Stein fillings of homology spheres with planar open books . - - PowerPoint PPT Presentation

. .

Stein fillings of homology spheres with planar open books

Takahiro Oba

Tokyo Institute of Technology

December 19, 2013 Nihon university

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. . Main Results (Roughly)

f : X4 → D2 : PALF w/ planar fiber (M3 = ∂X, ξ) : contact boundary of X . Results . . ∂X: homology sphere ? ⇝ ♯{crit. pts of f} (M, ξ): Stein fillable homology sphere ? ⇝ mapping class group ∃{(Mn, ξn)}: infinite sequence of Stein fillable homology spheres w/ planar open books s.t.    Mn ̸≈ Mm (n ̸= m) each Stein filling is a ”Mazur type mfd”

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. . Main Results (Roughly)

f : X4 → D2 : PALF w/ planar fiber (M3 = ∂X, ξ) : contact boundary of X . Results . . ∂X: homology sphere ? ⇝ ♯{crit. pts of f} (M, ξ): Stein fillable homology sphere ? ⇝ mapping class group ∃{(Mn, ξn)}: infinite sequence of Stein fillable homology spheres w/ planar open books s.t.    Mn ̸≈ Mm (n ̸= m) each Stein filling is a ”Mazur type mfd”

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. . The Plan of Talk §1. Definitions and Background §2. Positive allowable Lefschetz fibrations (PALFs) §3. Main Results and Ideas of the proofs

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. . §1.Definitions and Background

M: closed oriented 3-manifold, B: oriented link in M π : M\B → S1: smooth map . Definition . . (B, π) is called an open book (decomposition) if π is a fibration over S1 s.t. π−1(θ) = Int F (∀θ ∈ S1) where F is a cpt. surf. whose boundary ∂F = ∂π−1(θ) is B. B is called a binding and F is called a page of (π, B). M \ Int νB ≈ ([0, 1] × F)/(1, x) ∼ (0, φ(x)) (φ: monodromy) ⇝ We also denote (F, φ) to be an open book (B, π).

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. . §1.Definitions and Background

. Definition . . A 2-plane field ξ on M is called an (oriented) contact str. on M if ∃α ∈ Ω1(M) s.t. . .

1 ξ = kerα

. .

2 α ∧ dα > 0

. Definition . . An open book of (M, ξ = kerα) is called a supporting open book

• f ξ if

. .

1 dα is an area form of the page of the open book

.

2 α is positive on B 5 / 21

. . §1.Definitions and Background

. Theorem (Giroux 2002) . . { contact structure on M} / isotopy ↕ 1:1 { open book of M} / ”positive (de)stabilization”

C

( F, ) ( F', ' t )

C

○

F F'

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. . §1.Definitions and Background

. Theorem (Loi-Piergallini 2001, Akbulut-Ozbagci 2001) . . For ∀(M, ξ): Stein fillable contact manifold, ∃(F, φ): supporting open book of ξ s.t. φ has a positive fact. Conversely, for ∀(F, φ): open book of M s.t. φ has a positive fact., ∃ξ: Stein fillable contact str. on M s.t. ξ is supported by (F, φ). Furthermore, Stein filling = PALF . . Theorem (Wendl 2010) . . ξ is a Stein fillable contact structure on M supported by a planar

• pen book. Then, the monodromy of this open book has a positive

factorization.

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. . §1.Definitions and Background

. Problem . . (M, ξ): Stein fillable contact 3-mfd particularly homology sphere supported by a planar open book. Characterize the monodromy by a Stein filling of (M, ξ).

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. . §2.Positive allowable Lefschetz fibrations (PALF)

. Definition . . f : X4 → D2 is called a (positive) Lefschetz fibration (LF) if ∃{b1, b2, . . . , bm} =: Crit(f) ⊂ Int(D2) s.t. . .

1 Crit(f) is the set of critical values of f and

for ∀bi, ∃!pi ∈ f−1(bi) s.t. for ∀p ∈ f−1(bi) \ {pi} d fp : TpX → Tf(p)D2: onto, . .

2 f|f−1(D2 \ Crit(f)) is a fiber bundle over D2 \ Crit(f)

. .

3 for ∀pi (resp. ∀bi) ∃ (z1, z2) (resp. w) : local cpx. coordinate

• f X (resp. D2) s.t. w = f(z1, z2) = z2

1 + z2 2.

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. . §2.Positive allowable Lefschetz fibrations (PALF)

. Definition . . LF f : X → D2 is a positive allowable LF (PALF) if regular fiber

• f f is bounded and any vanishing cycle is homologically-nontrivial.

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. . §2.Positive allowable Lefschetz fibrations (PALF)

. Definition . . LF f : X → D2 is a positive allowable LF (PALF) if regular fiber

• f f is bounded and any vanishing cycle is homologically-nontrivial.

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. . §2.Positive allowable Lefschetz fibrations (PALF) Handle decomposition of PALF

Dn

PALF f : X → D2 w/ fiber Dn and m crit.pts. X ≈ (D2 × Dn) ∪ (∪m

i=1 H(2) i

) ≈ H(0) ∪ (∪n

j=1 H(1) j

) ∪ (∪m

i=1 H(2) i

) H(k): k-handle 2-handles attached to D2 × Dn along van. cycles w/ −1-framing.

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. . §2.Positive allowable Lefschetz fibrations (PALF) First homology of the total space X

Dn

C C C C

1 2 n-1 n

{C1, C2, . . . , Cn}: basis for H1(Dn) H1(X) ∼ = H1(Dn)/⟨γ1, γ2, . . . , γm : van.cycles⟩ [γi] : ”homology class” of van. cycle γi in H1(Dn) [γi] = εi1C1 + εi2C2 + · · · + εinCn (εij ∈ {0, 1}, ∀i) To compute H1(X), determine the SNF of (0, 1) matrix A = (εij).

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. . §2.Positive allowable Lefschetz fibrations (PALF) First homology of the total space X

Dn

C C C C

1 2 n-1 n

{C1, C2, . . . , Cn}: basis for H1(Dn) H1(X) ∼ = H1(Dn)/⟨γ1, γ2, . . . , γm : van.cycles⟩ [γi] : ”homology class” of van. cycle γi in H1(Dn) [γi] = εi1C1 + εi2C2 + · · · + εinCn (εij ∈ {0, 1}, ∀i) To compute H1(X), determine the SNF of (0, 1) matrix A = (εij).

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. . §3.Main Results

Suppose m ≥ n. Xi1,i2,...,in = (D2 × Dn) ∪ (∪n

k=1 H(2) ik ) : subhandlebody of X

. Theorem (O.) . . f : X → D2 : PALF w/ fiber Dn and m crit. pts. ∃Xi1,i2,...,in ⊂ X s.t. H1(Xi1,i2,...,in) = 0 Then, ∂X:homology sphere ⇔ n = m . Key Fact . . .

1 Y 4 = 0-handle ∪ 2-handles: 2-handlebody

QY : intersection form of Y Then, QY =linking matrix determined by the diagram of Y . . .

2 If H1(Y ) = 0, ∂Y : homology sphere ⇔ QY : unimodular. 13 / 21

. . §3.Main Results

Suppose m ≥ n. Xi1,i2,...,in = (D2 × Dn) ∪ (∪n

k=1 H(2) ik ) : subhandlebody of X

. Theorem (O.) . . f : X → D2 : PALF w/ fiber Dn and m crit. pts. ∃Xi1,i2,...,in ⊂ X s.t. H1(Xi1,i2,...,in) = 0 Then, ∂X:homology sphere ⇔ n = m . Key Fact . . .

1 Y 4 = 0-handle ∪ 2-handles: 2-handlebody

QY : intersection form of Y Then, QY =linking matrix determined by the diagram of Y . . .

2 If H1(Y ) = 0, ∂Y : homology sphere ⇔ QY : unimodular. 13 / 21

. . §3.Main Results

Idea of proof : Surgery on the cores of the 1-handles of X: X ⇝ X′: 2-handlebody surgered mfd X′ satisfies: H1(X′) = 0, ∂X = ∂X′ and QX′ = ( −Im

tA

A O ) ⇝ |detQX′|    = 1 if m = n > 1 if m > n ∴ ∂X: homology sphere ⇔ |detQX′| = 1 ⇔ m = n □

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. . §3.Main Results

Using ˇ Zivkovi´ c’s classification of (0, 1) matrices, we have the following corollaries. . Corollary (O.) . . Suppose n ∈ {1, 2, 3, 4}. f : X → D2 : PALF w/ fiber Dn and m crit. pts. Then, ∂X:homology sphere ⇔ n = m and H1(X) = 0. . Corollary (O.) . . Suppose n ∈ {1, 2, 3, 4}. (M, ξ): Stein fillable and supported planar open book (Dn, φ) X: Stein filling induced by positive fact. φ = tγ1 tγ2 · · · tγm Then, M: homology sphere ⇔ n = m and H1(X) = 0.

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. . §3.Main Results

Using ˇ Zivkovi´ c’s classification of (0, 1) matrices, we have the following corollaries. . Corollary (O.) . . Suppose n ∈ {1, 2, 3, 4}. f : X → D2 : PALF w/ fiber Dn and m crit. pts. Then, ∂X:homology sphere ⇔ n = m and H1(X) = 0. . Corollary (O.) . . Suppose n ∈ {1, 2, 3, 4}. (M, ξ): Stein fillable and supported planar open book (Dn, φ) X: Stein filling induced by positive fact. φ = tγ1 tγ2 · · · tγm Then, M: homology sphere ⇔ n = m and H1(X) = 0.

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. . §3. Main Results

Application : We consider PALF Xn (n > 2) whose van. cycles are as follows;

b b b b b c

1 2 3 n-3 n-2 16 / 21

To compute H1(Xn), determine the SNF of the following matrix:               1 1 1 1 · · · 1 . . . ... . . . 1 1 · · · 1 1 1               . ⇝ The SNF of this matrix is the identity matrix In. ∴ H1(Xn) = 0 By the theorem, ∂Xn is a homology sphere.

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. . §3. Main Results

. Definition . . A cpt. conn. orientable 4-mfd X is Mazur type if . .

1 X: contractible

. .

2 X = 0-handle∪ 1-handle∪ 2-handle

. .

3 ∂X ̸≈ S3

~

all framing -1

~

• n
• (n-2)

⇝ π1(Xn) = 1 and H∗(Xn) = 0 (∗ > 0) By Hurewicz theorem and Whitehead theorem, Xn is contractible.

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. . §3. Main Results

• n
• (n-2)

~ ~

1 n-2

Casson invariant λ is Z-valued invariant of homology spheres. . Property of Casson invariant . . λ(S3) = 0 (surgery formula) K ⊂ S3: knot, △K(t): Alexander poly. of K S3 + 1

qK: 3-mfd obtained from 1 q surgery on K

Then, λ(S3 + 1

qK) = q 2 △′′ K (1).

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. . §3. Main Results

Compute the Casson invariant of ∂Xn 1 n-2

K

△K(t) = t2 − 2t + 3 − 2t−1 + t−2, △′′

K(1) = 4.

By surgery formula, λ(∂Xn) = λ(S3 +

1 n−2K) = n−2 2

× 4 = 2(n − 2) > 0. ∴ Xn: Mazur type and ∂Xn ̸≈ ∂Xm(n ̸= m)

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Thank you for your attention !

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