June, 2013

Galois obstructions in Johnson cokernel

Naotake Takao (RIMS)
Johnson homomorphisms
Lecture Hall on the ground ﬂoor of the Mathematical Science Building, the University of Tokyoabstract
The purpose of this survey lecture is to introduce a number-theoretic approach to study
obstructions to the surjectivity of the Johnson homomorphism. In particular, theobstruction arising from the outer Galois action on the pro-` fundamental group of the
projective line minus three points over the rationals will be focused on. This obstruction
has been studied by Mamoru Asada, Makoto Matsumoto, Hiroaki Nakamura, Takayuki
Oda,and the lecturer. A recent celebrated theorem on the action of the Tannakian
fundamental group of the category of mixed Tate motives over the rational integers on the
motivic fundamental group of projective line minus three points by Francis Brown made it
possible to determine the “size” of this obstruction.
Acknowledgement
I would like to thank the organisers for giving me the opportunity of my lecture.

Contents
1.Background
1.1.Quick review of some results on the Johnson cokernel
1.2.pro-` universal monodromy representation
2.Main results
2.1.Oda conjecture
2.2.Deligne-Ihara Conjecture
3.A sketch of the proofs
4.Further interest
3.1. On the image of pro-` universal monodromy representaion
3.2. Towards to the integral Oda conjecture1

Notations(g, n) : a pair of non-negative integer such that 2g − 2 + n > 0
Πg,n := the fundamental group of a (hyperbolic) Riemann surface of type (g, n)
Mg,n : the mapping class group of type (g, n)
Πg,n [m] : the weight ﬁltration of Πg,n deﬁned by
Πg,n [1] = Πg,n ,Πg,n [2] = [Πg,n , Πg,n ] · h all inertia subgroups i
Πg,n [k] = h[Πg,n [k 0 ], Πg,n [k 00 ]] k 0 + k 00 = ki
Depending on the situation, g, n is often omitted .
Lk := grk Π = Π[k]/Π[k + 1] (Π = Πg,n )L := Gr Π = ⊕k≥1 Lk

2(k ≥ 3)

hk := {D ∈ Der(L)|D(Ld ) ⊂ Ld+k for any d ≥ 1,
D preserves each inertia Lie ideals}/InnLk (L)h = Derc (L)/Inn(L) := ⊕k≥1 hk
ϕ : M → Out+ (Π) : the Dehn-Nielsen map (isomorphism)
M[k] := Ker(M → Out+ (Π/Π[m + 1]))
grk M := M[k]/M[k + 1]
τk : grk M → hk : the k-th integral Johnson homomorphism in this lectureτk ⊗Z Q : grk M ⊗Z Q → hk ⊗Z Q : the k-th Johnson homomorphism in this lecture
Coker(τk ⊗Z Q) : the Johnson cokernel in this lecture3Mg,n : the moduli stack over Q of proper smooth geometrically connected curves of genus
g with disjoint ordered n sections (often refered as (g, n)-curve in this lecture)
GQ = Gal(Q/Q) : the absolute Galois group of rationals
π1 (X) : the étale fundamental group of a connected scheme/stack X` : a primeΠpro−` :the pro-` completion of Π =

lim
←−

Π/N

ΠBN,

(Π:N )=`-th power

This is often called the pro-` (geometric) fundamental group,
which is isomorphic to the maximal pro-` quotient of the étale fundamental group of acurve X over an algebraically closed ﬁeld of characteristic 0, whose kernel is acharacteristic subgroup of π1 (X).
Πpro−` [k] : the topological closure of Π[k]Note that grk (Πpro−` ) ' Lk ⊗Z Z` .
In what follows, depending on the situation, ` is omitted.
4

1 Background
1.1 Quick review of some results on the Johnson cokernel
A list of several (NOT ALL!) results on Coker(τk ⊗Z Q) and Coker(τk ⊗Z Q` ).
0.[Nakamura-Tsunogai] The Sp(2g, Q` ) × Sn -module structure of hk ⊗Z Q` is“computable”.
1.[Johnson, Morita] 1 ≤ k ≤ 2 ⇒ τk ⊗Z Q is isomorphism.
2.For 1 ≤ k ≤ 6, the Sp2g × Sn -module structure of grk Mg,n ⊗Z Q is as follows:[n]1[Johnson, Asada-Nakamura] gr Mg,n ⊗Z Q ' [1, 1, 1]Sp ⊗ 1 + [1]Sp ⊗ 1
[n]
[n]
2
[Morita, Asada-Nakamura] gr Mg,n ⊗Z Q ' [2, 2]Sp ⊗ 1 + [1, 1]Sp ⊗ 1 + [0]Sp ⊗ 2
[n]
[ ]
3
2 n
[Asada-Nakamura] gr Mg,n ⊗Z Q ' [3, 1, 1]Sp ⊗ 1 + [2, 1]Sp ⊗ 1 + [1]Sp ⊗ ∧ 1[Morita]
gr4 Mg,1 ⊗Z Q ' [4, 2]Sp + [3, 1, 1, 1]Sp + [2, 2, 2]Sp + 2[3, 1]Sp + 2[2, 1, 1]Sp + 2[2]Sp[Morita-Sakasai-Suzuki] gr5 Mg,1 ⊗Z Q ' · · · (sorry for not making on time) (g >> k)[Morita-Sakasai-Suzuki] gr6 Mg,1 ⊗Z Q ' · · · (sorry for not making on time) (g >> k)
5

In particular, together with Nakamura-Tsunogai’s results, when k = 3, 4, 5, 6, τk is notsurjective.
3.[Morita] For g ≥ 2, k ≡ 1 (mod 2),
in Coker(τ(g,1) k ⊗Z Q), [k]Sp appears with multiplicity 1.
4.[Enomoto-Sato] For g >> k, k ≡ 1 (mod 4),
in Coker(τ(g,1) k ⊗Z Q), [1k ]Sp appears with multiplicity 1.
5.[Conant-Kassabov-Voghtmann]
A generalization of the Morita’s traceinﬁnite obstructions in the h ⊗Z Q/[h ⊗Z Q, h ⊗Z Q]
6.[Oda, Ihara, Matsumoto, Nakamura, Ueno, T + Hain-Matsumoto, Brown] For
2g − 2 + n > 0, k ≡ 0 (mod 2), k 6= 2, 4, 8, 12, in Coker(τ(g,n) k ⊗Z Q` ), [0]Sp appearskwith multiplicity ≥ mk + 1.3 2 (k >> 1).

6

Conclusively, (roughly speaking,) when k is odd and n ≥ 1, then many kinds of
obstructions to surjectivity of τ(g,n) k exist.
but whenk is even or n = 0,
the lecturer knows little results. So I think that the results 6. is remarkable in this sense.
From now on, I introduce this result in this lecture.

7

1.2 pro-` universal monodromy representation
Roughly speaking, what is it?
The pro-` universal monodromy representation ϕarith : π1 (M) → Outc (Πpro−` ) can be
regarded as an arithmetic version of the Dehn-Nilsen map:

ϕ : M ' Out+ (Π),ϕarith : π1 (M) → Outc (Πpro−` ).
Outc (Πpro−` ) := {f ∈ Aut(Πpro−` )|f preserves each inertia subgroup }/Inn(Πpro−` )The Johnson homomorphism τk : grk M → hk is a Lie algebra version of Dehn-Nielsen
map ϕ : M → Out+ (Π).
τk ⊗Z Z` : grk M ⊗Z Z`↓
τkarith : grk π1 (M)

,→
ª
,→

hk ⊗Z Z`↓hk ⊗Z Z`(the left vertical arrow is injective and the right vertical arrow is the identity)8Formulation of ϕarith
Mg,n+1 → Mg,n : the universal family of (g, n)-curvex̄ → Mg,n : a geometric point, X̄ := Mg,n+1 ×Mg,n x̄Ã
1 → π1 (X̄) → π1 (Mg,n+1 ) → π1 (Mg,n ) → 1,Ã1→

π1 (X̄)→↓ª
1 → Inn(π1 (X̄)) →↓ª
1 → Inn(Πpro−` ) →π1 (Mg,n+1 )↓Aut(π1 (X̄))↓
Aut(Πpro−` )→π1 (Mg,n )ª↓→ Out(π1 (X̄))ª↓→ Out(Πpro−` )in which all rows are exact. And we have another exact sequencepg,n

1 → π1 (Mg,n ⊗ Q) → π1 (Mg,n ) → GQ → 1,
so we have
9

→1
→1
→1

Mg,n

→

\
M
g,n

1
↓
'

π1 (Mg,n ⊗ Q)↓
ϕarith:π1 (Mg,n )
g,n
↓
GQ
↓
1.

→

Outc (Πpro−`)g,n(ϕarithis called the pro-` universal monodromy representation.)
g,n
pro−`
π1 (Mg,n )[k] := Ker(π1 (Mg,n ) → OutC (Πpro−`/Π[k + 1])g,ng,npg,n (π1 (Mg,n )[k])
(k)
:=
Q̄
Q(`)
g,n
(`)
(`)
Gg,n
(k) := Gal(Q(`)
g,n (k + 1)/Qg,n (k))
(`)
(`)
Gg,n
:= ⊕k≥1 Gg,n(k);so we have10τk ⊗Z Z` :grk Mg,n ⊗Z Z`

1↓,→
grk π1 (Mg,n ⊗Q Q)↓
τkarith :grk π1 (Mg,n )↓(`)
Gg,n (k)↓1,→ hk ⊗Z Z`

Deﬁnition 1.1 (Galois obstruction) For each k ≥ 1, then we have(`)
Coker(τ(g,n) k ⊗Z Q` ) ←- Gg,n(k) ⊗Z` Q` .
(`)

Each element of Gg,n (k) can be regarded as a obstruction to the surjectivity of theJohnson homomorphism, which is called a Galois obstruction in this lecture.

11This obstruction appears as[0]Sp when k ∈ 2Z \ {2, 4, 8, 12} even if n = 0,
which is a central object in this lecture.

12

2 Main results
2.1 Oda conjecture
Theorem 2.1 (Ihara, Matsumoto, Nakamura, Takao)(`)(1) {Qg,n (k)}k≥1 is independent of nand almost independent of g and n in the following sense :
(`)

(`)

(`)

(`)

(`)

Q1,1 (k) ⊃ Qg,n (k) ⊃ Q0,3 (k),[Q1,1 (k) : Q0,3 (k)] < ∞.
(`)

(`)

(`)

(2) Qg,n is independent of g and n, where Qg,n = ∪k≥1 Qg,n (k).
In particular,
(`)

(`)
Gg,n
⊗Z` Q` ' G0,3 ⊗Z` Q` .

13

Corollary 2.2 (Galois obstruction) (If 2g − 2 + n > 0, then for k ≥ 1)(`)

Coker(τ(g,n) k ⊗Z Q` ) ←- G0,3 (k) ⊗Z` Q` .
(`)

In particular, writing rk := dimQ` G0,3 (k) ⊗Z` Q` ,dimQ` Coker(τ(g,n) k ⊗Z Q` ) ≥ rk .

14

2.2 Deligne-Ihara Conjecture(`)

Remark 2.3 (As M0,3 = {0},) G0,3 (k) is a free Z` -module of ﬁnite rank rk , so(`)

h0,3 ⊗Z Z` ←- G0,3 .
And G0,3 (k) ' Z` ( k2 )⊕rk as a Gal(Q(µ`∞ )/Q) ' Z×
` -module when k is even.
(`)

(`)

(`)

On the contrary G0,3 (k) = {0}, as h0,3 ⊗Z Z` = {0} when k is odd.
Theorem 2.4 (Soulé-Ihara) k ≡ 2 (mod 4) and k ≥ 6⇒∃ natural nontrivial characterk(`)κk : G0,3 (k) → Z` ( )
2

15

(`)

(`)

Take a σk ∈ G0,3 (k) such that G0,3 (k) = Ker(κk ) + Z` σk (k ≡ 2 (mod 4) and k ≥ 6)
Theorem 2.5 (Hain-Matsumoto, Brown)(`)
G0,3 ⊗Z` Q` is freely generated by σk (k ≡ 2 (mod 4) and k ≥ 6) as a graded Liealgebra/Q` .
Especially,
Corollary 2.6( 0) ∑
3∑1k
r2k0 = 0µ( (αid − 1 − (−1)d )),
k
d i=1
0
d|k

where αi (1 ≤ i ≤ 3) are the roots of x3 − x − 1.
In particular, if k 6= 2, 4, 8, 12 and k is even,then τ(g,n) k ⊗Z Q` is not surjective.

16

the calculated value of rk for small k;
k
rk

1
0

2
0

k
rk

11
0

k
rk
k
rk

3
0

4
0

5
0

6
1

12
0

13
0

14
1

21
0

22
2

23
0

31
0

32
5

33
0

7
0

8
0

9
0

10
1

15
0

16
1

17
0

18
1

19
0

20
1

24
2

25
0

26
3

27
0

28
3

29
0

30
4

34
7

35
0

36
7

37
0

38
10

39
0

40
12

···············
we have r2k + 1.3k for k is enough large172.3 A sketch of the proof
Oda conjecture
The proof consists of four parts:
Step1 [Nakamura-Takao-Ueno] Qg,n (k) is independent of n (n > 0).
Srep2 [Nakamura] Q1,1 (k) ⊃ Qg,n (k) ⊃ Q0,3 (k) (n > 0).
Step3 [Ihara-Nakamura]Qg,n = Q0,3 (n > 0),

Step4 [Takao] Qg,1 (k) = Qg,0 (k).

18

[Q1,1 (k) : Q0,3 (k)] < ∞.

Step1 [Nakamura-Ueno-Takao] Qg,n (k) is independent of n (n > 0).
A key idea is to consider “a braid version” of pro-` universal monodromy representation,that is, the pro-` monodromy representation associated to the universal family of the pureconﬁguration space of curves.
We write h(r) , Qg,n,r (k), · · · for the corresponding objects.
Then the proof can be reduced to the following three lemmas.
Lemma 2.7 Qg,n,r (k) ⊂ Qg,n+1,r (k).
Lemma 2.8 Qg,n,r (k) ⊂ Qg,n−1,r+1 (k).
Lemma 2.9 Qg,n,r (k) ⊂ Qg,n,r−1 (k).
The third lemma is shown by (tough) combinatorial Lie algebraic calculations in “braid Liealgebra”. A key theorem to show the last lemma is
Theorem 2.10 The Z` -Lie algebra homomorphismh(r+1) ⊗Z Z` −→ h(r) ⊗Z Z`
is injective.
19

Key principle of the proof is as follows:
Let D is the (canonical) lift of any element of the kernel of the above map.
Find A, B ∈ Lk+1 ⊗Z Z` such that [A, B] = 0 and D(B) = 0.
⇒ D(A) = 0.
; (omitted tough calculation) ;we can show that D = 0.

20

Step2 [Nakamura] Q1,1 (k) ⊃ Qg,n (k) ⊃ Q0,3 (k) (n > 0)
Key ingredients in this step are(1)Knudsen’s Clutching morphisms to relate (compactiﬁed) moduli stacks of diﬀerent type(2)Grothendieck-Murre theory to describe the fundamental group of the formal tubularneighborhood of the locus of a type of stable curves arising from the clutching, and tocombine the fundamental groups of moduli stacks of diﬀerent type(3)Nakamura’s invention to couple universal monodromy representations of diﬀerent types,
using Grothendieck-Murre sequences
Rough (and incorrect) sketch is as follows: Let M∗g,n be the Deligne-Mumford
compactiﬁcation of Mg,n . Taking, a boundary irreducible divisor D ' M∗g1 ,n1 × M∗g2 ,n2 inM∗g,n (g1 + g2 = g, n1 + n2 = n + 2), which is, roughly speaking, the image of the
clutching morphism. Then , by the theory of Grothendieck-Murre, we have1→Ẑ(1) →

π1 (]D[\D) →↓π1 (Mg,n )21

π1 (D)→↓
π1 (Mg1 ,n1 × Mg2 ,n2 ).

1

A Key theorem is
Theorem 2.11 (Coupling Theorem). There exists a closed subgroup N of π1 (]D[\D)which is (almost) surjectively mapped onto π1 (D), making the following diagram
commutes:
π1 (Mg1 ,n1 × Mg2 ,n2 )

→

c
pro−`
Outc (Πpro−`
g1 ,n1 ) × Out (Πg2 ,n2 )

%
↓

N
&
π1 (Mg,n )→Outc (Πpro−`).
g,n

A key idea of the proof is Nakamura’s invention to couple the universal monodromy
representation over Mg,n with the universal monodromy representation overMg1 ,n1 × Mg2 ,n2 . More precisely, we consider two another clutching morphisms:
c1 : M∗g1 ,n1 +1 × M∗g2 ,n2 → M∗g,n+1 ,c2 : M∗g1 ,n1 × M∗g2 ,n2 +1 → M∗g,n+1 .
Denoting the boundary component corresponding to c1 (resp.c2 ) by D1 (resp. D2 ), wehave
221 → Ẑ(1)→ π1 (]D1 [\D1 )↓π1 (Mg,n+1 )→π1 (D1 )→↓π1 (Mg1 ,n1 +1 × Mg2 ,n2 ),1

1 → Ẑ(1)→ π1 (]D2 [\D2 )↓π1 (Mg,n+1 )→π1 (D2 )→↓π1 (Mg1 ,n1 × Mg2 ,n2 +1 ),1andby Grothendieck-Murre theory.
By considering the formal tubular neighborhood ]D1 ∪ D2 [\D1 ∪ D2 , we get conclusion
after some group theoretical treatments.
Corollary 2.12 Qg1 ,n1 (k)Qg2 ,n2 (k) ⊃ Qg,n (k)Corollary 2.13 Q1,1 (k) ⊃ Qg,n (k) ⊃ Q0,3 (k) (n > 0)
Remark 2.14 Matsumoto also related g = 0 and g > 0 by another method.

23

Step3 [Ihara-Nakamura] - very rough sketch
Qg,n = Q0,3 (n > 0)
[Q1,1 (k) : Q0,3 (k)] < ∞ (n > 0)
Key ingredients in this step are(1)Tangential structure to construct a canonical universal deformation family of maximallydegenerate stable marked curves(2)Formal patching (Artin, Deligne, Harbarter) and Grothendieck-Murre theory to describe
the Galois action on the fundamental group of the formal tubular neighborhood of
maximally degenerate stable marked curves in the canonical deformation space(3)A certain weight argument to prove the latter statement24

A key theorem is
Theorem 2.15 If σ ∈ GQ acts trivially on the pro-` geometric fundamental groupoid of
P1 \ {0, 1, ∞} with respect to the set of Deligne’s tangential base points, then σ has anextension σ̃ ∈ GQ((q)) that acts trivially on the pro-` fundamental groupoid of the genericgeometric ﬁber of the family constructed in (1) with respect to the set of “tangential basepoints” which are associated with (ordinary double) singular points in the maximal
degenerated stable marked curve. In particular, the outer action of σ̃ on the pro-`fundamental group of the generic geometric ﬁber of the family constructed in (1) is trivial.
Very roughly speaking, the proof is done by “gluing Grothendieck-Murre sequence” using
formal patching. It can be shown that the pro-` fundamental group of the genericgeometric ﬁber of the family constructed in (1) is “almost” isomorphic to the free productof all the pro-` geometric fundamental groups of P1 \ {0, 1, ∞}.
A key idea of the proof of ﬁnite index part is to consider the nilpotency class. Then we
have Q1,1 (k) ⊂ Q0,3 (2k + 4). Using a certain weight argument, we get the conclusion.

25

Step4 [Takao] Qg,1 (k) = Qg,0 (k) (g ≥ 2) - very rough sketch
Enough to prove
Lemma 2.16 Qg,0,2 (k) ⊂ Qg,0,1 (k)A presentation of Lg,0,2 is as folllows:(j)generators
relations

Xi , Z(1 ≤ i ≤ 2g, 1 ≤ j ≤ 2),

g
∑

(j)

(j)

[Xi , Xi+g ] + Z = 0(1 ≤ j ≤ 2),

i=1(j)(j 0 )[Xi , Xi0 ] ={0(j 6= j 0 , i ≤ i0 and i0 6= i + g),Z(j 6= j 0 , i ≤ i0 and i0 = i + g).

In particular, we have
(1)

[Xi

(2)

+ Xi , Z] = 0(1 ≤ i ≤ 2g).

Now the outline is the same as in Step1.
But there is NOT! any generator A such that [A, Z] = 0.
(2)
(1)
We use Xi + Xi instead.
(1)(2)
C(Xi + Xi ) is not clear, but we can see
26

Lemma 2.17 For 1 ≤ i ≤ 2g,
(1)
(2)
(2)
C(Xi + Xi ) ∩ Gr N (2) = hXi , ZiLie .
Here, N (2) is the ﬁber subgroup obtained by forgetting the second component (i.e. thekernel of the natural projection of braid groups).
Key ingredients of the proof are· Elimination theorem for free Lie algebras· Degree discussion on free Lie algebras; (omitted tough calculation) ;we can show that D = 0.
Remark 2.18 Hoshi and Mochizuki proved the non-ﬁltered part of this step and step1 ina quite diﬀerent way from a quite diﬀerent motivation.

27

Deligne-Ihara conjecture - very very rough sketch· The Lie algebra of the pro-unipotent part of the motivic Galois group (=the Tannakian
fundamental group of the category of mixed Tate motives over Spec(Z)) is freely
generated by Soulè cyclotomic elements of the K-theory.
· The motivic Galois group acts on the motivic fundamental group of P1 \ {0, 1, ∞}· Brown proved that this action is faithful.
(`)· G0,3 is the the `-adic realization of the image of this action.
;
Theorem 2.5(Deligne-Ihara conjecture)28

By what kind of method did Brown prove the faithfulness?
In fact, Brown also proved that Hoﬀman conjecture on multiple zeta valuesζ(m1 , m2 , · · · , mr ) = Σ0<n1 ···<nr

1
mr
1
nm
·
·
·
n
r
1

(1 ≤ m1 , · · · , mr , 2 ≤ mr ).

Theorem 2.19 (Hoﬀman conjecture) The k-th graded piece of Q-graded algebra ofmultiple zeta values is generated by {ζ(m1 , m2 , · · · , mr )|mi = 2, 3, m1 + · · · + mr = k}
The proof was done by proving the motivic version of Hoﬀman conjecture.
The motivic version of Hoﬀman conjecture states that{ζ M (m1 , m2 , · · · , mr )|mi = 2, 3, m1 + · · · + mr = k} is a basis of the k-th graded pieceof the algebra of motivic multiple zeta values, which leads to faithfulness of the action.

29

3 Further interest
3.1 On the image of pro-` universal monodromy representaion
Im(τk ⊗Z Q` ) ∩ Gk ⊗Z` Q` = {0}.
But ϕarith (M) ⊂ φX (Gk ) if X/k is `-monodromically full deﬁned by Hoshi. HereφX : Gk → Out(Πpro−` ) is the pro-` outer Galois representation associated to X/k.
Matsumoto and Tamagawa proved that many such curves exist.
So the image of any mapping classes can be described as the image of some element of
Galois group. For example, How can Dehn twist be described ?
To the contrary, it seems interesting to characterize the image of mapping class group in
the image of Galois group. For example, Can some Forbenius element be described by
Mapping class?

30

3.2 Towards to the integral Oda conjecture
Finally speaking, the lecturer is now interested in the “integral Oda problem” and “integral
Deigne-Ihara problem”, which is, roughly speaking, the problem to investigate the(`)
structure of the associated graded Lie algebra Gg,n over Z` . That is because this structure
may has some arithmetic information. In fact, for example, this structure depends on `.
Conclusively, to solve the integral Oda problem, we must analyse “the integral Johnson(`)
homomorphism”. For example, if Coker(τ(g,n) k ) ⊗Z Z` is torsion-free, then Gg,n (k) is
torsion-free. So the lecturer is also interested in Integral Johnson homomorphism.

The lecture is end. Thank you for listening.

31