コンパクト Stein 曲面と写像類群

第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
Stein
)∗
(
Stein
(compact Stein surface)
Stein
CN
Stein
Stein
4
Stein
Stein
Stein
Eliashberg [El1] Gompf [Go]
Stein
Kirby
Stein
Loi
Piergallini [LP]
Akbulut
Ozbagci [AO]
Lefschetz
Stein
Lefschetz
Lefschetz
Lefschetz
Stein
1
Stein
Lefschetz
Stein
2
1 Lefschetz
Stein
Stein
1.1.
Stein
M
3
ture)
M
α ∧ dα > 0
(M, ξ), (M " , ξ " )
∗
1
M
ξ
(contact strucα
ξ = Kerα
M
α ξ
(contact form)
.
contactmorphic
(
15J05214)
152-8550
2-12-1
e-mail: [email protected]
89
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
H : M → M"
H∗ (ξ) = ξ "
(isotopic)
3
M
(M, ξ)
ξ, ξ "
4
Stein
ξ
Stein
(Stein fillable)
M
Stein
(X, J)
(M, ξ) (∂X, T ∂X ∩ J(T ∂X))
(X, J) (M, ξ) Stein
(Stein filling)
Stein
Stein
1
Eliashberg [El2]
Stein
4
B4
L(p, 1) (p $= 4)
S3
3
McDuff [Mc]
Stein
Euler
L(p, 1) (p $= 4)
Plamenevskaya
−p
Stein
Van Horn-Morris [PV]
Stein
1.2.
Lefschetz
Lefschetz
[FM]
Lefschetz
Σ := Σg,b
Σ
Diff+ (Σ, ∂Σ)
group) MΣ
α
Σ
Σ
tα
ϕ1 ϕ2 ∈ M Σ
[GS, Chapter 8] [OS, Chapter 10]
g
b
Diff+ (Σ, ∂Σ)
Σ
Diff+ (Σ, ∂Σ)
Dehn
[tα ] = [tα! ] ∈ MΣ
MΣ
ϕ1
1:
(mapping class
Σ
(right-handed Dehn twist) tα
α
1
1
α
α"
tα
[tα ]
ϕ1 , ϕ2 ∈ MΣ
ϕ2
α
Dehn
X
1.1. X
schetz fibration)
.
4
D2
D
2
f : X → D2
90
Lefschetz
(LefQf = {a1 , a2 , . . . , am }
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
1. f |f −1 (D2 − Qf )
2.
a i ∈ Qf
3.
D 2 − Qf
Σ
f −1 (ai )
f
pi ai
w = f (z1 , z2 ) = z12 + z22
1.2. f : X → D2
(isomorphic)
h◦f =f ◦H
f : X → D2
(s1 , s2 , . . . , sn )
a0
system)
(z1 , z2 ) w
f " : X " → D2
Lefschetz
a0 ai
a0
i = 1, 2, . . . , n
γi
(γ1 , γ2 , . . . , γn )
(
pi
Lefschetz
H : X → X"
a0
∂D2
i $= j
si
ai
f
X D2
f f"
h : D2 → D2
.
si ∩ sj = {a0 }
s1 , s2 , . . . , sn
Vi
Vi s i
{γ1 , γ2 , . . . , γn } π1 (D2 − Qf , a0 )
Hurwitz
(Hurwitz generating
2) Lefschetz
γi
2: Hurwitz
.
Σ
αi
(Σ × [0, 1])/((x, 1) ∼ (φ(x), 0)) → [0, 1]/(1 ∼ 0)
[φ] ∈ MΣ
αi ⊂ Σ
t αi
−1
f (ai )
(vanishing cycle)
(tα1 , tα2 , . . . , tαn ) f (γ1 , γ2 , . . . , γn )
(monodromy)
Hurwitz
(tα1 , . . . , tαi , tαi+1 , . . . , tαn ) ↔ (tα1 , . . . , tαi+1 , t−1
αi+1 tαi tαi+1 , . . . , tαn )
(tα1 , tα2 , . . . , tαn ) ↔ (t(α1 )ϕ , t(α2 )ϕ , . . . , t(αn )ϕ )
ϕ
MΣ
(tα1 , tα2 , . . . , tαn ) (tβ1 , tβ2 , . . . , tβn )
2
Hurwitz
(Hurwitz equivalent)
(tα1 , tα2 , . . . , tαn ) ≡ (tβ1 , tβ2 , . . . , tβn )
Lefschetz
Hurwitz
91
2
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
f : X → D2 allowable
αi
ALF allowable Lefschetz
1.3. Lefschetz
ALF
f
[αi ] ∈ H1 (Σ; Z)
Stein
1.4 (Loi-Piergallini [LP, Theorem 3], Akbulut-Ozbagci [AO, Theorem 5]). X
4
1. X
Stein
2. X
ALF
X
ALF
1.3.
[Et] [OS,
Chapter 9]
B M
1.5.
(B, π)
M
π : M − B → S1
Σθ ⊂ M
B
(B, π)
(page)
M
M −B
π
S1
(open book decomposition)
θ ∈ S1
IntΣθ = π −1 (θ)
∂Σθ = B
(binding) Σ ≈ Σθ (B, π)
π : (M − B) → S 1
(B, π)
ϕ ∈ MΣ
(monodromy)
Σ ϕ ∈ MΣ
∂Σ
r
∂Σ1 , ∂Σ2 , . . . , ∂Σr
Σϕ := (Σ × [0, 1])/((x, 1) ∼ (ϕ(x), 0))
3
1
r
r
S × D2
∂Σϕ
F
θ1 θ2
∂D2
[0, 1]
S 1 × {θ1 } ⊂ S 1 × ∂D2
!
∂Σi ×{θ2 } ⊂ ∂Σϕ
F
Σϕ ∪F ( r S 1 ×D2 )
3
Σ
(Σ, ϕ)
.
ξ M
M
ξ
(supporting open book decomposition)
ξ
ξ
α
dα
v
α(v) > 0
Thurston
Winkelnkemper
[TW]
M
Giroux
1.6 (Giroux [Gi]). M
• {M
• {M
ξ }/
3
2
(Σ, ϕ) }/
92
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
(Σ, ϕ)
(positive stabilization)
(Σ , tα ϕ)
Σ
"
h
a
h
"
α a∪a
"
a
Giroux
M
1Σ ∪ h Σ"
Σ
Stein
1.7 (
1. ξ
2. ξ
1.4
). M
3
ξ
M
Stein
(Σ, ϕ)
ϕ
Dehn
Dehn
f : X → D2
pr : (D2 −
ALF
Σ
(tα1 , tα2 , . . . , tαn )
ALF
2
{0}) → S
(r, θ) /→ θ
(r, θ) D − {0}
−1
1
−1
1
(pr ◦ f )|(pr ◦ f ) (S ) : (pr ◦ f ) (S ) → S 1 ∂X
(Σ, tα1 tα2 · · · tαn )
(M, ξ)
(Σ, ϕ)
ϕ
1.7 (2)
tα1 tα2 · · · tαn = ϕ
ϕ
Σ
(tα1 , tα2 , . . . , tαn )
ALF
(M, ξ) Stein
ALF
(Σ, ϕ)
ALF
1
1.4. Stein
manifold)
S
1.8 (Oba [Ob2]). M
ξ
(M, ξ)
Stein
Stein
Mazur
1.9.
2.
1.
1.8
Mazur
4
3
3
(Σ0,4 , ϕ)
ξ
(Mazur type
0- 1- 2M
M
4
Stein
S3
B4
S3
([Ob1])
1.8
3
S
3
Kirby
Stein
Stein
1.4 1.7
0
93
Stein
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
1.10 (Wendl[We, Theorem 1]). M
ξ
(M, ξ)
3
0
Stein
ξ
X
(Σ, ϕ)
Stein
(Σ, ϕ)
ALF
Stein
(Σ, ϕ)
ϕ
Dehn
Stein
1.8
Stein
1.11 ([Ob2]). M
ξ
(M, ξ) Stein
3
X
"
0 (i $= 0).
1.10
.
[Ob2]
(M, ξ)
Stein
(Σ, ϕ)
1.8
Mazur
1.10
ALF f : X → D2
(tα1 , tα2 , . . . , tαn )
Σ0,4
2
Σ0,4 ×D
X
χ(X) = 1
Stein fillable
Z (i = 0)
[Ob2]
X
M
0
Hi (X; Z) ∼
=
1.8
ξ
X
ξ
ALF
f
X
2χ(X) = χ(Σ0,4 × D2 ) + n = −2 + n
1.11
n = 3.
[α1 ], [α2 ], [α3 ] ∈ H1 (Σ0,4 ; Z)
3
(i) (ii)
Hurwitz
χ(X)
[α1 ], [α2 ], [α3 ] ∈ H1 (Σ0,4 ; Z)
3:
Kirby
X
ϕ
B4
tβ1 tβ2 tβ3
[α1 ] = [β1 ], [α2 ] = [β2 ], α3 = β3
2
94
.
(iii)
(tα1 , tα2 , tα3 ) ≡ (tβ1 , tβ2 , tβ3 )
ALF
Hurwitz
0tα1 , tα2 1 ⊂ MΣ0,4
[α1 ] = [β1 ], [α2 ] = [β2 ]
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
ψ1 , ψ2 ∈ 0tα1 , tα2 1
(αi )ψi = βi (i = 1, 2)
(tβ1 , tβ2 , tβ3 ) ≡ (t(α1 )ψ1 , t(α2 )ψ2 , tα3 ) ≡ (tα1 , t(α2 )ψ2 ψ1−1 , tα3 )
PSL(2; Z)
ψ2 ψ1−1 = tpα2 tqα1
p, q ∈ Z
p
q
(tα1 , t(α2 )ψ2 ψ1−1 , tα3 ) ≡ (tα1 , t(α2 )tα2 tα1 , tα3 ) ≡ (tα1 , tα2 , tα3 )
2
0tα1 , tα2 1 →
Stein
2.1.
Loi
Piergallini
Stein
2.1 (Loi-Piergallini [LP, Theorem 3]). X
1. X
2. X
Stein
B4
4
(
4
2.3)
Stein
S
S
Stein
2
S
2.2.
4
B4
[Ru], [Ka, Chapter 16 17], [APZ, Section 3]
[APZ, Section 5, 6]
D12 , D22
S D12 × D22
2.2. S ⊂ D12 × D22
surface)
m
m
pr1 : D12 × D22 → D12
((simple) braided
pS := pr1 |S : S → D12
pS
|QS | = n
a0 ∂D12
π1 (D12 − QS , a0 )
Hurwitz
(γ1 , γ2 , . . . , γn )
2
Sm m
ρ : π1 (D1 − QS , a0 ) → Sm
pS
2
ρ π1 (D1 − QS , a0 )
ρ(γi )
ρp S m
S
(braid
2
monodromy)ρS : π1 (D1 −QS , a0 ) → Bm
(ρS (γ1 ), ρS (γ2 ), . . . , ρS (γn ))
S
pS
ρS (γi ) = wi−1 σjεii wi
σji , wi ∈ Bm , εi ∈ {±1}
σ ji B m
QS
1.2
2.3.
ρS
i
S
(positive)
−1
ρS (γi ) = wi σji wi
95
S
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
p : X → B 4 ≈ D12 × D22
pr1 ◦ p : X → D2
ALF
−1
2
p ({a0 }×D2 ) (a0 ∈ ∂D12 )
S ∩ ({a0 } × D22 )
S
S
f :=
Q f = QS
f
f (a0 ) =
−1
2
−1
2
4) p|p ({a0 }×D2 ) : p ({a0 }×D2 ) → {a0 }×D22
f
[LP, Proposition 1]
−1
(
X, D12 × D22
4:
f , pS
B4
q : Σ → D2
d
Bm
m
[hβ ] ∈ MDm
m
β ∈ Bm
Qq
Dm
D − Qq Dm
2
(liftable)
MDm
Dm
Hβ : Σ → Σ
β ∈ Bm
2.4 ([Ob3]). S ⊂ D12 × D22
(w1−1 σj1 w1 , w2−1 σj2 w2 , . . . , wn−1 σjn wn )
a0
∂D12
{a0 } × D22 ⊂ D12 × D22 S ∩ ({a0 } × D22 )
d
−1
w i σj i w i ∈ B m q
S
4
2
p:X→B
pr1 ◦ p : X → D1
1.2
q
q ◦ H β = hβ ◦ q
q : Σ →
B 4 ≈ D12 × D22
Σ
ALF
. b0 ∈ {a0 } × ∂D22
π1 (({a0 } × D22 ) − (S ∩ ({a0 } × D22 )), (a0 , b0 ))
Hurwitz
(x1 , x2 , . . . , xm )
ρ : π1 (D4 −
S, (a0 , b0 )) → Sd q
ρq
π1 (D4 −
S, (a0 , b0 )) ι∗ (x1 ), ι∗ (x2 ), . . . , ι∗ (xm )
S
([Fo, p. 133], [Ya], [Ru, Proposition 4.1])
2
2
ι : (({a0 } × D2 ) − (S ∩ ({a0 } × D2 )), (a0 , b0 )) 0→ (D4 − S, (a0 , b0 ))
q
ρ
[Ob3]
96
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
2.3.
Stein
2.5. N
S
Stein
S ⊂ D12 ×D22
pi : Xi → B 4 ≈ D12 × D22 (i = 1, 2, . . . , N )
2
1. X1 , X2 , . . . , XN
2.
Xi
pi
Stein
Ji
Ji $= Jj (i $= j)
. N =2
a0 ∈ ∂D12 b0 ∈ {a0 } × ∂D22
B8
β1 , β2 , . . . , β5
β1 := σ5 β2 := (σ6−1 σ7−2 σ6−1 σ4 σ32 σ4 )−1 σ5 (σ6−1 σ7−2 σ6−1 σ4 σ32 σ4 ),
β3 := (σ6−1 σ5−1 σ4−1 σ3−1 σ2−1 σ1−2 σ2−1 σ7 σ6 σ5−1 σ4−1 σ3−1 σ4−2 σ3−1 σ4 )−1 σ7 ·
(σ6−1 σ5−1 σ4−1 σ3−1 σ2−1 σ1−2 σ2−1 σ7 σ6 σ5−1 σ4−1 σ3−1 σ4−2 σ3−1 σ4 ),
β4 := (σ4−1 σ5−2 σ4−1 σ2 σ12 σ2 )−1 σ3 (σ4−1 σ5−2 σ4−1 σ2 σ12 σ2 ),
β5 := (σ6 σ52 σ6 )−1 σ7 (σ6 σ52 σ6 ), β6 := (σ3 σ4 σ5 σ6 )−1 σ2 (σ3 σ4 σ5 σ6 )
8
S
(β1 , β2 , β3 , β4 , β5 , β6 )
q1 , q2 : Σ1,4 → {a0 } × D22
ρq1 , ρq2 : π1 ({a0 } × D22 , (a0 , b0 )) → S4
4
ρq1 (x1 ) = (1 2), ρq1 (x2 ) = (1 2), ρq1 (x3 ) = (2 3), ρq1 (x4 ) = (2 3)
ρq1 (x5 ) = (3 4), ρq1 (x6 ) = (3 4), ρq1 (x7 ) = (1 2), ρq1 (x8 ) = (1 2)
ρq2 (x1 ) = (1 2), ρq2 (x2 ) = (1 2), ρq2 (x3 ) = (3 4), ρq2 (x4 ) = (3 4)
ρq2 (x5 ) = (2 3), ρq2 (x6 ) = (2 3), ρq2 (x7 ) = (1 2), ρq2 (x8 ) = (1 2)
β1 , β2 , . . . , β5 q1 , q2
2.4
qi
pi : Xi → B 4 ≈ D12 × D22
pr1 ◦ pi : Xi → D12 ALF
Kirby
Kirby
X1 X2 Euler
−4
pi
Stein
Ji
1 Chern
Kirby
c1 (X1 , J1 ) $= 0
c1 (X2 , J2 ) = 0
(M, ξ)
x∈L
ξstd
2.6. N
L
2
2.6
L
(transverse link)
T x L + ξx = Tx M
S3
L ⊂ (S 3 , ξstd )
pi : Mi → S 3 (i = 1, 2, . . . , N )
1. M1 , M2 , . . . , MN
2.
Mi
ξi
62
97
i $= j
ξi
ξj
第６２回トポロジーシンポジウム講演集２０１５年８月於名古屋工業大学
[AO] S. Akbulut and B. Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom.
Topol. 5 (2001), 939–945.
[APZ] N. Apostolakis, R. Piergallini, and D. Zuddas, Lefschetz fibrations over the disc, Proc.
Lond. Math. Soc. (3) 107 (2013), no. 2, 340–390.
[El1] Y. Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Internat. J. Math. 1 (1990), no. 1, 29–46.
[El2] Y. Eliashberg, Filling by holomorphic discs and its applications, Geometry of Lowdimensional Manifolds: 2, Proc. Durham Symp. 1989, London Math. Soc. Lecture Notes
151, Cambridge Univ. Press, 1990, 45–67.
[Et] J. Etnyre, Lectures on open book decompositions and contact structures, Floer Homology,
Gauge Theory, and Low Dimensional Topology, Clay Math. Proc. 5, Amer. Math. Soc.,
Providence, RI, 2006.
[FM] B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical
Series 49, Princeton Univ. Press, Princeton, NJ, 2012.
[Fo] R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics
(Prentice-Hall, Englewood Cliffs, N.J., 1962), pp. 120–167.
[Gi] E. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures,
Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002),
405–414.
[Go] R. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998),
no. 2, 619–693.
[GS] R. Gompf and A. Stipsicz, 4-Manifolds and Kirby Calculus, Grad. Stud. Math. 20,
Amer. Math. Soc., Providence, RI, 1999.
[Ka] S. Kamada, Braid and knot theory in dimension four, Mathematical Surveys and Monographs, 95. American Mathematical Society, Providence, RI, 2002.
[LP] A. Loi and R. Piergallini, Compact Stein surfaces with boundary as branched covers of
B 4 , Invent. Math. 143 (2001), no. 2, 325–348.
[Mc] D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103
(1991), 651–671.
[Ob1] T. Oba, A note on Mazur type Stein fillings of planar contact manifolds,
arXiv:1405.3751.
[Ob2] T. Oba, Stein fillings of homology spheres and mapping class groups, arXiv:1407.5257.
[Ob3] T. Oba, Compact Stein surfaces as branched coverings with distinct covering monodromies, in preparation.
[OS] B. Ozbagci and A. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces, Bolyai
Soc. Math. Stud. 13, Springer-Verlag, 2004.
[PV] O. Plamenevskaya and J. Van Horn-Morris, Planar open books, monodromy factorizations and symplectic fillings, Geom. Topol. 14 (2010), 2077–2101.
[Ru] L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math.
Helv. 58 (1983), no. 1, 1–37.
[TW] W. Thurston and H. Winkelnkemper, On the existence of contact forms, Proc. Amer.
Math. Soc. 52 (1975), 345–347.
[We] C. Wendl, Strongly fillable contact manifolds and J-holomorphic foliations, Duke Math.
J. 151(3) (2010), 337–384.
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98

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