(D\otimes_{Q}H_{Q}^{[4_{2}2,1^{n-\}]})$.

Now it is clear that
. On the other hand, in our previous
paper [33], we showed that the cokemel of the rational Johnson homomorphism
is given by Coker. Hence we see thatis isomorphic toa submodule of. In the following, we show$\tau_{3,Q}(\overline{\mathcal{K}}_{n}^{Q})\subset H_{Q}^{*}\otimes_{Z}\mathcal{G}_{n}^{Q}$$\tau_{3,Q}$$(\tau_{3_{2}Q})=H_{Q}^{[3]}\oplus H_{Q}^{[1^{3}]}$$\tau_{3_{)}Q}(\overline{\mathcal{K}}_{n}^{Q})$$H_{Q}^{[2,1]}\oplus(D\otimes_{Q}H_{Q}^{[3_{2}2^{2},1^{n-4}]})$$H_{Q}^{[2,1]}\oplus(D\otimes_{Q}H_{Q}^{[3,2^{2},1^{n-4}]})$$\tau_{3,Q}(\overline{\mathcal{K}}_{n}^{Q})\cong$.

To show this, we prepare some elements ofsuch that $p>q,$ and $q>r$ , set$\{1,2, \ldots, n\}$$\mathcal{K}_{n}$. First, for any distinct$p,$ $q,$ $r,$$s\in$$r$$T(s,p, q, r):=[[K_{sp}^{-1}, K_{sr}^{-1}], K_{sqp}]\in IA_{n}$.

Since $T(s,p, q, r)$ satisfies$x_{t}\mapsto\{\begin{array}{ll}x_{s}[[x_{p}, x_{q}], [x_{p}, x_{r}]], if t=s,x_{t}, if t\neq s,\end{array}$$T(s,p, q, r)\in \mathcal{K}_{n}$any distinct

and$\tau_{3}(T(s,p, q, r))=x_{s}^{*}\otimes[[x_{p}, x_{q}], [x_{p}, x_{r}]]\in H^{*}\otimes z\mathcal{G}_{n}$$p,$ $q,$ $r,$ $s\in\{1,2,\ldots, n\}$such that $p>s$ , set. Next, for$E(s,p, q, r):=[[K_{sr}, K_{spq}], K_{rsq}](K_{rs}^{-1}[[K_{rs}, K_{spq}]^{-1}, K_{rq}^{-1}]K_{rs})\in IA_{n}$Then we have

Lemma 5.2. For any $n\geq 4$ ,(i)(ii) $E(s,p, q, r)\in \mathcal{K}_{n}$ .

$\tau_{3}(E(s,p, q, r))=x_{\epsilon}^{*}\otimes[[x_{p}, x_{q}], [x_{s}, x_{q}]]\in H^{*}\otimes_{Z}\mathcal{G}_{n}$Theorem 5.1. For $n\geq 4,$

Since$\tau_{3_{:}Q}$.

$\tau_{3_{1}Q}(\overline{\mathcal{K}}_{n}^{Q})\cong H_{Q}^{[2_{t}1]}\oplus(D\otimes_{Q}H_{Q}^{[3_{2}2^{2},1^{\mathfrak{n}-4}]})$.

is injective, this shows that$\overline{\mathcal{K}}_{n}^{Q}\cong H_{Q}^{[2,1]_{\oplus()}}D\otimes_{Q}H_{Q}^{[32^{2},1^{n-4}]})$

andCorollary 3. For $n\geq 4$ ,$rank_{Z}(H_{1}(\mathcal{K}_{n}, Z))\geq\frac{1}{3}n(n^{2}-1)+\frac{1}{8}n^{2}(n-1)(n+2)(n-3)$

17

.

.

112

.

5.2. Non surjectivity of the cup productIn this subsection, we also assume $n\geq 4$ . Here we show that the rational cup product: $\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow H^{2}(IAnM, Q)$ is not surjective. From the rational five-term exact
sequence$\bigcup_{Q}^{M}$$\bigcup_{Q}^{M}$$0arrow H^{1}$ $(IAnM, Q)arrow H^{1}(IA_{n}, Q)arrow H^{1}(\mathcal{K}_{n}, Q)^{IA_{n}}arrow H^{2}(IA_{n}^{M}, Q)arrow H^{2}($IA $n’ Q)$of (9), we have an exact sequence$0arrow H^{1}(\mathcal{K}_{n}, Q)^{IA_{n}}arrow H^{2}(IA_{n}^{M}, Q)arrow H^{2}(IA_{n}, Q)$

By Theorem 4.1, to show the non-surjectivity of the cup productIA .
that the non-triviality of$H^{1}(\mathcal{K}_{n}, Q)$

The natural projection$\mathcal{K}_{n}arrow\overline{\mathcal{K}}_{n}$.

$\bigcup_{Q}^{M}$

it suffices to show$n$

induces an injective homomorphism$H^{1}(\overline{\mathcal{K}}_{n}, Q)arrow H^{1}(\mathcal{K}_{n}, Q)^{IA_{\hslash}}$.

By Theorem 5.1, and the universal coefficients theorem, we see$H^{1}(\overline{\mathcal{K}}_{n}, Q)\cong Hom_{Z}(H_{1}(\overline{\mathcal{K}}_{n}, Z), Q)\neq 0$.

Therefore we obtain
Theorem 5.2. For $n\geq 4$ , the rational cup product$\bigcup_{Q}^{M}$:$\Lambda^{2}H^{1}(IA_{n}^{M}, Q)arrow H^{2}(IA_{n}^{M}, Q)$

is not surjective, and$\dim_{Q}(H^{2}(IA_{n}^{M}, Q))\geq\frac{1}{24}n(n-2)(3n^{4}+3n^{3}-5n^{2}-23n-2)$.

6. ACKNOWLEDGMENTS
The author would like to thank Professor Nariya Kawazumi for valuable advice anduseful suggestions. He would also like to express his thanks to the referee for helpfulcomments and correcting typos and grammatical mistakes. This research is supported
by JSPS Research Fellowships for Young Scientists.
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$IA_{n}$GRADUATE SCHOOL OF SCIENCES, DEPARTMENT OF MATHEMATICS, OSAKA UNIVERSITY, 1-16
MACHIKANEYAMA, TOYONAKA-CITY, OSAKA 560-0043, JAPANE-mail address: takaoQmath. sci. osaka-u. ac. jp

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