Low Dimensional Topology and
Symplectic Representations
Shigeyuki MORITA
.
.
based on jw/w Takuya SAKASAI and Masaaki SUZUKI
October 29, 2013
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Contents
Contents
1
Symplectic representations and characteristic classes
2
Structure of the space of symplectic invariant tensors
3
Structure of {symplectic derivation algebra}Sp
4
Prospects
.
.
.
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (1)
HQ = H1 (Σg ; Q) Σg : closed oriented surface, genus g (≥ 1)
µ : HQ ⊗ HQ → Q: intersection pairing
HQ : fundamental representation of Sp = Sp(2g, Q)
Sp-modules and characteristic classes in low dim. topology
(I) ρab : π1 Σg → HQ = H1 (Jac; Q)
ρ∗ab : H ∗ (HQ ) ∼
= ∧∗ HQ → H ∗ (Σg ; Q)
(∧2 HQ )Sp ∼
= Q 3 ω0 (symplectic class) 7→ g [Σg ]∗
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (2)
(II)
Mapping class group
Mg = π0 Diff + Σg ,
(II − 1)
Mg,1 = π0 Diff(Σg , D2 )
Torelli group
Ig = Ker (Mg → Sp(2g, Z))
Theorem (Johnson)
H1 (Ig ; Q) ∼
= ∧3 HQ /HQ
(II − 2)
= irrep. [13 ]Sp
(g ≥ 3)
Johnson filtration of Mg,1
Mal’cev completion of π1 Σg,1 :
.
· · · → Nd+1 → Nd → · · · → N1 = HZ → 0
.
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (3)
representation of Mg,1 :
ρ∞ = {ρd }d : Mg,1 → lim Aut0 Nd
←−
(ρd : Mg,1 → Aut0 Nd )
d→∞
induced embedding of Lie algebras:
τg,1 :
∞
⊕
Mg,1 (d)/Mg,1 (d + 1)
small
⊂
hZ,+
g,1
ideal
⊂
hZg,1
d=1
Mg,1 (d) := Ker ρd
hZg,1 =
∞
⊕
Johnson filtration
hZg,1 (k): symplectic derivation algebra of L(HZ )
k=0
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (4)
(II − 3)
representation of Mg,1 :
ρ̃2 : Mg,1 → H1 (Ig ; Q) o Sp(2g, Q)
(M.)
⇒
Φ : H ∗ (∧3 HQ /HQ )Sp → H ∗ (Mg ; Q)
Theorem (Kawazumi-M.)
Im Φ = R∗ (Mg ) = Q[MMM-classes]/relations
tautological algebra in cohomology
Madsen-Weiss: H ∗ (M∞,1 ; Q) = Q[MMM-classes]
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (5)
(II − 4)
Infinitesimal presentation of Ig
Theorem (Hain)
• Gr Ig ∼
= Free Lie h∧3 HQ /HQ i/quadratic relations (g ≥ 4)
• Im τg,1 ⊗ Q is generated by the degree 1 part
(III)
.
Graph homology of Kontsevich
cg = ham02g
.
commutative
(rational form)
hg,1 = hZg,1 ⊗ Q
lie
ag = symplectic derivations of T0 HQ
associative
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (6)
Theorem (Kontsevich)
(n+1)
Sp ∼
• P Hk (c+
)
∞ )2n = Hk (G∗
• P Hck (b
h∞,1 )2n ∼
= H2n−k (Out Fn+1 ; Q)
Sp ∼ ⊕
2n−k (Mm ; Q)Sm
• P Hk (a+
2g−2+m=n H
∞ )2n =
g
.
m>0
(n+1)
G∗
: graph complex
b
h∞,1 : completion of
h∞,1 = lim hg,1
g→∞
Out Fn acts on Outer Space of Culler-Vogtmann p. discont.
.
Mm
g : moduli space of genus g curves with m marked points
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (7)
(III − 1) H1 (h+
g,1 ) and characteristic classes in H∗ (Out Fn ; Q)
Split extension hg,1 ∼
= h+
g,1 o sp(2g, Q) gives rise to
Sp → ⊕∞ H (Out F ; Q)
Φ : Hc∗ (H1 (h+
∗
n
n=2
g,1 ))
∞
⊕
3
trace maps : h+
S 2k+1 HQ ⇒ many classes
g,1 → ∧ HQ ⊕
k=1
K.
H 2 (S 2k+1 HQ )Sp ∼
h∞,1 )4k+2 ∼
= Q 7→ t2k+1 ∈ Hc2 (b
= H4k (Out F2k+2 ; Q)
µk ∈ H4k (Out F2k+2 ; Q) (k = 1, 2, . . .) important classes
non-triviality: µ1 M., µ2 Conant-Vogtmann, µ3 Gray
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (8)
Theorem (Conant-Kassabov-Vogtmann)
∼ 3
H1 (h+
0-loop)
∞,1 ) =(∧ HQ (Johnson,
)
∞
2k+1
⊕ ⊕k=1 S
HQ (M., trace maps: 1-loop)
⊕ (⊕∞
[2k
+
1,
1]Sp ⊕ other part) (2-loops) .
k=1
⊕ non-trivial ? (3, 4, . . .-loops) ? : deep question
Hairy graph complex and generalized trace map
⇒ many more classes, e.g. ∈ H46 (Out F25 ; Q)
.
(III − 2) Associative case
Theorem (Sakasai-Suzuki-M.)
( 2
)
3
3
∼
H1 (a+
∞ ) = lim ∧ HQ ⊕ S HQ ⊕ ∧ HQ /hω0 i
g→∞
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (9)
Corollary (vanishing of the top cohomology)
H 4g−5 (Mg ; Q) = 0
.
(g ≥ 2)
.
.
Harer (unpublished), another proof: Church-Farb-Putman
Theorem (Enomoto-Satoh)
The homomorphism
ESk : hg,1 (k) → ag (k − 2)
defined as (
∑k
i
i=1 (12 · · · k) )
◦ K12 satisfies ESk ◦ τg,1 (k) = 0.
other approaches: Kawazumi-Kuno, Conant
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.
Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (10)
(IV) Relation with number theory
(IV − 1)
Galois obstruction
1980’s, Oda predicted: Gal(Q/Q) (Soulé p-adic regulators)
should “appear” in (Cok τg,1 )Sp ⊗ Zp
(p: prime)
Nakamura, Matsumoto: proof and related many works
precise description of the Galois image: unknown
important both in topology and number theory
earlier foundational works (g = 0) of Ihara, Deligne
and more recent works (g = 1) of Hain-Matsumoto, Nakamura
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (11)
Fundamental Lie algebra
f := Free Liehσ3 , σ5 , · · · i
f → hSp
g,1
Nakamura, Matsumoto, ... , Brown
f → hSp
1,1
Hain-Matsumoto, Pollack
Important problem: identify the image of σk in each case
(IV − 2)
H 1 (SL(2, Z); S 2k HQ ) appears in many ways
• Eichler-Shimura Eisenstein series, cusp forms
• M.
H ∗ (BDiff + T 2 ; Q) ch. classes of torus bundles
• Conant-Kassabov-Vogtmann H1 (h+
g,1 ), H4k+3 (Aut F2k+3 )
• Hain-Matsumoto, Pollack h1,1
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (12)
(V) Group of homology cobordism classes of homology cylinders
Garoufalidis-Levine (based on Goussarov and Habiro):
Hg,1 = {(homology Σg,1 × I, ϕ); ϕ : Σg,1 ∼
= Σg,1 × {1}}
/homology cobordism
enlargement of Mg,1 ,
(a theorem of Stallings ⇒)
Theorem (Garoufalidis-Levine, Habegger)
There exists a homomorphism
ρ̃∞ : Hg,1 → lim Aut0 Nd
←−
d→∞
which extends ρ∞ , each finite factor ρ̃d : Hg,1 → Aut0 Nd is
surjective for any d ≥ 1
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (13)
Define a quotient group Hg,1 by the following central extension
0 → Θ3 = H0,1 → Hg,1 → Hg,1 → 1
Θ3 := Group of homology cobordism classes of homology 3-spheres
infinite rank by Furuta, Fintushel-Stern, also ρ̃∞ (Θ3 ) = {1}
Problem
Study the Euler class
χ(Hg,1 ) ∈ H 2 (Hg,1 ; Θ3 )
Freedman
acy
top
Θ3 → Hg,1 → Hg,1 −−−−−−→ Hg,1
→ Aut0 F2g
→ lim Aut0 Nd
←−
d
.
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (14)
One of the foundational results of Freedman:
Theorem (Freedman)
Any homology 3-sphere bounds a contractible topological
.
4-manifold so that Θ3 (top) = 0
top
It follows that Hg,1 → Hg,1
factors through Hg,1
.
Theorem (Sakasai, Dehn-Nielsen type theorem for Hg,1 )
top
Hg,1
→ lim Aut0 Nd factors through
←−
d→∞
alg
acy
Hg,1
= Aut0 F2g
: Sakasai’s algebraic version
acy
by using acyclic closure F2g
of Levine
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (15)
A representation of Hg,1 and classes in H 2 (Hg,1 ; Q)
Theorem (M.)
There exists a stable homomorphism
(
ρ̃ : Hg,1 −→ lim Aut0 Nd −→
←−
d
∧ 3 HQ ×
∞
∏
)
S 2k+1 HQ oSp(2g, Q)
k=1
Zariski dense at any finite level, S k HQ : k-th symmetric .power
An extension of the homomorphism obtained earlier:
ρ : Mg,1 −→ ∧3 HQ o Sp(2g, Q)
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Low Dimensional Topology and Symplectic Representations
Symplectic representations and characteristic classes (16)
Definition (characteristic classes for homology cylinders)
ρ̃∗ on H ∗ yields many stable cohomology classes of Hg,1
top
t̃2k+1 ∈ H 2 (Hg,1 ; Q), H 2 (Hg,1
; Q)
(k = 1, 2, . . .)
.
most important classes coming from H 2 (S 2k+1 HQ )Sp ∼
=Q
candidates for χ(Hg,1 ) ∈ H 2 (Hg,1 ; Θ3 )
group version of
t2k+1 ∈ H 2 (hg,1 ; Q) defined earlier
.
works for Hg,1 : survey by Sakasai and Habiro-Massuyeau
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Low Dimensional Topology and Symplectic Representations
Structure of the space of symplectic invariant tensors (1)
(HQ⊗2k )Sp : Sp-invariant subspace of the tensor product HQ⊗2k
We analyze the structure of this space completely
µ⊗2k : HQ⊗2k ⊗ HQ⊗2k → Q defined by
(u1 ⊗ · · · ⊗ u2k ) ⊗ (v1 ⊗ · · · ⊗ v2k ) 7→ Π2k
i=1 µ(ui , vi ) (ui , vi ∈ HQ ).
Clearly µ⊗2k is a symmetric bilinear form.
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Structure of the space of symplectic invariant tensors (2)
Theorem (M.)
µ⊗2k on (HQ⊗2k )Sp is positive definite for any g
⇒ it defines a metric on this space
Furthermore, ∃ an orthogonal direct sum decomposition
⊕
(HQ⊗2k )Sp ∼
Uλ
=
|λ|=k, h(λ)≤g
λ: Young diagram
|λ|: number of boxes, h(λ): number of rows
Uλ ∼
= (λδ )S2k as an S2k -module
Shigeyuki MORITA
.
Low Dimensional Topology and Symplectic Representations
Structure of the space of symplectic invariant tensors (3)
{λ; |λ| = k}
bijective
⇔
{µλ ; |λ| = k} eigenvalues
Table: Orthogonal decomposition of (HQ⊗8 )Sp
λ
[14 ]
[212 ]
[22 ]
[31]
[4]
total
µλ (eigen value of Uλ )
(2g − 6)(2g − 4)(2g − 2)2g
(2g − 4)(2g − 2)2g(2g + 1)
(2g − 2)(2g − 1)2g(2g + 1)
(2g − 1)2g(2g + 1)(2g + 2)
2g(2g + 1)(2g + 2)(2g + 3)
Shigeyuki MORITA
dim Uλ
1
20
14
56
14
105
g
g
g
g
g
g
for Uλ 6= {0}
= 4, 5, · · ·
= 3, 4, 5, · · ·
= 2, 3, 4, 5, · · ·
= 2, 3, 4, 5, · · ·
= 1, 2, 3, 4, 5, · · ·
Low Dimensional Topology and Symplectic Representations
Structure of {symplectic derivation algebra}Sp (1)
Theorem (Sakasai-Suzuki-M.)
Canonical metric ⇒ orthogonal decomposition
⊕
hg,1 (2k)Sp ∼
Hλ
=
|λ|=k+1, h(λ)≤g
in terms of certain subspaces Hλ with
dim Hλ =
1
(2k + 2)!
∑
χ2k (γ)χλδ (γ)
γ∈S2k+2
χ2k : Kontsevich’s character
χλδ : S2k+2 -irrep. corresponding to λδ
.
⇒ any Sp-invariant tensor ∈ hg,1 (2k)Sp has its “coordinates”
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Low Dimensional Topology and Symplectic Representations
Structure of {symplectic derivation algebra}Sp (2)
Table: Orthogonal decompositions of hg,1 (6)Sp , hg,1 (10)Sp
hg,1 (6)Sp
g
g
g
g
g
=1
=2
=3
=4
≥5
dim
1
4
5
Hλ
[4]
2[31][22 ]
[212 ]
dim
3
51
97
107
108
hg,1 (10)Sp
Hλ
3[6]
15[51]26[42]7[32 ]
19[412 ]24[321]3[23 ]
7[313 ]3[22 12 ]
[214 ]
Theorem (Sakasai-Suzuki-M.)
We have determined the coordinates of Im τg,1 (6)Sp , as well as
the Galois elements in hg,1 (6)Sp for both g = 1 and g ≥ 2
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Structure of {symplectic derivation algebra}Sp (3)
Theorem (Sakasai-Suzuki-M.)
We have determined the structure of
hg,1 (k) = jg,1 (k) ⊕ Lg (k) ⊕ Im τg (k) ⊕ Cok τg (k)
.
for k = 5 and k = 6 completely.
k=1:
Johnson hg,1 (1) ∼
= ∧ 3 HQ
k=2:
Hain, M.
k=3:
relation with l.c.s. of Ig,1 , Casson inv.
.
Hain, Asada-Nakamura trace [3] ∈ Cok τg (3)
k=4:
M. Asada-Nakamura [2] ∈ jg,1 (4) appears
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Low Dimensional Topology and Symplectic Representations
Structure of {symplectic derivation algebra}Sp (4)
Table: Structure of Cok τg (k)
k
1
2
3
4
5
6
Cok τg (k)
H1 (h+
g,1 )(k)
[3]
[212 ]
[5][32][22 1][15 ][21][13 ][1]
2[412 ][32 ][321][313 ][22 12 ]
[4][31][31][22 ]
2[212 ][14 ]2[12 ][0]
tr [3]
Ker ESk \ Im τg,1 (k)
tr [5]
CKV Tr [31]
[14 ]
[12 ]
Galois [0]
Theorem (Satoh)
0
Ker Satohk = Im τn (k) for Aut Fn in a certain stable range
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Structure of {symplectic derivation algebra}Sp (5)
Big differences between the MCG Mg and Aut Fn :
• Satoh + Sakasai-Massuyeau: Satohk is complete, i.e.
Ker Satohk = Im τn (k) for Aut Fn while
Galois appears in the gap between Ker ESk and Im τg,1 (k)
• M. found a gap (over Q) between Johnson filtration and the
l.c.s. of Ig (manifestation of the Casson invariant) while
there may be no gap? between Andreadakis filtration and
the l.c.s of IAn (Andreadakis conjecture)
We may say that symplectic representations reveal the above
differences, namely the Casson invariant and the Galois
obstructions
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Low Dimensional Topology and Symplectic Representations
Prospects (1)
(I) conjectural geometric meaning of classes
µk ∈ H4k (Out F2k+2 ; Q)
secondary classes associated with the difference between
two reasons for the vanishing of Borel regulator classes
βk ∈ H 4k+1 (GL(N, Z); R)
(1) βk = 0 ∈ H 4k+1 (Out FN ; R) (Igusa, Galatius)
?
(2) βk = 0 ∈ H 4k+1 (GL(Nk∗ , Z); R) critical Nk∗ = 2k + 2
Theorem (Bismut-Lott)
βk = 0 ∈ H 4k+1 (GL(2k + 1, Z); R)
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Low Dimensional Topology and Symplectic Representations
Prospects (2)
(II) Abelianizations of h∞,1 and Hg,1
Conjecture
H1 (h∞,1 ) = lim H1 (h+
g,1 )Sp = 0
g→∞
equivalently, the top homology w.r.t. vcd vanishes
H2n−3 (Out Fn ; Q) = 0 for any n ≥ 2
vcd(Out Fn ) = 2n − 3
.
(Culler-Vogtmann)
Very important because it is related to another mystery:
top
can show ∃ H1 (Hg,1
) → H1 (h∞,1 ): surjection while
.
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Prospects (3)
Important Question
What is the abelianization H1 (Hg,1 ) of Hg,1 ? In particular
H1 (Hg,1 ; Q) = 0?
Theorem (Cha-Friedl-Kim)
H1 (Hg,1 ) contains (Z/2)∞ as a direct summand
even the composition seems unknown to be trivial or not:
Θ3 → Hg,1 → H1 (Hg,1 )
.
.
Shigeyuki MORITA
Low Dimensional Topology and Symplectic Representations
Prospects (4)
(III) Ultimate goal
From 3-dimensional point of view, elements of hg,1 measure
higher and higher Massey products:
Sullivan, Johnson, Kitano, Garoufalidis-Levine
⇒
geometrical meaning of the class t̃2k+1 is:
intersection number of higher Massey products
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Low Dimensional Topology and Symplectic Representations
Prospects (5)
The classes t̃2k+1 would detect the differences between
topological versus smooth categories in dimension 4:
Conjecture
In the central extension
0 → Θ3 → Hg,1 → Hg,1 → 1
Θ3 “transgresses” to the classes t̃2k+1 ∈ H 2 (Hg,1 ; Q).
More precisely, these classes are all non-trivial in
top
H 2 (Hg,1 ; Q), and hence in H 2 (Hg,1
; Q) as well, while
they are trivial in H 2 (Hg,1 ; Q).
Shigeyuki MORITA
.
Low Dimensional Topology and Symplectic Representations