ON THE STABLE COHOMOLOGY OF THE
IA-AUTOMORPHISM GROUPS OF FREE GROUPSKAZUO HABIRO AND MAI KATADA

Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba
Meguro-ku Tokyo 153-8914, Japan
KH’s email address: habiro@ms.u-tokyo.ac.jp
KH’s ORCID No.: 0000-0002-2263-3832
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba
Meguro-ku Tokyo 153-8914, Japan
Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, Japan
819-0395
MK’s email address: katada@math.kyushu-u.ac.jp
MK’s ORCID No.: 0000-0002-7094-175X

Abstract. Borel’s stability and vanishing theorem gives the stable cohomology of GL(n, Z) with coefficients in algebraic GL(n, Z)-representations. By
combining the Borel theorem with the Hochschild–Serre spectral sequence, we
compute the twisted first cohomology of the automorphism group Aut(Fn ) of
the free group Fn of rank n. We also study the stable rational cohomology of
the IA-automorphism group IAn of Fn . We propose a conjectural algebraic
structure of the stable rational cohomology of IAn , and consider some relations
to known results and conjectures. We also consider a conjectural structure of
the stable rational cohomology of the Torelli groups of surfaces.

Contents
1.

Introduction

2

2.

Algebraic GL(n, Z)-representations

7

3.

Borel’s stability and vanishing theorem for cohomology of GL(n, Z)

94.

The low-degree twisted cohomology of Aut(Fn ) and Out(Fn )115.

Stable families of polynomial Aut(Fn )-modules

15

6.

GL(n, Z)-invariant part of the stable rational cohomology of IAn

17

7.

Stable rational cohomology of IAn

21

8.

Stable rational cohomology of IOn

26

9.

Conjectures about the stable cohomology of the Torelli groups

29

Date: January 17, 2024 (First version: November 24, 2022).
2020 Mathematics Subject Classification. 20F28, 20J06.
Key words and phrases. Automorphism groups of free groups, General linear groups, IAautomorphism groups of free groups, Group cohomology, Torelli groups.
1

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KAZUO HABIRO AND MAI KATADA

References

34

1. Introduction
Let Fn be the free group of rank n, Aut(Fn ) the automorphism group of Fn andGL(n, Z) the general linear group of rank n. Let IAn denote the IA-automorphismgroup of Fn , which is defined by the following exact sequence(1.0.1)

1 → IAn → Aut(Fn ) → GL(n, Z) → 1.

The Hochschild–Serre spectral sequence for the above exact sequence gives a strong
connection between the twisted cohomology of Aut(Fn ), the twisted cohomology ofGL(n, Z) and the rational cohomology of IAn .
Borel [2] proved that the rational cohomology of GL(n, Z) stabilizes, which meansthat the map on the rational cohomology in each degree induced by the inclusion
GL(n, Z) ,→ GL(n+1, Z) is an isomorphism for sufficiently large n, and he computed
the stable rational cohomology of GL(n, Z). Borel [3] also computed the stablecohomology of GL(n, Z) with coefficients in algebraic GL(n, Z)-representations. Li
and Sun [35] obtained an improved stable range that is independent of the types ofalgebraic GL(n, Z)-representations.
Hatcher and Vogtmann [21, 22] showed that the rational homology of Aut(Fn )stabilizes, and Galatius [14] showed that the stable rational homology is trivial. Let
HZ = H1 (Fn , Z), HZ∗ = HomZ (H, Z), H = HZ ⊗Q and H ∗ = Hom(H, Q). Satoh [49,50, 51] computed the low-degree (co)homology with coefficients in HZ , HZ∗ and HZ∗ ⊗V2
HZ . The stable homology with coefficients in HZ⊗p and (H ∗ )⊗q was computedby Djament and Vespa [10], Vespa [53] and Randal-Williams [47]. Kawazumi and
Vespa [30] proposed a conjectural structure of the stable cohomology of Aut(Fn )with coefficients in H ⊗p ⊗ (H ∗ )⊗q , which has been proven by Lindell [37] recently.
By the exact sequence (1.0.1), we have a GL(n, Z)-action on the rational cohomology of IAn . Cohen and Pakianathan [7], Farb [11] and Kawazumi [28] independently determined the first cohomology of IAn . Pettet [46] and the second-named
author [27] determined the second and third Albanese cohomology of IAn , which is
the GL(n, Z)-subrepresentation of the rational cohomology of IAn obtained as theimage of the rational cohomology of the abelianization of IAn under the induced
map by the abelianization map. Day and Putman [8] showed that the GL(n, Z)invariant part of H 2 (IAn , Z) vanishes.
Unlike the case of GL(n, Z) and Aut(Fn ), the rational cohomology of IAn does
not stabilize. In what follows, by the stable rational cohomology of IAn we mean
the rational cohomology of IAn for sufficiently large n. The rational cohomology
of IAn is expected to stabilize as a family of GL(n, Z)-representations, which iscalled representation stability [6]. In fact, the first rational cohomology of IAn isrepresentation stable. The stable rational cohomology of IAn in degree greater than
1 has not been determined. The second-named author [27] proposed a conjectural
structure of the stable Albanese cohomology of IAn . To the best of our knowledge,
no conjecture about the structure of the whole stable rational cohomology of IAn
has been given in the literature.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

3

In this paper, we consider the cohomology in a stable range of the families ofgroups Aut(Fn ) and IAn with coefficients in algebraic GL(n, Z)-representations. By
using the improved Borel theorem and the Hochschild–Serre spectral sequence associated to the exact sequence (1.0.1), we compute the first cohomology of Aut(Fn )with coefficients in algebraic GL(n, Z)-representations and the Aut(Fn )-modules ofJacobi diagrams and their duals. We propose a conjectural algebraic structure of
the stable rational cohomology of IAn . We also consider the stable rational cohomology of IOn , which is an analogue of IAn to the outer automorphism group
Out(Fn ) of Fn , and the stable rational cohomology of the Torelli groups of surfaces.
Here we make some conventions about stability. We say that some statement
about cohomology of groups depending on n holds stably if it holds for n sufficiently large with respect to cohomological degree. A stable isomorphism means amorphism that is stably an isomorphism. Note that if we say “we stably have an
isomorphism”, then we do not assume a morphism in an unstable range, while if
we say “we have a stable isomorphism”, then we assume that we have a morphism
in both stable and unstable ranges.
1.1. First cohomology of Aut(Fn ) with coefficients in algebraic GL(n, Z)representations. Let H = H1 (Fn , Q) and H ∗ = H 1 (Fn , Q) = HomQ (H, Q).
Let Vλ denote the irreducible algebraic GL(n, Z)-representation corresponding to
the bipartition λ, i.e., a pair of partitions. See Section 2 for algebraic GL(n, Z)representations.
It is well known that we have H 1 (Aut(Fn ), Q) = 0. We consider the twisted firstcohomology of Aut(Fn ). We have the following result of Satoh [51, 49].
Theorem 1.1 (Satoh [51, 49]). We have
H 1 (Aut(Fn ), H ∗ ) = 0,H 1 (Aut(Fn ), H) = Q

1

H (Aut(Fn ), V12 ,1 ) = Q

for n ≥ 2,

for n ≥ 5.

Djament and Vespa [10] showed the vanishing of the stable homology of Aut(Fn )with coefficients induced by reduced polynomial functors on the category of finitely
generated free groups. In particular, the stable homology with coefficients in H ⊗pvanishes for any p ≥ 1. By using Djament’s result [9], Vespa [53] obtained the stablehomology of Aut(Fn ) with coefficients in (H ∗ )⊗q for any q ≥ 1. Randal-Williams[47, Theorem A] obtained the following stable range by using geometric techniques
H 1 (Aut(Fn ), (H ∗ )⊗q ) = 0,
1

H (Aut(Fn ), H⊗q

) = 0,

for n ≥ q + 5,
for n ≥ q + 5, q ̸= 1.

We obtain the vanishing of the first cohomology of Aut(Fn ) with coefficients inalgebraic GL(n, Z)-representations Vλ for any bipartitions λ ̸= (1, 0), (12 , 1).
Theorem 1.2 (Theorem 4.1). For any bipartition λ ̸= (1, 0), (12 , 1), we have
H 1 (Aut(Fn ), Vλ ) = 0

for n ≥ 3.

H 1 (Aut(Fn ), V1,0 ) = Q

for n ≥ 4,

For λ = (1, 0), (12 , 1),
1

H (Aut(Fn ), V12 ,1 ) = Q

for n ≥ 4.

4

KAZUO HABIRO AND MAI KATADALet H p,q = H ⊗p ⊗ (H ∗ )⊗q . Since H p,q is decomposed into a direct sum of somecopies of Vλ for bipartitions λ, we obtain the first cohomology of Aut(Fn ) withcoefficients in H p,q .
Theorem 1.3 (Theorem 4.4). Let p, q ≥ 1.
(1) If p − q ̸= 1, then we have
H 1 (Aut(Fn ), H p,q ) = 0

for n ≥ 3.

(2) If p − q = 1, then we have1

H 1 (Aut(Fn ), H q+1,q ) = Q 2 (q+2)!

for n ≥ 4.

Kawazumi and Vespa [30] proposed a conjectural structure of the stable cohomology of Aut(Fn ) with coefficients in H p,q for p, q ≥ 0. The above theorem gives a
proof of their conjecture in cohomological degree 1. Lindell [37] has recently proved
their conjecture in all degrees (see Theorem 1.8) after the first version of the present
paper appeared.
1.2. Conjectural structure of the stable rational cohomology of IAn . In
this subsection, we make a conjecture about a complete algebraic structure of thestable rational cohomology of IAn . To the best of our knowledge, no such conjecture
has been given in the literature.
Cohen and Pakianathan [7], Farb [11] and Kawazumi [28] independently determined the first cohomology of IAn . However, the rational cohomology of IAn ofdegree greater than 1 has not been determined.
∗(IAn , Q) of IAn is defined as the image of the
The Albanese cohomology HA
rational cohomology of the abelianization of IAn under the induced map by theabelianization map. This terminology was introduced in Church, Ellenberg andFarb [5], but the notion had been studied earlier. Pettet [46] and the second-named
2(IAn , Q) andauthor [27] determined the stable GL(n, Q)-module structure of HA
3
HA (IAn , Q), respectively.

Let W∗ denote the traceless part of the graded-symmetric algebra of U∗ =Vi+1
H). Then the second-named author [27] madei≥1 Ui , where Ui = Hom(H,the following conjecture.

L

Conjecture 1.4 ([27], see Conjecture 7.1). We stably have an isomorphism ofgraded GL(n, Q)-representations∗
HA(IAn , Q) ∼= (W∗ )∗ ,where (W∗ )∗ is the graded-dual of W∗ .
Let H ∗ (IAn , Q)GL(n,Z) denote the GL(n, Z)-invariant part of H ∗ (IAn , Q). Then
we have a graded algebra homomorphism induced by the cup product∗ωn : HA(IAn , Q) ⊗ H ∗ (IAn , Q)GL(n,Z) → H ∗ (IAn , Q).

Conjecture 1.5 (Conjecture 7.4). The morphism ωn is a stable isomorphism ofgraded algebras with GL(n, Z)-actions.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

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Day and Putman [8] showed that the GL(n, Z)-invariant part of H 2 (IAn , Z)
vanishes. We also propose a conjectural structure of the GL(n, Z)-invariant part ofthe stable rational cohomology of IAn .
Conjecture 1.6 (Conjecture 6.1). The GL(n, Z)-invariant part of the rational
cohomology of IAn stabilizes, and we stably have an isomorphism of graded algebrasH ∗ (IAn , Q)GL(n,Z) ∼= Q[z1 , z2 , . . . ], deg zi = 4i.
By Conjectures 1.4, 1.5 and 1.6, we obtain the following conjecture about acomplete algebraic structure of H ∗ (IAn , Q).
Conjecture 1.7 (Conjecture 7.5). We stably have an isomorphism of graded algebras with GL(n, Z)-actionsH ∗ (IAn , Q) ∼= (W∗ )∗ ⊗ Q[z1 , z2 , . . . ], deg zi = 4i.
1.3. Relation to Church and Farb’s conjecture. Kawazumi and Vespa [30]proposed a conjectural structure of the stable cohomology H ∗ (Aut(Fn ), H p,q ), which
was previously known to hold for p = 0 or q = 0 by [10, 9, 53, 47]. After the firstversion of the present paper appeared on arXiv, Lindell [37] proved the conjectureof Kawazumi and Vespa.
Theorem 1.8 (Lindell [37], see Theorem 7.3). Let p, q ≥ 0. If i ̸= p − q, then wehave
H i (Aut(Fn ), H p,q ) = 0
for sufficiently large n. Otherwise, we have
H i (Aut(Fn ), H q+i,q ) = CP ⟳ (q + i, q)0

for sufficiently large n, where CP ⟳ is a certain wheeled PROP (see Remark 4.5).
0

We consider the following stability of the family {H i (IAn , Q)}n of GL(n, Z)representations.
(SFi ): For some r ≥ 0, {H i (IAn , Q)}n≥r is a stable family of GL(n, Z)-representations,that is, there are finitely many bipartitions λj (possibly with repetitions)such that for all n ≥ r we have an isomorphism of GL(n, Z)-representationsM
H i (IAn , Q) ∼Vλj (n).
=j

Conjecture 1.9 (Church–Farb [6], see Conjecture 6.3). For each i ≥ 0, the hypothesis (SFi ) holds.
By using Theorem 1.8, we have the following relation between the above conjectures.
Theorem 1.10 (see Theorems 6.4 and 7.6). We have the following.
• Conjecture 1.9 implies Conjecture 1.6.
• Conjectures 1.4 and 1.9 imply Conjecture 1.5.
Therefore, Conjectures 1.4 and 1.9 imply Conjecture 1.7.

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KAZUO HABIRO AND MAI KATADA

1.4. Conjectural structure of the stable rational cohomology of IOn . Similarly to the case of IAn , we make the following conjecture about a complete algebraic structure of the stable rational cohomology of IOn , which is an analogue ofConjecture 1.7.
Conjecture 1.11 (Conjecture 8.7). We stably have an isomorphism of gradedalgebras with GL(n, Z)-actionsH ∗ (IOn , Q) ∼= (W∗O )∗ ⊗ Q[z1 , z2 , . . . ], deg zi = 4i,
L
where W∗O is the traceless part of the graded-symmetric algebra of U∗O = i≥1 UiO ,where U1O = U1 /H and UiO = Ui for i ≥ 2.
A conjectural relation between the stable rational cohomology of IAn and IOn
is the following.
Conjecture 1.12 (Conjecture 8.13). We have a stable isomorphismψ : H ∗ (IOn , Q) ⊗ H ∗ (Fn , Q) → H ∗ (IAn , Q)of graded algebraic GL(n, Z)-representations.
1.5. Stable rational cohomology of the Torelli groups of surfaces. Let Ig(resp. Ig,1 ) denote the Torelli group of a compact oriented surface of genus g (resp.
with one boundary component). Then Sp(2g, Z) acts on the rational cohomologyof the Torelli groups.
We have injective graded algebra maps∗
ιg : HA(Ig , Q) ,→ H ∗ (Ig , Q),∗ιg,1 : HA(Ig,1 , Q) ,→ H ∗ (Ig,1 , Q).

Then the following conjecture determines an algebraic structure of the stable cohomology of the Torelli groups.
Conjecture 1.13 (Conjecture 9.13). The graded algebra maps ιg and ιg,1 are stableisomorphisms.
Let ig,1 : Ig,1 ,→ IA2g denote the Dehn–Nielsen embedding. We make thefollowing conjecture about the invariant parts of the stable rational cohomology of
IAn and Ig,1 .
Conjecture 1.14 (Conjecture 9.10). The graded algebra homomorphismi∗g,1 : H ∗ (IA2g , Q)GL(2g,Z) → H ∗ (Ig,1 , Q)Sp(2g,Z)
induced by ig,1 is a stable isomorphism.
Remark 1.15. After the first version of the present paper was circulated, Oscar
Randal-Williams informed us of the result of Li and Sun about the improvementof the Borel theorem [35]. We have removed our results about Borel’s stable range
in the first version since they are weaker than Li and Sun’s result. Our results on
Borel’s stable range has been separated into another paper [16] since they give anapproach different from Li and Sun’s. Also, we have improved the stable ranges ofthe low degree cohomology of Aut(Fn ) and Out(Fn ) by using Li and Sun’s result.
Moreover, we realized that the assumption in Theorem 1.10, i.e. Conjecture 1.9,
can be replaced with a weaker assumption that the family {Hi (IAn ; Q)}n is stably
algebraic in the sense of Remark 3.6.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

7

Remark 1.16. After the first version of the present paper was circulated, Erik
Lindell informed us of his proof of the conjecture of Kawazumi–Vespa, and later his
preprint appeared on arXiv [37]. In Section 7, we have decided to rely on Lindell’s
result, which was a conjecture in the previous version, so that our results about therelations between various conjectures become simpler statements.
1.6. Organization of the paper. The rest of this paper is organized as follows. In
Section 2, we recall some facts about representation theory of GL(n, Q). In Section
3, we recall Borel’s stability and vanishing theorem for the cohomology of GL(n, Z)
with coefficients in irreducible algebraic representations Vλ corresponding to bipartitions. In Section 4, we compute the low-degree cohomology of Aut(Fn ) withcoefficients in Vλ by using the Hochschild–Serre spectral sequence. In Section 5, wecompute the first cohomology of Aut(Fn ) with coefficients in the Aut(Fn )-moduleof Jacobi diagrams, which is not semisimple. In Section 6, we propose a conjectural
structure of the GL(n, Z)-invariant part of the stable cohomology H ∗ (IAn , Q) andprove it under the assumption (SFi ) for each i. In Section 7, we propose a conjectural structure of the stable cohomology H ∗ (IAn , Q) and study relations to someother known conjectures. In Section 8, we study the stable cohomology H ∗ (IOn , Q)
as in the case of H ∗ (IAn , Q). In Section 9, we make conjectures about the stablerational cohomology of the Torelli groups of surfaces.
Acknowledgements. The authors thank Nariya Kawazumi, Erik Lindell, Geoffrey Powell, Oscar Randal-Williams, Takuya Sakasai and Christine Vespa for helpful comments. K.H. was supported in part by JSPS KAKENHI Grant Number
18H01119 and 22K03311. M.K. was supported in part by JSPS KAKENHI Grant
Number JP22J14812.

2. Algebraic GL(n, Z)-representations
By a polynomial GL(n, Q)-representation, we mean a finite-dimensional Q[GL(n, Q)]module V such that after choosing a basis for V , the (dim V )2 coordinate functions
are polynomial functions of the n2 variables. Similarly, a GL(n, Q)-representationis algebraic if the coordinate functions are rational functions. We refer the reader
to [13] for some standard facts from representation theory.
It is well known that irreducible polynomial GL(n, Q)-representations are classified by partitions with at most n parts. A partition λ = (λ1 , λ2 , . . . , λl ) is a sequenceof non-negative integers such that λ1 ≥ λ2 ≥ · · · ≥ λl . Let l(λ) = max({0} ∪ {i |λi > 0}) denote the length of λ and |λ| = λ1 + · · · + λl(λ) the size of λ.
Let H = H(n) = Qn denote the standard representation of GL(n, Q). We usuallyomit (n) in the following. For a partition λ, let S λ denote the Specht module forλ, which is an irreducible representation of S|λ| . Define a GL(n, Q)-representation
Vλ = Vλ (n) = H ⊗|λ| ⊗Q[S|λ| ] S λ .
If l(λ) ≤ n, then Vλ is an irreducible polynomial GL(n, Q)-representation. If l(λ) >n, then we have Vλ = 0.

8

KAZUO HABIRO AND MAI KATADALet p and q be non-negative integers. Let H p,q = H ⊗p ⊗ (H ∗ )⊗q . For a pair(i, j) ∈ {1, . . . , p} × {1, . . . , q}, the contraction map
ci,j : H p,q → H p−1,q−1
is defined by
ci,j ((v1 ⊗ · · · ⊗ vp ) ⊗ (f1 ⊗ · · · ⊗ fq ))= ⟨vi , fj ⟩(v1 ⊗ · · · vbi · · · ⊗ vp ) ⊗ (f1 ⊗ · · · fbj · · · ⊗ fq )for v1 , . . . , vp ∈ H and f1 , . . . , fq ∈ H ∗ , where vbi (resp. fbj ) denotes the omission ofvi (resp. fj ), and where ⟨−, −⟩ : H ⊗ H ∗ → Q denotes the dual pairing.
Let H ⟨p,q⟩ denote the traceless part of H p,q , which is defined by\
H ⟨p,q⟩ =ker ci,j ⊂ H p,q .
(i,j)∈{1,...,p}×{1,...,q}By a bipartition, we mean a pair λ = (λ, λ′ ) of two partitions λ and λ′ . The
length of λ is l(λ) = l(λ) + l(λ′ ). We define the degree of λ by deg λ = |λ| − |λ′ | ∈ Z,
and the size of λ by |λ| = |λ| + |λ′ |. The dual of λ is defined by λ∗ = (λ′ , λ).
Let λ be a bipartition, and set p = |λ| and q = |λ′ |. Let′

Vλ = Vλ (n) = H ⟨p,q⟩ ⊗Q[Sp ×Sq ] (S λ ⊗ S λ ).
If l(λ) ≤ n, then Vλ is an irreducible algebraic GL(n, Q)-representation. If l(λ) >n, then we have Vλ = 0. We have the following decomposition of H ⟨p,q⟩ as a
Q[GL(n, Q) × (Sp × Sq )]-moduleM′H ⟨p,q⟩ =(2.0.1)
Vλ ⊗ (S λ ⊗ S λ ).
λ=(λ,λ′ ):bipartition withl(λ)≤n, |λ|=p, |λ′ |=q(See [31, Theorem 1.1].) It is well known that irreducible algebraic GL(n, Q)representations are classified by bipartitions with at most n parts.
For 0 ≤ l ≤ min(p, q), let
1 <i2 <···<il+1 ≤p,
Λp,q (l) = {((i1 , j1 ), . . . , (il+1 , jl+1 )) ∈ ([p] × [q])l+1 | 1≤i
j1 ,j2 ,...,jl+1 : distinct }.

For I = ((i1 , j1 ), . . . , (il+1 , jl+1 )) ∈ Λp,q (l), let
cI : H p,q → H p−l−1,q−l−1
be the composition of contraction maps defined by(v1 ⊗ · · · ⊗ vp ) ⊗ (f1 ⊗ · · · ⊗ fq )
!
l+1
Y
c
[
7→
⟨vir , fjr ⟩ (v1 ⊗ · · · vc
i1 · · · vd
il+1 · · · ⊗ vp ) ⊗ (f1 ⊗ · · · fj1 · · · fjl+1 · · · ⊗ fq ).
r=1

Let k = min(p, q). Define an increasing filtration F ∗ = {F l }0≤l≤k
H ⟨p,q⟩ = F 0 ⊂ F 1 ⊂ · · · ⊂ F l ⊂ F l+1 ⊂ · · · ⊂ F k = H p,qof H p,q by
F l = ker MI∈Λp,q (l)cI : H p,q →MI∈Λp,q (l)H p−l−1,q−l−1  .

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

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If n > p + q − 1, then we have the following exact sequencepq

0 → F l−1 → F l → (H ⟨p−l,q−l⟩ )⊕( l )( l )l! → 0(2.0.2)for 1 ≤ l ≤ k.

Let V
det denote the (1-dimensional) determinant representation. Then we have
n
det ∼
V ∼
=
= V(1n ) , where (1n ) = (1, . . . , 1) consists of n copies of 1. For any
bipartition λ = (λ, λ′ ) with l(λ) ≤ n, we haveVλ ∼= Vµ ⊗ detk ,where µ is a partition with at most n parts and k is an integer satisfying(λ1 , . . . , λl(λ) , 0, . . . , 0, −λ′l(λ′ ) , . . . , −λ′1 ) = (µ1 + k, . . . , µn + k).
By an algebraic GL(n, Z)-representation, we mean the restriction to GL(n, Z) ofan algebraic GL(n, Q)-representation. Note that the square of the determinant representation, det2 , is trivial as a GL(n, Z)-representation. Therefore, any irreducible
algebraic GL(n, Z)-representation is equivalent to the restriction of an irreducible
polynomial GL(n, Q)-representation to GL(n, Z).
3. Borel’s stability and vanishing theorem for cohomology ofGL(n, Z)In [2, 3], Borel computed the cohomology H p (Γ, V ) of an arithmetic group Γwith coefficients in an algebraic Γ-representation V in a stable range. Recently, Li
and Sun [35] gave an improved Borel’s stable range, which is independent of thetype of the representation V .
3.1. The stable cohomology of GL(n, Z). Here we recall known results about
the stable cohomology of GL(n, Z).
Theorem 3.1 (Borel [2, 3], Li–Sun [35]). (1) For each integer n ≥ 1, the algebramap
H ∗ (GL(n + 1, Z), Q) → H ∗ (GL(n, Z), Q)
induced by the inclusion GL(n, Z) ,→ GL(n + 1, Z) is an isomorphism in ∗ ≤ n − 2.
Moreover, we have an algebra isomorphism^lim H ∗ (GL(n, Z), Q) ∼(x1 , x2 , . . .), deg xi = 4i + 1
=
←
−
Q
n
in degree ∗ ≤ n − 2.
(2) Let V be an algebraic GL(n, Q)-representation such that V GL(n,Q) = 0. Then
we haveH p (GL(n, Z), V ) = 0

for p ≤ n − 2.

Remark 3.2. Borel’s original result V
is about the real cohomology H ∗ (SL(n, Z), R).
∗
We stably have H (SL(n, Z), R) = R (b1 , b2 , . . . ), where bk ∈ H 4k+1 (SL(n, Z), R)
is the Borel class. Since the natural map between the cohomology of SL(n, Z) andGL(n, Z) is stably isomorphism, we consider the case of GL(n, Z) in Theorem 3.1.
The element xk ∈ H 4k+1 (GL(n, Z), Q) is a scalar multiple ck bk , where ck ∈ R \ {0}.
The choice of ck does not matter in what follows.

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KAZUO HABIRO AND MAI KATADA

Kupers–Miller–Patzt [32] improved the stable range in Theorem 3.1 in the caseof the rational cohomology of GL(n, Z). The following theorem gives the best stable
range of the cohomology of GL(n, Z) that we know.
Theorem 3.3 (Borel [2, 3], Li–Sun [35], Kupers–Miller–Patzt [32]). Let p ≥ 0.
Then we have the following.
(1) For n ≥ p + 1, we haveH ∗ (GL(n, Z), Q) ∼=^Q

(x1 , x2 , . . .),deg xi = 4i + 1

in cohomological degree ∗ ≤ p.
(2) Let λ ̸= (0, 0) be a bipartition. For n ≥ p + 2, we haveH p (GL(n, Z), Vλ ) = 0.
3.2. Stable families of algebraic GL(n, Z)-representations. Here we consider
families of GL(n, Z)-representations indexed by n ∈ N that are stable with respect
to the types of representations.
Definition 3.4. Let 0 ≤ s ≤ r be integers. A family W∗ = {Wn }n≥s of GL(n, Z)representations Wn is called stable in ranks ≥ r if there are finitely many bipartitions λi (possibly with repetitions) such that for all n ≥ r we have a GL(n, Z)module isomorphismMWn ∼Vλi (n).
=i

A stable family of GL(n, Z)-representations defined above is almost the sameas a “uniformly multiplicity-stable sequence” in the sense of Church and Farb [6].
(Here we consider only families of modules that are finite dimensional in the stable
range.)
Let W∗ be aL
stable family of GL(n, Z)-representations in ranks ≥ r such that
we have Wn ∼= i Vλi (n) for n ≥ r. For each n ≥ s and p ≥ 0, we have the cup
product map
θn,p : H p (GL(n, Z), Q) ⊗ WnGL(n,Z) → H p (GL(n, Z), Wn ).
By the work of Borel [3] and Li–Sun [35], it is known that the cup product map
θn,p is an isomorphism in a stable range. We will improve the stable range. For a
non-negative integer p, set(Lmax(r, p + 1) (W∗ = finite Q)
n0 (W∗ , p) =max(r, p + 2) (otherwise).
By Theorem 3.3, we have the following proposition.
Proposition 3.5. Let W∗ be a stable family of GL(n, Z)-representations. Thenθn,p is an isomorphism for n ≥ n0 (W∗ , p).
Remark 3.6. Using Theorem 3.3, we can prove that Proposition 3.5 also holds forstably algebraic families, which are weaker variants of stable families. Here a familyW∗ = {Wn }n≥s is stably algebraic in ranks ≥ r for some r ≥ s if Wn is an algebraic
GL(n, Z)-representation for all n ≥ r. In this case, the sources and targets of theGL(n,Z)
stable isomorphisms θn,p do not stabilize since they depend on Wn.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

11

4. The low-degree twisted cohomology of Aut(Fn ) and Out(Fn )In this section, we compute the low-degree twisted cohomology of Aut(Fn ) andOut(Fn ) by using Theorem 3.3 and the Hochschild–Serre spectral sequence (see [4]for example). Let HZ = H1 (Fn , Z), HZ∗ = H 1 (Fn , Z), H = H1 (Fn , Q) ∼= Qn and∗1
H = H (Fn , Q).
4.1. Known results about the first (co)homology of Aut(Fn ) and Out(Fn ).
Here we list what is known about the first (co)homology of Aut(Fn ) and Out(Fn ).
It is well known that H1 (Aut(Fn ), Q) = H1 (Out(Fn ), Q) = 0 for all n ≥ 1.
Satoh [51, 49] obtained
H1 (Aut(Fn ), HZ ) = 0,
H1 (Aut(Fn ), HZ∗ ) = Z,2
H 1 (Aut(Fn ), IAabn )=Z ,H1 (Out(Fn ), HZ ) = 0

for n ≥ 4,

H1 (Out(Fn ), HZ∗ ) = Z/(n − 1)Z for n ≥ 4,
H 1 (Out(Fn ), IAabn ) = Z for n ≥ 5,

and for coefficients H and H ∗ , the results hold for n ≥ 2 by tensoring Q.
Randal-Williams [47, Theorem A] obtained
H 1 (Aut(Fn ), (HZ∗ )⊗q ) = 0,
1

H (Aut(Fn ), H⊗q

) = 0,H 1 (Out(Fn ), (H ∗ )⊗q ) = 0
1

H (Out(Fn ), H

⊗q

)=0

for n ≥ q + 5,

for n ≥ q + 5, q ̸= 1.

He obtained similar results for higher degree cohomology as well (see Section 7.1).
4.2. First cohomology of Aut(Fn ) and Out(Fn ) with algebraic coefficients.
In this section, we compute the low-degree cohomology of Aut(Fn ) and Out(Fn )with coefficients induced by irreducible algebraic GL(n, Q)-representations.
Let λ be a bipartition. For the exact sequence
1 → IAn → Aut(Fn ) → GL(n, Z) → 1,
we consider the Hochschild–Serre spectral sequence
E2p,q = H p (GL(n, Z), H q (IAn , Vλ )) ⇒ H p+q (Aut(Fn ), Vλ ).
Since IAn acts trivially on Vλ , we haveE20,q = H 0 (GL(n, Z), H q (IAn , Vλ )) = (H q (IAn , Q) ⊗ Vλ )GL(n,Z) .
By Theorem 3.3, if n ≥ p + 2, then we haveE2p,0 = H p (GL(n, Z), H 0 (IAn , Vλ )) = H p (GL(n, Z), Vλ ) = 0
for 1 ≤ p ≤ 4. By Farb [11], Cohen–Pakianathan [7] and Kawazumi [28], we have
(4.2.1)H 1 (IAn , Q) = V0,1 ⊕ V1,12

for n ≥ 3.

By the above argument, we obtain the following theorem.
Theorem 4.1. For any bipartition λ ̸= (1, 0), (12 , 1), we have
H 1 (Aut(Fn ), Vλ ) = 0

for n ≥ 3.

12

KAZUO HABIRO AND MAI KATADAFor λ = (1, 0), (12 , 1),
H 1 (Aut(Fn ), V1,0 ) = Q
1

H (Aut(Fn ), V12 ,1 ) = Q

for n ≥ 4,
for n ≥ 4.

Proof. For a bipartition λ ̸= (1, 0), (12 , 1), we haveE20,1 = (H 1 (IAn , Q) ⊗ Vλ )GL(n,Z) = ((V0,1 ⊕ V1,12 ) ⊗ Vλ )GL(n,Z) = 0.
By Theorem 3.3, for n ≥ 3, we have E21,0 = 0 and thus we have
H 1 (Aut(Fn ), Vλ ) = 0.
For λ = (1, 0), we haveE20,1 = (H 1 (IAn , Q) ⊗ V1,0 )GL(n,Z) = ((V0,1 ⊕ V1,12 ) ⊗ V1,0 )GL(n,Z) = Q
for n ≥ 3. By Theorem 3.3, for n ≥ 4, we have E21,0 = E22,0 = 0 and thus we have
H 1 (Aut(Fn ), V1,0 ) = E20,1 = Q.
For λ = (12 , 1), we haveE20,1 = (H 1 (IAn , Q) ⊗ V12 ,1 )GL(n,Z) = ((V0,1 ⊕ V1,12 ) ⊗ V12 ,1 )GL(n,Z) = Q
for n ≥ 3. By Theorem 3.3, for n ≥ 4, we have E21,0 = E22,0 = 0 and thus we have
H 1 (Aut(Fn ), V12 ,1 ) = E20,1 = Q.
□
In a similar way, we have the following theorem for Out(Fn ).
Theorem 4.2. For λ ̸= (12 , 1), we have
H 1 (Out(Fn ), Vλ ) = 0

for n ≥ 3.

In particular, we have H 1 (Out(Fn ), V1,0 ) = 0 for n ≥ 3.
For λ = (12 , 1), we have
H 1 (Out(Fn ), V12 ,1 ) = Q

for n ≥ 4.

Proof. Let IOn be the subgroup of Out(Fn ) appearing in the short exact sequence(4.2.2)

1 → IOn → Out(Fn ) → GL(n, Z) → 1.

The proof is the same as that of Theorem 4.1 except that we use Kawazumi’s
computation [28]H 1 (IO(Fn ), Q) = V1,12
instead of (4.2.1) and the Hochschild–Serre spectral sequences for the short exact
sequence (4.2.2).
□
By Theorems 4.1 and 4.2, we obtain the following corollary, which partially
recovers (and in a few cases improves) results of Satoh [51, 49] and Randal-Williams[47] that we mentioned in Section 4.1.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

13

Corollary 4.3 (Cf. [51, 49, 47]).
H1 (Aut(Fn ), H) = 0,∗

H1 (Aut(Fn ), H ) = Q
1H1 (Out(Fn ), H) = 0∗ ⊗q

H (Aut(Fn ), (H )1

H (Aut(Fn ), H⊗q

) = 0,) = 0,

for n ≥ 3,

H (Out(Fn ), H1 (IAn , Q)) = Q

for n ≥ 4,

1

H (Aut(Fn ), H1 (IAn , Q)) = Q ,1

H1 (Out(Fn ), H ∗ ) = 0

for n ≥ 4,

2

for n ≥ 3,

1

∗ ⊗q

1

⊗q

H (Out(Fn ), (H )H (Out(Fn ), H

)=0

)=0

for n ≥ 3,

for n ≥ 3, q ̸= 1.

Proof. The proof is straightforward. For coefficients in H ⊗q and (H ∗ )⊗q , we useLλthe decomposition H(n)⊗q = λ Vλ (n)⊕ dim S , where λ runs through all partitions
with |λ| = q and l(λ) ≤ n.
□
The following result complements Randal-Williams’s result for twisted cohomology of Aut(Fn ) and Out(Fn ) [47, Theorem A] in cohomological degree 1.
Theorem 4.4. (1) Let p, q ≥ 1. If p − q ̸= 1, then we have
H 1 (Aut(Fn ), H p,q ) = 0

for n ≥ 3.

For Out(Fn ), the same result holds.
(2) Let p, q ≥ 1. If p − q = 1, then we have1

H 1 (Aut(Fn ), H q+1,q ) = Q 2 (q+2)!

for n ≥ 4.

For Out(Fn ), we haveq+1

H 1 (Out(Fn ), H q+1,q ) = Q( 2 )q!

for n ≥ 4.

Proof. (1) By (2.0.1) and (2.0.2), H p,q is a direct sum of some copies of Vλ ’s for
λ = (λ, λ′ ) with deg λ = p − q and |λ| ≤ p + q. Therefore, the statements follow
from Theorems 4.1 and 4.2.
(2) If p = q + 1, then we need to count the multiplicities of V12 ,1 and V1,0 in
H q+1,q since we have
H 1 (Aut(Fn ), H q+1,q ) = H 1 (IAn , H q+1,q )GL(n,Z)
for n ≥ 4 by the fact that E21,0 = E22,0 = 0 for n ≥ 4. By the exact sequenceq(2.0.2), the multiplicity of V1,0 in H q+1,q is q+1
q
q q! = (q + 1)!, and the mul

q+1
q
⟨2,1⟩
q+1,q
tiplicity of V
in H
is q−1 q−1 (q − 1)! = q+1
by the2 q!. Therefore,

q+1
q+1,q
decomposition (2.0.1), the multiplicity of V12 ,1 in H
is 2 q! since we have2
dim(S 1 ) dim(S 1 ) = 1. Therefore, we haveq+1

H 1 (Aut(Fn ), H q+1,q ) = Q((q+1)!+( 2 )q!) = Q 2 (q+2)! .
1

For Out(Fn ), we only need the multiplicity of V12 ,1 in H q+1,q , which isby the above argument. Therefore, we have

q+1
2 q!



q+1

H 1 (Out(Fn ), H q+1,q ) = H 1 (IOn , H q+1,q )GL(n,Z) = Q( 2 )q!
for n ≥ 4.

□

14

KAZUO HABIRO AND MAI KATADA

Remark 4.5. The vector spaces H 1 (Aut(Fn ), H p,q ) and H 1 (Out(Fn ), H p,q ) has
Sp × Sq -module structure. Conjecture 6 of [30] (see Theorem 7.3) implies thatfor sufficiently large n, we have
H 1 (Aut(Fn ), H q+1,q ) = CP ⟳ (q + 1, q),0where CP ⟳ is the wheeled PROP associated to the wheeled completion of the operad0P0 , which is the operadic suspension of the operad Com (see also [27, Conjecture112.2]). We can check that CP ⟳ (q + 1, q) = Q 2 (q+2)! . Therefore, the above theorem
0
gives a proof of the degree 1 part of their conjecture. Note that Lindell [37] has
recently proved their conjecture in all degrees.
4.3. Second cohomology of Aut(Fn ) and Out(Fn ) with algebraic coefficients. Here we consider the second cohomology of Aut(Fn ) and Out(Fn ) andapply it to the second cohomology of IAn and IOn .
Theorem 4.6. (1) For any bipartition λ = (λ, λ′ ) ̸= (0, 0), we have
H 2 (Aut(Fn ), Vλ ) ∼= (H 2 (IAn , Q) ⊗ Vλ )GL(n,Z)

for n ≥ 5.

For the trivial coefficient, we have
H 2 (Aut(Fn ), Q) ∼= H 2 (IAn , Q)GL(n,Z)

for n ≥ 4.

(2) For any bipartition λ = (λ, λ′ ) ̸= (0, 0), we have
H 2 (Out(Fn ), Vλ ) ∼= (H 2 (IOn , Q) ⊗ Vλ )GL(n,Z)

for n ≥ 5.

For the trivial coefficient, we have
H 2 (Out(Fn ), Q) ∼= H 2 (IOn , Q)GL(n,Z)

for n ≥ 4.

Proof. (1) For n ≥ 5, we have E22,0 = E23,0 = 0. For n ≥ 4, we have E21,1 = E22,1 = 0.
Therefore, we have0,2
H 2 (Aut(Fn ), Vλ ) ∼= E2 = (H 2 (IAn , Q) ⊗ Vλ )GL(n,Z) .

The case of the trivial coefficient follows from Theorem 3.3 in a similar way.
(2) The proof is similar to that of (1).

□Day and Putman [8] showed that H2 (IAn , Z)GL(n,Z) = 0 for n ≥ 6. The rationalversion of this result follows from Theorem 4.6.
Corollary 4.7. For n ≥ 4, we have
H 2 (IAn , Q)GL(n,Z) = H 2 (IOn , Q)GL(n,Z) = 0.
Proof. We have H 2 (Aut(Fn ), Q) = H 2 (Out(Fn ), Q) = 0 for n ≥ 1 by Hatcher–
Vogtmann [22]. Therefore, by Theorem 4.6, we have
H 2 (IAn , Q)GL(n,Z) = H 2 (IOn , Q)GL(n,Z) = 0
for n ≥ 4.

□

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

15

5. Stable families of polynomial Aut(Fn )-modules
In this section, we apply the results of the previous sections to non-semisimple
Aut(Fn )-modules.
5.1. Polynomial Aut(Fn )-modules. Here we introduce polynomial Aut(Fn )-modules
which may relate to other well-known notions of polynomiality.
Definition 5.1. By a polynomial Aut(Fn )-module, we mean an Aut(Fn )-module
M which has a composition series of finite length
M = M 0 ⊋ M 1 ⊋ M2 ⊋ · · · ⊋ Mk = 0such that each composition factor Mj /Mj+1 is induced by an irreducible algebraic
GL(n, Z)-representation.
A family M (∗) = {M (n)}n≥0 of polynomial Aut(Fn )-modules M (n) is stable inranks ≥ r if there is a decreasing filtration
M (∗) = M0 (∗) ⊋ M1 (∗) ⊋ M2 (∗) ⊋ · · · ⊋ Mk (∗) = 0
such that each Mj (∗)/Mj+1 (∗) is a stable family of GL(n, Z)-representations in
Lk−1
ranks ≥ r. Let gr M (∗) = j=0 Mj (∗)/Mj+1 (∗). We set
n0 (M (∗), p) = n0 (gr M (∗), p).
Polynomial Out(Fn )-modules and stable families of polynomial Out(Fn )-modules
can be defined just in the same way as Aut(Fn ).
Proposition 5.2. Let M (∗) be a stable family of polynomial Aut(Fn )-modules withno composition factors that are isomorphic to either V1,0 or V12 ,1 . Then we have
H 1 (Aut(Fn ), M (n)) = 0(5.1.1)for n ≥ n0 (M (∗), 1).

Proof. Let n ≥ n0 (M (∗), 1). We use the induction on the length k of the decreasing filtration of M (∗). If k = 1, then M (n) is an irreducible algebraic GL(n, Z)representation Vλ such that λ ̸= (1, 0), (12 , 1). Therefore, (5.1.1) follows from Theorem 4.1. Suppose that (5.1.1) holds for k − 1. For M (∗) with length k, the exact
sequence of Aut(Fn )-modules
0 → M1 (n) → M (n) → M (n)/M1 (n) → 0
induces a long exact sequence· · · → H 1 (Aut(Fn ), M1 (n)) → H 1 (Aut(Fn ), M (n)) → H 1 (Aut(Fn ), M (n)/M1 (n)) → · · · .
By the hypothesis of the induction and Theorem 4.1, we have
H 1 (Aut(Fn ), M (n)/M1 (n)) = H 1 (Aut(Fn ), M1 (n)) = 0
for n ≥ n0 (M (∗), 1), so we have H 1 (Aut(Fn ), M (n)) = 0.
For stable families of polynomial Out(Fn )-modules, we obtain the following.

□

16

KAZUO HABIRO AND MAI KATADA

Proposition 5.3. Let M (∗) be a stable family of polynomial Out(Fn )-modules withno composition factors that are isomorphic to V12 ,1 . Then we have
H 1 (Out(Fn ), M (n)) = 0
for n ≥ n0 (M (∗), 1).
5.2. Applications to the Aut(Fn )-modules of Jacobi diagrams. Massuyeau
and the first-named author [17] extended the Kontsevich integral of links to afunctorial invariant defined on the category of bottom tangles in handlebodies. Thisinvariant takes values in the category A of Jacobi diagrams in handlebodies, where
Jacobi diagrams are certain uni-trivalent graphs.
The second-namedL
author [25, 26] studied the Aut(Fn )-module structure on thevector space A(n) = d≥0 Ad (n) of Jacobi diagrams on n oriented arcs, where theaction is induced by the composition of morphisms in the category A. For d = 0,
we have A0 (n) = Q. For d > 0, Ad (n) admits a filtration
Ad (n) = Ad,0 (n) ⊃ Ad,1 (n) ⊃ · · · ⊃ Ad,2d−1 (n) = 0,
whose associated graded Ad,k (n)/Ad,k+1 (n) is identified with the vector space Bd,k (n)of open Jacobi diagrams of degree d with univalent vertices colored by elementsof the GL(n, Q)-module (Qn )∗ . It follows that {Bd,k (n)}n is a stable family ofGL(n, Q)-representations. Hence {Ad (n)}n is a stable family of polynomial Aut(Fn )modules. Note here that the Aut(Fn )-module Ad (n) does not factor through aGL(n, Z)-module if d ≥ 2.
We have the following result.
Proposition 5.4. For each d ≥ 0, we have
H 1 (Aut(Fn ), Ad (n)) = H 1 (Aut(Fn ), Ad (n)∗ ) = 0
for n ≥ 3.
Proof. For d > 0, since each irreducible component of Bd (n) is of type V0,λ with
2 ≤ |λ| ≤ 2d, it is easily seen that Ad (n) and Ad (n)∗ do not include composition
factors isomorphic to V0,1 or V1,12 . For example, A1 (n) is isomorphic to V0,2 , and
the Aut(Fn )-module A2 (n) has the following composition series
A2 (n) ⊋ A′′2 (n) ⊋ A2,1 (n) ⊋ A2,2 (n) ⊋ 0(n ≥ 3)whose composition factors are V0,4 , V0,22 , V0,13 and V0,2 (see [25]). Hence Proposition 5.2 implies the desired result.
□
Remark 5.5. Let d ≥ 0 and i ≥ 1. By [10] and the fact that Ad is a contravariant
polynomial functor on the category F of finitely generated free groups and homomorphisms [25], we obtain limn H i (Aut(Fn ), Ad (n)) = 0. Considering the stable←−
range, one can show that
H i (Aut(Fn ), Ad (n)) = 0,n ≥ 2i + 2d + 3

and if i ≥ 2d + 1, then one can also show that
H i (Aut(Fn ), Ad (n)∗ ) = 0,n ≥ 2i + 2d + 3

by using Theorem 7.3 below. If i < 2d + 1, then the latter twisted cohomology canbe non-trivial. For example, we have H 3 (Aut(Fn ), A2 (n)∗ ) ̸= 0 for n ≥ 13.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

17

Remark 5.6. Vespa [54] generalized the functor Ad to another contravariant polynomial functor Ad (m, −)0 of degree 2d on F by considering “beaded” Jacobi diagrams on oriented arcs. If m > 0, then {Ad (m, n)0 }n≥0 is not a stable family ofpolynomial Aut(Fn )-modules since Ad (m, n)0 is infinite-dimensional for d, n > 0.
However, each composition factor of Ad (m, n)0 is of the form Vλ (n)∗ , |λ| ≤ 2d.
Similarly to Remark 5.5, one can show that
H i (Aut(Fn ), Ad (m, n)0 ) = 0,n ≥ 2i + 2d + 3.

The second-named author [25] proved that the Aut(Fn )-module structure on
Ad (n) induces an Out(Fn )-module structure on Ad (n). Therefore, {Ad (n)}n is alsoa stable family of polynomial Out(Fn )-modules. Then by Proposition 5.3, we obtainthe first cohomology of Out(Fn ) in a similar way.
Proposition 5.7. For each d ≥ 0, we have
H 1 (Out(Fn ), Ad (n)) = H 1 (Out(Fn ), Ad (n)∗ ) = 0
for n ≥ 3.
6. GL(n, Z)-invariant part of the stable rational cohomology of IAn
So far, no non-trivial elements of the GL(n, Z)-invariant part of the stable rational cohomology H ∗ (IAn , Q) have been found. Igusa [23] constructed higher Franz–
Reidemeister torsion classes in H 4i (IOn , Q)GL(n,Z) . Morita, Sakasai and Suzuki [44]constructed the “secondary classes” in H 4i (IOn , Q)GL(n,Z) and H 4i (IAn , Q)GL(n,Z)(i ≥ 1). Their classes for IOn coincide with Igusa’s torsion classes. It is not known
whether or not these classes are non-trivial.
In this section, we propose a conjectural structure of the GL(n, Z)-invariant
part of the stable rational cohomology H ∗ (IAn , Q), and prove it under Church andFarb’s stability conjecture (Conjecture 6.3).
6.1. Conjectural stable structure of H ∗ (IAn , Q)GL(n,Z) . Let us recall somefacts and conjectures about H ∗ (IAn , Q)GL(n,Z) . We have H 1 (IAn , Z)GL(n,Z) = 0
by [7, 11, 28]. As we mentioned in Section 4.3, we also have H 2 (IAn , Z)GL(n,Z) = 0
by [8]. Church and Farb [6, Conjecture 6.5] conjectured that the GL(n, Z)-invariant
part of H ∗ (IAn , Q) vanishes stably. We make the following conjecture, which contradicts the above conjecture of Church and Farb.
Conjecture 6.1. The GL(n, Z)-invariant part of the rational cohomology of IAnstabilizes, that is, for each i ≥ 0, there is an integer N ≥ 0 such that for all n ≥ N
the inclusion map IAn ,→ IAn+1 induces an isomorphism∼=

H i (IAn+1 , Q)GL(n+1,Z) → H i (IAn , Q)GL(n,Z) .
Moreover, we stably have an isomorphism of graded algebrasH ∗ (IAn , Q)GL(n,Z) ∼= Q[z1 , z2 , . . . ],deg zi = 4i.

Remark 6.2. We expect that the generators zi of our conjectural structure of theGL(n, Z)-invariant part of H ∗ (IAn , Q) can be constructed by using the Borel classes.
See Proposition 6.7.

18

KAZUO HABIRO AND MAI KATADA

We will prove Conjecture 6.1 by assuming the following conjecture, which follows
from [6, Conjectures 6.3, 6.6 and 6.7].
Conjecture 6.3 (Church–Farb [6]). For each i ≥ 0, the following hypothesis (SFi )holds.
(SFi ) {H i (IAn , Q)}n is a stable family of GL(n, Z)-representations.
It is known that (SFi ) holds for i = 0, 1.
The proof of the following theorem will be given in Section 6.5.
Theorem 6.4. Conjecture 6.3 implies Conjecture 6.1.
Remark 6.5. In Theorem 6.4, Conjecture 6.3 can be replaced with the weakerassumption that the family {H i (IAn , Q)}n is stably algebraic. See Remark 6.8.
6.2. Antitransgression maps. Here we discuss “antitransgression” maps for the
Hochschild–Serre spectral sequences for certain group extensions.
Let G be a group such that H i (G, Q) = 0 for i = k, k + 1, where k ≥ 1. Consider
an exact sequence of groups
1→N →G→Q→1
and the Hochschild–Serre spectral sequence
E2p,q = H p (Q, H q (N, Q)) ⇒ H p+q (G, Q).
0,k
k+1,0
Since H k (G, Q) = H k+1 (G, Q) = 0, we have Ek+2
= Ek+2= 0 and hence the
0,k
0,k
k+1,0
map dk+1 : Ek+1 → Ek+1 is an isomorphism. Define a Q-linear map−1
φk = ιk (d0,kπk+1 : E2k+1,0 → E20,k ,
k+1 )
k+1,0
0,k
where πk+1 : E2k+1,0 ↠ Ek+1
and ιk : Ek+1,→ E20,k are the edge maps. We callthe map φk the kth antitransgression map since it runs in the opposite direction ofa transgression map (if it exists).

6.3. Graded-commutative multiplicative spectral sequences. Here we use
Zeeman’s comparison theorem to obtain certain isomorphism of graded algebras.
L
L
Let V =k≥2 Vk be a graded vector space. Let V [−1] =k≥1 V [−1]k ,V [−1]k = VL
obtainedfromVbydegree-shiftby −1.
k+1 , be the graded vector spaceL
Let SV = i≥0 S i (V ) and S(V [−1]) = i≥0 S i (V [−1]) be the graded-symmetricalgebras on V and V [−1], respectively.
A multiplicative spectral sequence E = {Erp,q } is a spectral sequence with as′ ′′′sociative, unital multiplication maps Erp,q ⊗ Erp ,q → Erp+p ,q+q . A multiplica′′
tive spectral sequence E is graded-commutative if ab = (−1)(p+q)(p +q ) ba for any′ ′a ∈ Erp,q and b ∈ Erp ,q . For example, the cohomological Hochschild–Serre spectral sequences for exact sequences of groups are graded-commutative multiplicative
first-quadrant spectral sequences.
We have the following theorem, which is closely related to Borel’s result [1] (see[39, Theorem 3.27]).

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

19

Theorem 6.6. Let N, P, Q ∈ {0, 1, 2, . . . , ∞}. Let E be a graded-commutative,
multiplicative first-quadrant spectral sequence over Q satisfying the following conditions(1) the multiplication map E2p,0 ⊗E20,q → E2p,q is a linear isomorphism for p ≤ P ,q ≤ Q,
p,q(2) E∞= 0 if 0 < p + q ≤ N ,L(3) there is a graded vector space V = k≥2 Vk such that we have an isomorphism of graded algebras E2∗,0 ∼= SV for ∗ ≤ P .
Then the restriction φk |Vk+1 of the antitransgression map φk : E2k+1,0 → E20,k is
an isomorphism for each k ≥ 1, and we have an isomorphism of graded algebras
E 0,∗ ∼= S(V [−1])2for ∗ ≤ min(P − 2, Q, N − 2). Here, V [−1]k is identified with the image φk (Vk+1 ).
Proof. We first consider the case where at least one of P , Q and N are finite. Let
K = 2+min(P −2, Q, N −2). For 2 ≤ k ≤ K, let k E denote the graded-commutative
multiplicative first-quadrant spectral sequence given byk

E2p,q = S(Vk )p ⊗ S(V [−1]k−1 )q∼=

with dj = 0 for j ̸= k and dk0,k−1 = idVk : k Ek0,k−1 → k Ekk,0 . Since the differential
is determined by the Leibniz rule, we have an exact sequence
0 → S i (V [−1]k−1 ) → S 1 (Vk ) ⊗ S i−1 (V [−1]k−1 ) → · · · → S i (Vk ) → 0p,qp,q= 0 for p + q > 0. Set
for each i ≥ 1. Therefore, we have k Ek+1
= · · · = k E∞Ni
E = 2≤i≤k E for 2 ≤ k ≤ K.

≤k

We claim that the restriction of φk−1 : E2k,0 → E20,k−1 to Vk is an isomorphism
for k = 2, . . . , K, and that there are morphisms of multiplicative spectral sequenceskf : ≤k E → E

for k = 2, . . . , K such that k f2p,0 is an isomorphism for 0 ≤ p ≤ k, and such thatk+1
f |≤k E = k f for each k = 2, . . . , K − 1.
The proof of this claim is done by induction on k. For k = 2, the differential
d0,1: E20,1 → E20,2 is an isomorphism by conditions (2) and (3). Therefore, the2inverse map φ1 : E22,0 = V2 → E20,1 is an isomorphism, and we have such a morphism2f : ≤2 E → E.
Let k ≥ 2. Suppose that such a morphism k f exists. Then since k f2p,0 is anp,qisomorphism for 0 ≤ p ≤ k and k f∞
is an isomorphism for any 0 < p + q ≤ N
by the condition (2), it follows from Zeeman’s comparison theorem [55] that k f20,q
is an isomorphism for 0 ≤ q ≤ k − 2. Therefore, by the condition (1), k f2p,q is an
isomorphism for 0 ≤ p ≤ k, 0 ≤ q ≤ k − 2. In particular, k f2p,k−p is an isomorphismfor 2 ≤ p ≤ k − 1. Since we have E21,0 = 0, by condition (1), we have E21,k−1 = 0.
k+1,0Therefore, the restriction of the edge map πk : E2k+1,0 → Ek+1
to Vk+1 is an0,k −1k+1,0
isomorphism. Then the restriction of φk = ιk (dk+1 ) πk : E2
→ E20,k to Vk+1
is an isomorphism. Hence, we have a morphism k+1 f extending k f such that k+1 f2p,0
is an isomorphism for 0 ≤ p ≤ k + 1.

20

KAZUO HABIRO AND MAI KATADA

We now have a morphism K f : ≤K E → E. By Zeeman’s comparison theorem,
K 0,qf2 : ≤K E20,q → E20,q is an isomorphism for q ≤ K − 2, which implies E20,∗ ∼=S(V [−1]) for ∗ ≤ K − 2.
Let us consider the case where P = Q = N = ∞. The above proof for the finitek≤kcase shows that there
E → E for k ≥ 2. Let
N isk a family of morphisms f :≤k′
E = limk E = k≥2 E be the direct limit of multiplicative spectral sequence.
−→
There is a morphism ′ f : ′ E → E extending k f for all k ≥ 2. By Zeeman’s□
comparison theorem, we obtain E20,∗ ∼= S(V [−1]) in a similar way.
6.4. The classes yk ∈ H 4k (IAn , Q)GL(n,Z) . We apply the antitransgression map
to the exact sequence
1 → IAn → Aut(Fn ) → GL(n, Z) → 1.
By Theorem 3.3, we have for n ≥ p + 1

^E2p,0 = the degree p part of(x1 , x2 , . . . ) ,Q

where deg xk = 4k + 1.
For k ≥ 1, set
yk = φ4k (xk ) ∈ E20,4k = H 4k (IAn , Q)GL(n,Z) .
We expect that yk is a nonzero scalar multiple of the “secondary cohomology class”
0
T β2k+1∈ H 4k (IAn , R) appearing in Morita, Sakasai and Suzuki [44].
6.5. Proof of Theorem 6.4. Let yk be the elements defined in Section 6.4. Theorem 6.4 follows from the following.
Proposition 6.7. Let P, Q ≥ 0. Suppose (SFi ) holds for any i ≤ Q. Let n ≥max{n0 ({H i (IAn , Q)}n , P )}i≤Q . Then we have a stable isomorphism of algebrasH ∗ (IAn , Q)GL(n,Z) ∼= Q[y1 , y2 , . . . ]in degree ∗ ≤ min(P − 2, Q, n2 − 3).
Proof. Consider the cohomological Hochschild–Serre spectral sequence for the exact
sequence 1 → IAn → Aut(Fn ) → GL(n, Z) → 1, which is graded-commutative andmultiplicative. By the assumption (SFi ) for i ≤ Q, it follows from Proposition 3.5
that we have
E p,q = H p (GL(n, Z), H q (IAn , Q)) ∼= H p (GL(n, Z), Q) ⊗ (H q (IAn , Q))GL(n,Z)
2

for 0 ≤ p ≤ P , 0 ≤ q ≤ Q. By Galatius [14], we have H i (Aut(Fn ), Q) = 0 forp,q
we have E∞
0 < i ≤ n−2= 0 for 0 < p + q ≤ n−2
2 . Therefore,
2 . By Theorem 3.3,∗,0 ∼ V
we have E2 = Q (x1 , . . .) for ∗ ≤ n − 1. Therefore, by Theorem 6.6, we have an
isomorphism of graded algebrasH ∗ (IAn , Q)GL(n,Z) ∼= S(V [−1])Lnfor ∗ ≤ min(P − 2, Q, 2 − 3), where V =k≥1 Qxk . Since we have V [−1] =
L
Qy
,
S(V
[−1])
=
Q[y
,
y
,
.
.
.].
□
k
1 2
k≥1
Remark 6.8. By Remark 3.6, we can deduce Proposition 6.7 with the assumption(SFi ) replaced with the following weaker assumption:

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

21(SAi ) {H i (IAn , Q)}n is a stably algebraic family.
7. Stable rational cohomology of IAn
The whole structure of the stable rational cohomology of IAn is still mysterious
in degree ≥ 2. In this section, we propose a conjectural structure of the stablerational cohomology of IAn , and study some relations to other known conjectures.
7.1. Known results and conjectures. In [27], the second-named author studied
the Albanese homology H∗A (IAn , Q) of IAn , which is predual to the Albanese cohomology in the sense of Church, Ellenberg and Farb [5]. The Albanese homology
H∗A (IAn , Q) of IAn is defined byτ∗H∗ (U, Q) ,H∗A (IAn , Q) = im H∗ (IAn , Q) −→where U = H1 (IAn , Q) and τ∗ is induced by the abelianization map τ : IAn → U .
∗The Albanese cohomology HA(IAn , Q) of IAn is defined by


τ∗
∗
HA
(IAn , Q) = im H ∗ (U, Q) −→ H ∗ (IAn , Q) ,which is dual to H∗A (IAn , Q).
∗∗(IAn , Q) is a quotient of(IAn , Q)GL(n,Z) = 0 since HA
Note that we have HA∗
GL(n,Z)
H (U, Q) and since we have H (U, Q)
= 0.
Vi+1
Let Ui = Hom(H,H) for i ≥ 1. Let S ∗ (U∗ ) denotethe graded-symmetricL
algebra of the graded GL(n, Q)-representation U∗ = i≥1 Ui . The traceless partL∗W∗ =i≥0 Wi of S (U∗ ) is defined by using contraction maps (see [27] for details). The second-named author [27] proved that HiA (IAn , Q) includes a GL(n, Q)subrepresentation which is isomorphic to Wi , and made the following conjecture,
which holds for i = 1 [7, 11, 28], for i = 2 [46] and for i = 3 [27].
∗Conjecture 7.1 ([27]). Let i ≥ 0. We have a GL(n, Q)-isomorphism
HiA (IAn , Q) ∼= Wi
for sufficiently large n.
For any algebraic GL(n, Q)-representation V , the inclusion map IAn ,→ Aut(Fn )induces a morphism of graded vector spaces(7.1.1)i∗ : H ∗ (Aut(Fn ), V ) → H ∗ (IAn , V )GL(n,Z) .

The second-named author [27] also made the following conjecture about the relation
between the Albanese cohomology of IAn and the twisted cohomology of Aut(Fn ).
ii
For any algebraic GL(n, Q)-representation V , let HA(IAn , V ) = HA
(IAn , Q) ⊗ V .
Conjecture 7.2 ([27]). For any algebraic GL(n, Q)-representation V , the map i∗∗
is stably injective and its image stably equals to HA(IAn , V )GL(n,Z) . That is, we
stably have an isomorphism of graded vector spaces∼=∗i∗ : H ∗ (Aut(Fn ), V ) −→ HA(IAn , V )GL(n,Z) .

22

KAZUO HABIRO AND MAI KATADA

Kawazumi and Vespa [30] proposed a conjectural structure of the cohomologyof Aut(Fn ) with coefficients in H p,q , the degree 1 part of which we have mentioned
in Remark 4.5, and which was previously known to hold for p = 0 or q = 0 by
Djament–Vespa [10], Djament [9], Vespa [53] and Randal-Williams [47]. Recently,
Lindell [37] proved their conjecture.
Theorem 7.3 (Lindell [37]). Let p, q ≥ 0. If i ̸= p − q, then we have
H i (Aut(Fn ), H p,q ) = 0
for sufficiently large n. Otherwise, we have
H i (Aut(Fn ), H q+i,q ) = CP ⟳ (q + i, q)0

for sufficiently large n. (Here CP ⟳ is the wheeled PROP associated to the wheeled0
completion of the operad P0 , see Remark 4.5.)
By combining Theorem 7.3 and the result of the second-named author [27, Proposition 12.7], it follows that Conjectures 7.1 and 7.2 are equivalent.
7.2. Conjectural stable structure of H ∗ (IAn , Q). Here we propose a conjectural structure of the whole stable rational cohomology of IAn .
We have the restrictions of the cup product maps
M jωn,i :HA (IAn , Q) ⊗ H k (IAn , Q)GL(n,Z) → H i (IAn , Q),
j+k=iwhich form a graded algebra map∗
ωn : HA(IAn , Q) ⊗ H ∗ (IAn , Q)GL(n,Z) → H ∗ (IAn , Q).

The map ωn,i is an isomorphism in degree i = 0, 1, since we havei
H i (IAn , Q) = HA
(IAn , Q)(i = 0, 1).

Conjecture 7.4. The graded algebra map ωn is a stable isomorphism.
Conjectures 7.4, 7.1 and 6.1 imply the following conjecture, which would give acomplete algebraic structure for the stable rational cohomology of IAn .
Conjecture 7.5. We stably have an isomorphism of graded algebras with GL(n, Z)actionsH ∗ (IAn , Q) ∼= (W∗ )∗ ⊗ Q[z1 , z2 , . . . ], deg zi = 4i,
where (W∗ )∗ denotes the graded dual of W∗ .
Conjecture 7.4 is related to some other conjectures as follows.
Theorem 7.6. Suppose that Conjectures 6.3 and 7.1 hold. Then Conjecture 7.4
holds.
Proof. By the assumptions (SFi ), the graded algebra map∗
ωn : HA(IAn , Q) ⊗ H ∗ (IAn , Q)GL(n,Z) → H ∗ (IAn , Q)

is a stable isomorphism if and only if for any bipartition λ, the cup product map∪∗
HA(IAn , Vλ )GL(n,Z) ⊗ H ∗ (IAn , Q)GL(n,Z) −→ H ∗ (IAn , Vλ )GL(n,Z)

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

23

is a stable isomorphism. Therefore, since Conjectures 7.1 and 7.2 are equivalent,
Conjecture 7.4 holds by Conjecture 7.2 and Lemma 7.7 below.
□
We have a morphism of graded vector spacesξ : H ∗ (Aut(Fn ), Vλ ) ⊗ H ∗ (IAn , Q)GL(n,Z) → H ∗ (IAn , Vλ )GL(n,Z)
defined by ξ(u⊗v) = i∗ (u)∪v for u ∈ H ∗ (Aut(Fn ), Vλ ) and v ∈ H ∗ (IAn , Q)GL(n,Z) ,where i∗ is defined in (7.1.1).
Lemma 7.7. Suppose that Conjecture 6.3 holds. Then for each bipartition λ, themap ξ is a stable isomorphism.
Proof. Let λ be a bipartition. In what follows, we assume that n is sufficiently large
with respect to the cohomological degrees of cohomology groups that we consider.
We use induction on the cohomological degree i. The case i = 0 is trivial. Leti ≥ 1. Suppose that we haveM∼=
(7.2.1) ξ :H j (Aut(Fn ), Vλ ) ⊗ H k (IAn , Q)GL(n,Z) −→ H q (IAn , Vλ )GL(n,Z)
j+k=qfor 0 ≤ q ≤ i − 1. We consider the Hochschild–Serre spectral sequencep+qp
qE(λ)p,q(Aut(Fn ), Vλ ).
2 = H (GL(n, Z), H (IAn , Vλ )) ⇒ H

If we have H j (Aut(Fn ), Vλ ) = 0 for any j ≤ i−1, then we have H q (IAn , Vλ )GL(n,Z) =0 for any q ≤ i−1 by (7.2.1). By the assumptions (SFj ) for j ≤ i−1 and Proposition
3.5, we have E(λ)p,q2 = 0 for any q ≤ i−1. Therefore, by the convergence of the spec∼=
tral sequence, one can check that ξ = i∗ : H i (Aut(Fn ), Vλ ) −→ H i (IAn , Vλ )GL(n,Z) .
Otherwise, there is 0 ≤ q0 ≤ i − 1 such that we have H q0 (Aut(Fn ), Vλ ) ̸= 0.
Then by the first half of Theorem 7.3, we have H q (Aut(Fn ), Vλ ) = 0 for q ̸= q0 .
Therefore, by (7.2.1), ξ gives the following isomorphisms(7.2.2)0q
GL(n,Z) ∼H (IAn , Vλ )= H q0 (Aut(Fn ), Vλ ) q0
H (IAn , Vλ )GL(n,Z) ⊗ H q−q0 (IAn , Q)GL(n,Z)(q < q0 )(q = q0 )(q0 < q ≤ i − 1).

Recall the Hochschild–Serre spectral sequence E2p,q = H p (GL(n, Z), H q (IAn , Q))
that we used in the proof of Proposition 6.7. Let Ê = {Êr }r≥2 denote the spectral
sequence concentrated in bidegree (0, q0 ) with Ê20,q0 = H q0 (IAn , Vλ )GL(n,Z) . Then
we have a canonical morphism of spectral sequences ψ : Ê → E(λ). Consider thetensor product Ê ⊗ E of two spectral sequences Ê and E. We have a morphism ofspectral sequences defined as the composition
ψ⊗id∪

ϕ : Ê ⊗ E −−−→ E(λ) ⊗ E −→ E(λ).

24

KAZUO HABIRO AND MAI KATADA

We have the following commutative diagram
H q0 (IA, Vλ )GL ⊗ H q−q0 (IA, Q)GL ⊗ H p (GL, Q)∪⊗id/ H q (IA, Vλ )GL ⊗ H p (GL, Q)

id ⊗∪H q0 (IA, Vλ )GL ⊗ H p (GL, H q−q0 (IA, Q))∪=ϕp,q2

∪/ H p (GL, H q (IA, Vλ )),where we write GL = GL(n, Z) and IA = IAn . Here the vertical maps are isomorphisms for q ≤ i − 1 by Proposition 3.5, and the upper horizontal map is an
is also an isomorphism forisomorphism for q ≤ i − 1 by (7.2.2). Therefore, ϕp,q2q ≤ i − 1.
0,i0,iBy Lemma 7.8 below, the map ϕ0,i2 : (Ê ⊗ E)2 → E(λ)2 is an isomorphism
since we have

• ϕp,q2 is an isomorphism for q ≤ i − 1,0,qj−q0 ∼
0,q0⊗E∞• for j = i, i+1, (Ê ⊗E)j∞ ∼= Ê2 0 ⊗H j−q0 (Aut(Fn ), Q) = 0= Ê∞by [14], and• for j = i, i + 1, E(λ)j∞ ∼= H j (Aut(Fn ), Vλ ) = 0.
Therefore, we haveMξ:
H j (Aut(Fn ), Vλ ) ⊗ H k (IAn , Q)GL(n,Z)
j+k=ii∗ ⊗id−−∼−→ H q0 (IAn , Vλ )GL(n,Z) ⊗ H i−q0 (IAn , Q)GL(n,Z)=∪=ϕ0,i2−−−−→ H i (IAn , Vλ )GL(n,Z) ,∼=which completes the proof.

□

Lemma 7.8. Let ϕ : ′ E → E be a morphism between first-quadrant spectral sequence. Let i ≥ 1 be an integer. Suppose that
p,q
′ p,q
(1) ϕp,q
is an isomorphism for 0 ≤ q ≤ i − 1 and 0 ≤ p ≤ 2i.
2 : E2 → E2
p,q
′ p,q
(2) E∞ = E∞ = 0 for p, q with p + q = i, i + 1.

Then ϕ0,i2 is an isomorphism.
and ϕp+r,q−r+1
are isomorphisms, then we have
Proof. Firstly, note that if ϕp,q
r
r
′ p,q
p,q
ϕp,q
r (ker dr ) = ker dr ,p,qϕrp+r,q−r+1 (im ′ dp,qr ) = im dr .

p+r,q−r+1
Therefore, if all of the three maps ϕrp−r,q+r−1 , ϕp,q
are isomorphisms,
r , ϕrp,qthen ϕr+1 is also an isomorphism. By the assumption (1), it follows inductively
that the maps ϕp,q
are isomorphisms for any q ≤ i − r + 1, p ≤ 2i − 2q.
r

Let Λ = {(p, q) | p + q = i + 1, p ≥ 2} ∪ {(0, i)}. We will show, by a backward
induction on r, that for 2 ≤ r ≤ i + 2, the map ϕp,qis an isomorphism for eachr(p, q) ∈ Λ. For each (p, q) ∈ Λ, the map ϕp,qi+2 is an isomorphism since we have
p,q
p,q
′ p,q
p,q
E∞ (= ′ Ei+2
) = 0 and E∞(= Ei+2) = 0 by the assumption (2). Suppose thatp,qϕr+1 is an isomorphism for each (p, q) ∈ Λ. We consider the following commutativeON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

25

diagram′(7.2.3)

′

′

ϕ0,i
r

Er0,i

/ Er0,i

′ 0,i
dr



Err,i−r+1


ϕr,i−r+1r/ Err,i−r+1′ r,i−r+1
dr

Er2r,i−2r+2d0,ir

ϕ2r,i−2r+2r

dr,i−r+1
r/ Er2r,i−2r+2 .

As we mentioned in the first paragraph, ϕr,i−r+1
and ϕ2r,i−2r+2
are isomorphisms.
rr
Therefore, we have an isomorphism∼=

ϕrr,i−r+1 |ker ′ drr,i−r+1 : ker ′ drr,i−r+1 −→ ker dr,i−r+1.
rr,i−r+1
By the induction hypothesis, ϕr+1
is also an isomorphism, thus, we have∼=r,i−r+1′ r,i−r+1
ker ′ dr,i−r+1/ im ′ d0,i−→Er+1= ker dr,i−r+1/ im d0,irr = Er+1
r
r .
∼
=

0,i
Therefore, we have an isomorphism ϕrr,i−r+1 |im ′ d0,i
: im ′ d0,ir −→ im dr . We also
r
have
=
0,i
′ 0,i ∼
0,iker ′ d0,ir = Er+1 −→Er+1 = ker dr0,isince ϕ0,ir+1 is also an isomorphism. Hence, ϕr is also an isomorphism. We canp,qalso check that ϕr are isomorphisms for p + q = i + 1, i − r + 1 < q ≤ i − 1 byconsidering the commutative diagram starting at ′ Erp,q as in (7.2.3).

Therefore, by induction, ϕ20,i is an isomorphism.

□

Remark 7.9. In the statements of Theorem 7.6 and Lemma 7.7, we can replace theassumption, Conjecture 6.3, with the assumption (SAi ), that we considered in Remark 6.8. Consequently, if we assume Conjecture 7.1, the following are equivalent:• Conjecture 7.5,
• Conjecture 6.3,
• for each i ≥ 0, the hypothesis (SAi ) holds.
7.3. Types of irreducible components of H ∗ (IAn , Q). Here we obtain some irreducible algebraic GL(n, Q)-representations which are not included in H ∗ (IAn , Q).
Proposition 7.10. Let q ≥ 0 and suppose that (SFi ) holds for i ≤ q so that wehaveM ⊕c(q,λ)H q (IAn , Q) ∼
V
=
λ

λ

for integers c(q, λ) ≥ 0 for all sufficiently large n. Then we have c(q, λ) = 0 for
bipartitions λ such that either deg λ ≤ −q − 1 or 0 ≤ deg λ with λ ̸= (0, 0).
Proof. Let λ be a bipartition such that either deg λ ≤ −q − 1 or 0 ≤ deg λ with
λ ̸= (0, 0). By Theorem 7.3, for any 0 ≤ i ≤ q, we have
(7.3.1)

H i (Aut(Fn ), Vλ∗ ) = 0

26

KAZUO HABIRO AND MAI KATADA

for sufficiently large n. We will prove by induction on i that for any 0 ≤ i ≤ q, wehave
H i (IAn , Vλ∗ )GL(n,Z) = 0
GL(n,Z)for sufficiently large n. For i = 0, we have H 0 (IAn , Vλ∗ )GL(n,Z) = Vλ∗= 0.
Suppose that for any k ≤ i − 1, we have H k (IAn , Vλ∗ )GL(n,Z) = 0 for sufficiently
large n. Then by the assumption (SFi ) and by Proposition 3.5, for any j ≤ i + 1,
we have E2j,k = H j (GL(n, Z), H k (IAn , Vλ∗ )) = 0 for sufficiently large n. Therefore,by (7.3.1), we have H i (IAn , Vλ∗ )GL(n,Z) = 0, which implies that Vλ is not includedin H q (IAn , Q).
□
8. Stable rational cohomology of IOn
In this section, we study the stable rational cohomology of IOn as in the case ofIAn .
8.1. GL(n, Z)-invariant part of H ∗ (IOn , Q). We make the following conjecture
about the GL(n, Z)-invariant part of H ∗ (IOn , Q).
Conjecture 8.1. We stably have an isomorphism of graded algebrasH ∗ (IOn , Q)GL(n,Z) ∼= Q[z1 , z2 , . . .],deg zi = 4i.

We will prove Conjecture 8.1 by assuming the following conjecture, which isanalogous to Conjecture 6.3.
Conjecture 8.2. For each i ≥ 0, the following hypothesis (SF′i ) holds.
(SF′i ) {H i (IOn , Q)}n is a stable family of GL(n, Z)-representations.
It is known that (SF′i ) holds for i = 0, 1 [28].
Remark 8.3. Similarly to the case of IAn , we can replace the assumption in Theorems 8.4 and 8.11, i.e. Conjecture 8.2, by the weaker assumption that {H i (IOn , Q)}n
is a stably algebraic family.
Theorem 8.4. Conjecture 8.2 implies Conjecture 8.1.
Proof. We have an exact sequence
1 → IOn → Out(Fn ) → GL(n, Z) → 1
and by Galatius [14], we have H i (Out(Fn ), Q) = 0 for n sufficiently large withrespect to i. Therefore, the proof is similar to that of Proposition 6.7.
□
8.2. Conjectural stable structure of H ∗ (IOn , Q). The whole structure of thestable rational cohomology of IOn has not been determined in degree ≥ 2. We
propose a conjectural stable structure of H ∗ (IOn , Q) as in the case of IAn .
We have the restriction of the cup product maps
M j
′
ωn,i
:
HA (IOn , Q) ⊗ H k (IOn , Q)GL(n,Z) → H i (IOn , Q),
j+k=i

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

27

which form a graded algebra map∗
ωn′ : HA
(IOn , Q) ⊗ H ∗ (IOn , Q)GL(n,Z) → H ∗ (IOn , Q).
′
is an isomorphism in degree i = 0, 1.
The map ωn,i

Conjecture 8.5. The graded algebra map ωn′ is a stable isomorphism.
The second-named author [27] also defined the traceless part W∗O of the
L gradedsymmetric algebra S ∗ (U∗O ) of the graded GL(n, Q)-representation U∗O = i≥1 UiO ,where UiO = Ui for i ≥ 2 and U1O = U1 /H ∼= V12 ,1 , and proposed a conjectural
structure of the stable Albanese homology H∗A (IOn , Q).
Conjecture 8.6 ([27]). Let i ≥ 0. We have a GL(n, Q)-isomorphism
H A (IOn , Q) ∼
= WO
i

i

for sufficiently large n.
By combining Conjecture 8.6 with Conjectures 8.1 and 8.5, we obtain the following conjecture, which would give a complete algebraic structure of the stablerational cohomology of IOn .
Conjecture 8.7. We stably have an isomorphism of graded algebras with GL(n, Z)actionsH ∗ (IOn , Q) ∼= (W∗O )∗ ⊗ Q[z1 , z2 , . . . ], deg zi = 4i.
8.3. Twisted stable cohomology of Out(Fn ). The stable cohomology of Out(Fn )with coefficients in H p,0 and H 0,q has been determined as follows.
Theorem 8.8 (Randal-Williams [47]). We have
H i (Out(Fn ), (HZ∗ )⊗q ) = 0i

H (Out(Fn ), H

⊗q

)=0

for

2i ≤ n − q − 3,for2i ≤ n − q − 3, i ̸= q,

and for n ≥ 4q + 3, H q (Out(Fn ), H ⊗q ) ⊗ sgnSq is the permutation module on theset of partitions of {1, . . . , q} having no parts of size 1.
Here we make the following conjecture about the relation between the twisted
stable cohomology of Out(Fn ) and the stable Albanese cohomology of Out(Fn ),which is an analogue of Conjecture 7.2.
Conjecture 8.9. Let V be an algebraic GL(n, Q)-representation. We stably have
an isomorphism of graded algebrasH ∗ (Out(Fn ), V ) ∼= H ∗ (IOn , V )GL(n,Z) .
A

Conjectures 8.9 and 8.6 imply the following.
Conjecture 8.10. Let i ≥ 0 and let λ be a bipartition such that i ̸= deg λ. Then
we haveH i (Out(Fn ), Vλ ) = 0
for sufficiently large n.
We obtain the following proposition which is analogous to Theorem 7.6.
Theorem 8.11. Conjectures 8.2, 8.9 and 8.10 imply Conjecture 8.5.

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KAZUO HABIRO AND MAI KATADA

8.4. Relation between H ∗ (IOn , Q) and H ∗ (IAn , Q). Here we study the relation
between H ∗ (IOn , Q) and H ∗ (IAn , Q).
Let n ≥ 2. The projection p : IAn ↠ IOn induces a morphism of gradedalgebras with GL(n, Z)-actions p∗ : H ∗ (IOn , Q) → H ∗ (IAn , Q). The morphism ofgraded GL(n, Z)-representations H ∗ (IAn , Q) → H ∗ (Fn , Q) induced by the inclusion
Fn ,→ IAn has a splittingt : H ∗ (Fn , Q) → H ∗ (IAn , Q)
since H ∗ (Fn , Q) = Q[0] ⊕ H ∗ [1] and H 1 (IAn , Q) ∼= (U/H)∗ ⊕ H ∗ = V1,12 ⊕ V0,1 .
Define a morphism of graded GL(n, Z)-representations
ψ : H ∗ (IOn , Q) ⊗ H ∗ (Fn , Q) → H ∗ (IAn , Q)by ψ(u ⊗ v) = p∗ (u) ∪ t(v).
Proposition 8.12. Let n ≥ 2. We have an isomorphism of graded GL(n, Z)representations
gr H ∗ (IAn , Q) ∼= H ∗ (IOn , Q) ⊗ H ∗ (Fn , Q)
where the left hand side denotes the associated graded of H ∗ (IAn , Q) with respect
to a filtration (of length 2) of the GL(n, Z)-representation H ∗ (IAn , Q).
Proof. Consider the Hochschild–Serre spectral sequence
E2p,q = H p (IOn , H q (Fn , Q)) ⇒ H p+q (IAn , Q)
for the short exact sequence 1 → Fn → IAn → IOn → 1. Since IOn acts triviallyon H ∗ (Fn , Q) we haveE2p,q = H p (IOn , Q) ⊗ H q (Fn , Q).
For q > 1, we have E2p,q = 0 since H q (Fn , Q) = 0. We have d0,1
= 0 : E20,1 =
2
2,0
H 1 (Fn , Q) → E2 = H 2 (IOn , Q) since otherwise it would follow that H 1 (IAn , Q) ∼=E21,0 = H 1 (IOn , Q). By multiplicativity of the spectral sequence, we see thatd2∗,∗ = 0. Hence E∞ = E2 . Therefore the result follows.
□
Conjectures 6.3 and 8.2 and Proposition 8.12 imply the following conjecture.
Conjecture 8.13. The map ψ is a stable isomorphism of graded algebraic GL(n, Z)representations.
Note that for n ≥ 2, we have an isomorphism of graded algebraic GL(n, Z)representations
∼
=

∗
∗
∗
ψ A : HA
(IOn , Q) ⊗ HA
(Fn , Q) −→ HA(IAn , Q)[27, Proposition 9.8]. The projection p : IAn ↠ IOn induces a morphism of graded∼=algebras p∗ : H ∗ (IOn , Q)GL(n,Z) −→ H ∗ (IAn , Q)GL(n,Z) . Then we have the following

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

29

commutative diagram∗∗
HA(IOn , Q) ⊗ HA
(Fn , Q) ⊗ Q[y1 , y2 , . . .]ψA∼=

/ H ∗ (IAn , Q) ⊗ Q[y1 , y2 , . . .]
A

id ⊗ϕ

id ⊗ id ⊗φψA ⊗p∗∗∗/ H ∗ (IAn , Q) ⊗ H ∗ (IAn , Q)GL(n,Z)
HA(IOn , Q) ⊗ HA
(Fn , Q) ⊗ H ∗ (IOn , Q)GL(n,Z)
A
′
ωn
⊗id


H ∗ (IOn , Q) ⊗ H ∗ (Fn , Q)

ωn

ψ


/ H ∗ (IAn , Q),

where ϕ (resp. φ) is the canonical map that sends yi to yi ∈ H 4i (IAn , Q)GL(n,Z)(resp. yi ∈ H 4i (IOn , Q)GL(n,Z) ). Note that we have conjectured that the verticalmaps are all stable isomorphisms (see Proposition 6.7, Theorem 8.4 and Conjectures
7.4 and 8.5). By Conjecture 8.13 and the fact that ψA is an isomorphism, it isnatural to make the following conjecture as well.
Conjecture 8.14. The morphisms p∗ : H i (IOn , Q)GL(n,Z) → H i (IAn , Q)GL(n,Z)
are isomorphisms for n sufficiently large with respect to i.
Question 8.15. For coefficients in Z, are the morphisms p∗ : H i (IOn , Z)GL(n,Z) →H i (IAn , Z)GL(n,Z) always isomorphisms?
9. Conjectures about the stable cohomology of the Torelli groups
In this section, we discuss some conjectures about the stable rational cohomologyof the Torelli groups and the relation to the stable rational cohomology of the IAautomorphism groups of free groups.
9.1. Stable cohomology of the mapping class groups and the symplectic
groups. Let Σg (resp. Σg,1 ) denote a compact oriented surface of genus g (resp.
with one boundary component). Let Mg (resp. Mg,1 ) denote the mapping classgroup of Σg (resp. Σg,1 ). Let Ig (resp. Ig,1 ) denote the Torelli group, which isdefined as the kernel of the canonical surjective homomorphism from Mg (resp.
Mg,1 ) to the symplectic group Sp(2g, Z). Then we have exact sequences(9.1.1)

1 → Ig → Mg → Sp(2g, Z) → 1,

1 → Ig,1 → Mg,1 → Sp(2g, Z) → 1.

Then Sp(2g, Z) acts on the cohomology of the Torelli groups. See e.g. [43] for more
precise definitions.
Madsen and Weiss [38] proved the Mumford conjecture [45], which determines
the stable rational cohomology of the mapping class groups.
Theorem 9.1 (Madsen–Weiss [38]). We stably have an algebra isomorphism
H ∗ (Mg , Q) ∼= H ∗ (Mg,1 , Q) ∼= Q[e1 , e2 , . . .],where ei is the Mumford–Morita–Miller class of degree 2i [45, 41, 40].
Borel’s stability and vanishing theorem can also be applied to Sp(2g, Z). In thecase of Sp(2g, Z), the stable range given by Li and Sun [35] is equal to Borel’s stable
range that Borel remarked in [3], which was computed by Tshishiku [52].

30

KAZUO HABIRO AND MAI KATADA

Theorem 9.2 (Borel [2, 3], Li–Sun [35], Tshishiku [52]). (1) We have an algebra
isomorphism
H ∗ (Sp(2g, Z), Q) ∼= Q[u1 , u2 , . . .], deg ui = 4i − 2
in ∗ ≤ g − 1.
(2) Let V be an algebraic Sp(2g, Q)-representation such that V Sp(2g,Q) = 0. Then
we haveH p (Sp(2g, Z), V ) = 0 for p ≤ g − 1.
Here we recall some relations between the stable rational cohomology of Sp(2g, Z)
and that of Mg described in the introduction of Morita’s paper [42]. The surjective
map p : Mg → Sp(2g, Z) induces an injective map(9.1.2)p∗ : lim H ∗ (Sp(2g, Z), Q) ,→ lim H ∗ (Mg , Q).
←−
←−
g

∗

g

∗

The image of limg H (Sp(2g, Z), Q) under p is Q[e1 , e3 , . . .]. For i ≥ 1 and g suffi←−
ciently large with respect to i, let e′2i−1 ∈ H 4i−2 (Sp(2g, Z), Q) denote the element
such that p∗ (e′2i−1 ) = e2i−1 .
9.2. Sp(2g, Z)-invariant part of the stable rational cohomology of the Torelligroups. Kawazumi and Morita made the following conjecture about a stable structure of the Sp(2g, Z)-invariant part of the stable rational cohomology of Ig [29,Conjecture 13.8] (see also [43, Conjecture 3.4]).
Conjecture 9.3 (Kawazumi–Morita [29]). We stably have
∼ Q[e2 , e4 , . . . ],H ∗ (Ig , Q)Sp(2g,Z) =where e2i ∈ H 4i (Ig , Q)Sp(2g,Z) is the image of e2i under the map induced by the
inclusion Ig ,→ Mg .
It is natural to make the following conjecture, which is a variant of Conjecture
9.3.
Conjecture 9.4. The Sp(2g, Z)-invariant part of the rational cohomology of Ig,1
stabilizes, that is, for each i ≥ 0, there is an integer N ≥ 0 such that for all g ≥ N
the inclusion map Ig,1 ,→ Ig+1,1 induces an isomorphism∼=

H i (Ig+1,1 , Q)Sp(2g+2,Z) → H i (Ig,1 , Q)Sp(2g,Z) .
Moreover, we stably have an algebra isomorphism
H ∗ (Ig,1 , Q)Sp(2g,Z) ∼= Q[e2 , e4 , . . . ],where e2i ∈ H 4i (Ig,1 , Q)Sp(2g,Z) is the image of e2i under the map induced by the
inclusion Ig,1 ,→ Mg,1 .
For the second cohomology, we have the following Torelli group version of Corollary 4.7. This result in some stable range has probably been known to some experts
regarding Conjecture 9.3, but we include it here because we could not find such a
result in the literature.
Proposition 9.5. For g ≥ 4, we have
H 2 (Ig,1 , Q)Sp(2g,Z) = H 2 (Ig , Q)Sp(2g,Z) = 0.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

31

Proof. Associated to the exact sequences (9.1.1), we have the Hochschild–Serre
spectral sequences
E2p,q = H p (Sp(2g, Z), H q (Ig , Q)) ⇒ H p+q (Mg , Q),
E2p,q = H p (Sp(2g, Z), H q (Ig,1 , Q)) ⇒ H p+q (Mg,1 , Q),
respectively. Since we have E22,0 = Q, E21,1 = E23,0 = E22,1 = 0 by Theorem 9.2, wehave E20,2 = 0 by Theorem 9.1. The stable range of the cohomology of Mg and
Mg,1 are given by Harer [20] (see also [12, Theorem 5.8]) and the stable range ofthe cohomology of Sp(2g, Z) is given by Tshishiku [52] (see Theorem 9.2).
□
Similarly to Definition 3.4 for GL(n, Z)-representations, we make the following
definition.
Definition 9.6. By a stable family of Sp(2g, Z)-representations, we mean a familyW∗ = {Wg }g≥0 of Sp(2g, Z)-representations Wg such that there are finitely many
partitions λi (possibly with repetitions) such that, for sufficiently large g, we havean Sp(2g, Z)-module isomorphismM Sp
Vλi (g),Wg ∼=iwhere for a partition λ (with l(λ) ≤ g), VλSp (g) denotes the irreducible Sp(2g, Z)representation corresponding to λ.
The following conjecture would follow from [6, Conjectures 6.1, 6.6 and 6.7], andis analogous to Conjecture 6.3.
Conjecture 9.7 (Church–Farb [6]). For each i ≥ 0, the following hypothesis (SFIi )holds.
(SFIi ) {H i (Ig,1 , Q)}g is a stable family of Sp(2g, Z)-representations.
We have (SFIi ) for i = 0, 1. The cases i ≥ 2 are not known.
We will prove the following theorem which is analogous to Theorem 6.4. Morita[43] gave a related argument.
Theorem 9.8. Conjecture 9.7 implies Conjecture 9.4. Similarly, the variant ofConjecture 9.7 for {H i (Ig , Q)}g implies Conjecture 9.3.
Proof. We consider only the case of Ig,1 ; the case of Ig can be proved in the sameway.
We have the Hochschild–Serre spectral sequence
E2p,q (g) = H p (Sp(2g, Z), H q (Ig,1 , Q)) ⇒ H p+q (Mg,1 , Q)
associated to the exact sequence (9.1.1).
The map induced by the inclusion Ig,1 ,→ Mg,1 can be decomposed into thefollowing maps
0,4j
ϕ : H 4j (Mg,1 , Q) ↠ E∞(g) ,→ E20,4j (g) = H 4j (Ig,1 , Q)Sp(2g,Z) .

Let e′2i = ϕ(e2i ) ∈ H 4j (Ig,1 , Q)Sp(2g,Z) .

32

KAZUO HABIRO AND MAI KATADALet Q[ee1 , ee2 , . . .] be a bigraded polynomial algebra, where deg ee2i−1 = (4i − 2, 0)
and deg ee2i = (0, 4i) for each i ≥ 1. Consider the following hypothesis(Pm,n ) There is gm,n ≥ 0 such that for all g ≥ gm,n , the algebra map
fm,n : Q[ee1 , ee3 , . . . , ee2[ m+2 ]−1 , ee2 , ee4 , . . . , ee2[ n4 ] ] → E2∗,∗ (g)4which maps each eei to e′i , is an isomorphism in bidegree (p, q) with 0 ≤ p ≤m, 0 ≤ q ≤ n, and we have isomorphisms
E p,q (g) ∼= E p,q (g)2∞

for any 0 ≤ p ≤ m, 0 ≤ q ≤ n.
p,qNote that we have E2p,q (g) ∼(g) if and only if the differentials to and from= E∞
p,q
Er are zero for all r ≥ 2.

We have (Pm,0 ) for any m ≥ 0 since we have p∗ (e′2i−1 ) = e2i−1 for any i ≥ 1 andwe have limg p∗ (H ∗ (Sp(2g, Z), Q)) = Q[e1 , e3 , . . .], where p∗ is the injective map in←−(9.1.2) with Mg replaced by Mg,1 .
We prove (P0,n ) by induction on n. For n = 0, we have already seen that (P0,0 )holds. Let n ≥ 0 and suppose that (P0,n ) holds. We will first show that (Pm,n )holds for any m ≥ 0. Let 0 ≤ p ≤ m, 0 ≤ q ≤ n and g sufficiently large with respect
to m and n. By the assumption (SFIq ) and Theorem 9.2, we havep,00,q
E2p,q (g) ∼= E2 (g) ⊗ E2 (g),and thus fm,n in bidegree (p, q) can be identified with fm,0 ⊗f0,n : Q[ee1 , ee3 , . . . ]p,0 ⊗
p,0
0,q
0,q
Q[e
e2 , ee4 , . . . ] → E2 (g) ⊗ E2 (g), which is an isomorphism by the hypotheses(Pm,0 ) and (P0,n ). Moreover, by (Pm,0 ) and (P0,n ), we have
p,0
0,q
p,0
0,q
E2p,q (g) ∼(g) ⊗ E∞(g).
= E2 (g) ⊗ E2 (g) ∼= E∞p,q(g) is a subquoBy the definition of the Hochschild–Serre spectral sequence, E∞p,q∗tient of E2 (g). Since limg H (Mg,1 , Q) has no relation for the generators ei by←−∗,∗(g) is compatible with the
Theorem 9.1, and since the multiplication map of E∞∗
multiplication map of H (Mg,1 , Q), it follows that if g is large enough, then the bi∗,∗
graded algebra E∞(g) has no relation in bidegree (p, q) with 0 ≤ p ≤ m, 0 ≤ q ≤ n,p,qp,q
and thus we have E∞(g) ∼= E2 (g). Therefore, (Pm,n ) holds.

We will next show that (P0,n+1 ) holds. Since the Hochschild–Serre spectral
sequence is first-quadrant, the differentials to Er0,n+1 (g) are zero. The differentials
from Er0,n+1 (g) are also zero since (Pn+2,n ) holds. Therefore, we have E20,n+1 (g) ∼
=
0,n+1
p,q
E∞
(g). By the hypothesis (Pn+2,n ), we have E∞(g) ∼e1 , ee2 , . . .]p,q for p+q =
= Q[e
0,n+1
n + 1, 1 ≤ p ≤ n + 1. By the argument in the above paragraph, E∞(g) includes
0,n+1
′
′
. Therefore, by using Theorem 9.1, we haveQ[e2 , . . . , e2[ n ] ]4(
1 (n + 1 ≡ 0 (mod 4))
0,n+1
′
′
0,n+1
dim E2(g) − dim Q[e2 , . . . , e2[ n ] ]=40 (otherwise).
0,n+1
Since E20,n+1 (g) ∼(g), the map ϕ : H n+1 (Mg,1 , Q) → H n+1 (Ig,1 , Q)Sp(2g,Z)= E∞
is surjective. Therefore, by dimension counting, we have e′n+1 ̸= 0 if n + 1 ≡ 0
2(mod 4) and thus the algebra map f0,n+1 is an isomorphism in bidegree (0, q) forq ≤ n + 1. This implies (P0,n+1 ).

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

33

We now have (P0,n ) for all n ≥ 0. Thus we stably have an algebra isomorphism
H ∗ (Ig,1 , Q)Sp(2g,Z) ∼= Q[e′2 , e′4 , . . . ].
□Remark 9.9. Similarly to the case of IAn , we can replace the assumption, Conjecture
9.7, with a weaker assumption that {H p (Ig,1 , Q)}g is a stably algebraic family ofSp(2g, Z)-representations.
The Dehn–Nielsen–Baer embedding maps the mapping class group Mg,1 into
Aut(F2g ) and restricts to an embedding ig,1 : Ig,1 → IA2g . Then Conjectures 6.1
and 9.4 imply the following conjecture.
Conjecture 9.10. The graded algebra homomorphismi∗g,1 : H ∗ (IA2g , Q)GL(2g,Z) → H ∗ (Ig,1 , Q)Sp(2g,Z)
induced by ig,1 is a stable isomorphism.
Remark 9.11. In Section 6.4, we defined yk ∈ H 4k (IA2g , Q), which is expected to0
be a nonzero scalar multiple of T β2k+1∈ H 4k (IA2g , R). Morita, Sakasai and Suzuki∗
0[44] proved that ig,1 (T β2k+1 ) is a nonzero scalar multiple of e2i ∈ H 4i (Ig,1 , Q)Sp(2g,Z) .
Therefore, Conjecture 9.10 may follow from Proposition 6.7 and Conjecture 9.4.
9.3. The stable rational cohomology of the Torelli groups. Here we makea conjecture about the relation between the stable rational cohomology and thestable Albanese cohomology of the Torelli groups Ig and Ig,1 .
The stable Albanese cohomology of the Torelli groups of degree ≤ 3 has been
determined by Johnson [24], Hain [18], Sakasai [48] and Kupers–Randal-Williams[33]. Lindell [36] detected subquotient Sp(2g, Z)-representations of the higher degree stable Albanese cohomology of the Torelli groups.
By [19, 34] it follows that the stable Albanese cohomology of the Torelli groups
is a quotient of the cohomology of the associated graded Lie algebra of the Torelligroups with respect to the lower central series. The latter cohomology has beenstudied by Hain [18], Garoufalidis–Getzler [15] and Kupers–Randal-Williams [34].
Recently, the second-named author [27] proposed conjectural structures of thestable Albanese cohomology of the Torelli groups in a way similar to the Albanese
cohomology of IAn , which is true in degree ≤ 3 by [18, 48, 33]. In particular, shemade the following conjecture about the Sp(2g, Z)-invariant part.
Conjecture 9.12 ([27]). We stably have isomorphisms of graded algebras∗∗
HA(Ig , Q)Sp(2g,Z) ∼(Ig,1 , Q)Sp(2g,Z) ∼= HA= Q[z1 , z2 , . . . ],where deg zi = 4i.
By the definition of the Albanese cohomology, we have inclusion mapsiιig : HA
(Ig , Q) ,→ H i (Ig , Q),i
ιig,1 : HA
(Ig,1 , Q) ,→ H i (Ig,1 , Q),

which form graded algebra maps∗
ι g : HA
(Ig , Q) ,→ H ∗ (Ig , Q),∗ιg,1 : HA(Ig,1 , Q) ,→ H ∗ (Ig,1 , Q).

34

KAZUO HABIRO AND MAI KATADAThe maps ιig and ιig,1 are isomorphisms in degree i = 0, 1. It seems natural to
make the following conjecture, which says that the rational cohomology and the
Albanese cohomology of the Torelli groups stably coincide.
Conjecture 9.13. The graded algebra maps ιg and ιg,1 are stable isomorphisms.
References[1] Armand Borel. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes degroupes de Lie compacts. Ann. of Math. (2), 57:115–207, 1953.
[2] Armand Borel. Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup.
(4), 7:235–272 (1975), 1974.
[3] Armand Borel. Stable real cohomology of arithmetic groups. II. In Manifolds and Lie groups(Notre Dame, Ind., 1980), volume 14 of Progr. Math., pages 21–55. Birkhäuser, Boston,
Mass., 1981.
[4] Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics.
Springer-Verlag, New York-Berlin, 1982.
[5] Thomas Church, Jordan S. Ellenberg, and Benson Farb. FI-modules and stability for representations of symmetric groups. Duke Math. J., 164(9):1833–1910, 2015.
[6] Thomas Church and Benson Farb. Representation theory and homological stability. Adv.
Math., 245:250–314, 2013.
[7] F. Cohen and J. Pakianathan. On automorphism groups of free groups, and their nilpotent
quotients. in preparation.
[8] Matthew Day and Andrew Putman. On the second homology group of the Torelli subgroupof Aut(Fn ). Geom. Topol., 21(5):2851–2896, 2017.
[9] Aurélien Djament. Décomposition de Hodge pour l’homologie stable des groupes
d’automorphismes des groupes libres. Compos. Math., 155(9):1794–1844, 2019.
[10] Aurélien Djament and Christine Vespa. Sur l’homologie des groupes d’automorphismes desgroupes libres à coefficients polynomiaux. Comment. Math. Helv., 90(1):33–58, 2015.
[11] Benson Farb. Automorphisms of Fn which act trivially on homology. in preparation.
[12] Benson Farb and Dan Margalit. A primer on mapping class groups. Princeton Math. Ser.,49, Princeton University Press, Princeton, NJ, 2012.
[13] William Fulton and Joe Harris. Representation theory. A first course. Reading in Mathematics, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991.
[14] Søren Galatius. Stable homology of automorphism groups of free groups. Ann. of Math. (2),
173(2):705–768, 2011.
[15] Stavros Garoufalidis and Ezra Getzler. Graph complexes and the symplectic character of theTorelli group. arXiv preprint arXiv:1712.03606, 2017.
[16] Kazuo Habiro and Mai Katada. On Borel’s stable range of the twisted cohomology ofGL(n, Z). arXiv preprint arXiv:2212.09074, 2022, to appear in Osaka J. Math.
[17] Kazuo Habiro and Gwénaël Massuyeau. The Kontsevich integral for bottom tangles in handlebodies. Quantum Topol., 12(4):593–703, 2021.
[18] Richard Hain. Infinitesimal presentations of the Torelli groups. J. Amer. Math. Soc.,
10(3):597–651, 1997.
[19] Richard Hain. Johnson homomorphisms. EMS Surv. Math. Sci., 7(1):33–116, 2020.
[20] John Harer. The second homology group of the mapping class group of an orientable surface.
Invent. Math., 72(2):221–239, 1983.
[21] Allen Hatcher and Karen Vogtmann. Cerf theory for graphs. J. London Math. Soc. (2),58(3):633–655, 1998.
[22] Allen Hatcher and Karen Vogtmann. Rational homology of Aut(Fn ). Math. Res. Lett.,
5(6):759–780, 1998.
[23] Kiyoshi Igusa. Higher Franz-Reidemeister torsion. AMS/IP Studies in Advanced Mathematics, 31, American Mathematical Society, Providence, RI; International Press, Somerville, MA,
2002.
[24] Dennis Johnson. The structure of the Torelli group. III. The abelianization of T . Topology,
24(2):127–144, 1985.
[25] Mai Katada. Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I.
Ann. Inst. Fourier (Grenoble), 73(4):1489–1532, 2023.

ON THE STABLE COHOMOLOGY OF (IA-)AUTOMORPHISM GROUPS

35[26] Mai Katada. Actions of automorphism groups of free groups on spaces of Jacobi diagrams.
II. J. Inst. Math. Jussieu, published online, 2022.
[27] Mai Katada. Stable rational homology of the IA-automorphism groups of free groups. arXiv
preprint arXiv:2207.00920, 2022.
[28] Nariya Kawazumi. Cohomological aspects of Magnus expansions. arXiv preprint
math/0505497, 2005.
[29] Nariya Kawazumi and Shigeyuki Morita. The primary approximation to the cohomology ofthe moduli space of curves and cocycles for the Mumford-Morita-Miller classes. preprint,
1996.
[30] Nariya Kawazumi and Christine Vespa. On the wheeled PROP of stable cohomology ofAut(Fn ) with bivariant coefficients. arXiv preprint arXiv:2105.14497, 2021, to appear in
Algebr. Geom. Topol.
[31] Kazuhiko Koike. On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters. Adv. Math., 74(1):57–86, 1989.
[32] Alexander Kupers, Jeremy Miller, and Peter Patzt. Improved homological stability for certain
general linear groups. Proceedings of the London Mathematical Society, 125(3):511–542, 2022.
[33] Alexander Kupers and Oscar Randal-Williams. On the cohomology of Torelli groups. Forum
Math. Pi, 8:e7, 83, 2020.
[34] Alexander Kupers and Oscar Randal-Williams. On the Torelli Lie algebra. arXiv preprint
arXiv:2106.16010, 2021.
[35] Jian-Shu Li and Binyong Sun. Low degree cohomologies of congruence groups. Sci. China
Math., 62(11):2287–2308, 2019.
[36] Erik Lindell. Abelian cycles in the homology of the Torelli group. J. Inst. Math. Jussieu,
published online, 2021.
[37] Erik Lindell. Stable cohomology of Aut(Fn ) with bivariant twisted coefficients. arXiv preprint
arXiv:2212.11075, 2022.
[38] Ib Madsen and Michael Weiss. The stable moduli space of Riemann surfaces: Mumford’s
conjecture. Ann. of Math. (2), 165(3):843–941, 2007.
[39] John McCleary. A user’s guide to spectral sequences, volume 58 of Cambridge Studies inAdvanced Mathematics. Cambridge University Press, Cambridge, second edition, 2001.
[40] Edward Y. Miller. The homology of the mapping class group. J. Differential Geom., 24(1):1–
14, 1986.
[41] Shigeyuki Morita. Characteristic classes of surface bundles. Invent. Math., 90(3):551–577,
1987.
[42] Shigeyuki Morita. A linear representation of the mapping class group of orientable surfaces
and characteristic classes of surface bundles. In Topology and Teichmüller spaces (Katinkulta,
1995), pages 159–186. World Sci. Publ., River Edge, NJ, 1996.
[43] Shigeyuki Morita. Structure of the mapping class groups of surfaces: a survey and a prospect.
In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr.,
pages 349–406. Geom. Topol. Publ., Coventry, 1999.
[44] Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki. Secondary characteristic classes for
subgroups of automorphism groups of free groups. arXiv preprint arXiv:1512.06365, 2015.
[45] David Mumford. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and geometry, Vol. II, volume 36 of Progr. Math., pages 271–328. Birkhäuser Boston,
Boston, MA, 1983.
[46] Alexandra Pettet. The Johnson homomorphism and the second cohomology of IAn . Algebr.
Geom. Topol., 5:725–740, 2005.
[47] Oscar Randal-Williams. Cohomology of automorphism groups of free groups with twisted
coefficients. Selecta Math. (N.S.), 24(2):1453–1478, 2018.
[48] Takuya Sakasai. The Johnson homomorphism and the third rational cohomology group of
the Torelli group. Topology Appl., 148(1-3):83–111, 2005.
[49] Takao Satoh. Twisted first homology groups of the automorphism group of a free group. J.
Pure Appl. Algebra, 204(2):334–348, 2006.
[50] Takao Satoh. Twisted second homology groups of the automorphism group of a free group.
J. Pure Appl. Algebra, 211(2):547–565, 2007.
[51] Takao Satoh. First cohomologies and the Johnson homomorphisms of the automorphismgroup of a free group. J. Pure Appl. Algebra, 217(1):137–152, 2013.

36

KAZUO HABIRO AND MAI KATADA[52] Bena Tshishiku. Borel’s stable range for the cohomology of arithmetic groups. J. Lie Theory,29(4):1093–1102, 2019.
[53] Christine Vespa. Extensions between functors from free groups. Bull. Lond. Math. Soc.,
50(3):401–419, 2018.
[54] Christine Vespa. On the functors associated with beaded open Jacobi diagrams. arXiv
preprint arXiv:2202.10907, 2022.
[55] E. C. Zeeman. A proof of the comparison theorem for spectral sequences. Proc. Cambridge
Philos. Soc., 53:57–62, 1957.