. Stein fillings of homology spheres with planar open books . Takahiro Oba Tokyo Institute of Technology December 19, 2013 Nihon university 1 / 21
. Main Results (Roughly) . f : X 4 → D 2 : PALF w/ planar fiber ( M 3 = ∂X, ξ ) : contact boundary of X . Results . ∂X : homology sphere ? ⇝ ♯ { crit. pts of f } ( M, ξ ) : Stein fillable homology sphere ? ⇝ mapping class group ∃{ ( M n , ξ n ) } : infinite sequence of Stein fillable homology spheres w/ planar open books M n ̸≈ M m ( n ̸ = m ) s.t. each Stein filling is a ”Mazur type mfd” . 2 / 21
. Main Results (Roughly) . f : X 4 → D 2 : PALF w/ planar fiber ( M 3 = ∂X, ξ ) : contact boundary of X . Results . ∂X : homology sphere ? ⇝ ♯ { crit. pts of f } ( M, ξ ) : Stein fillable homology sphere ? ⇝ mapping class group ∃{ ( M n , ξ n ) } : infinite sequence of Stein fillable homology spheres w/ planar open books M n ̸≈ M m ( n ̸ = m ) s.t. each Stein filling is a ”Mazur type mfd” . 2 / 21
. The Plan of Talk . § 1 . Definitions and Background § 2 . Positive allowable Lefschetz fibrations (PALFs) § 3 . Main Results and Ideas of the proofs 3 / 21
. § 1 . Definitions and Background . M : closed oriented 3-manifold, B : oriented link in M π : M \ B → S 1 : smooth map . Definition . ( B, π ) is called an open book (decomposition) if π is a fibration over S 1 s.t. π − 1 ( θ ) = Int F ( ∀ θ ∈ S 1 ) 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 ) . 4 / 21
. § 1 . Definitions and Background . . Definition . A 2 -plane field ξ on M is called an (oriented) contact str. on M if ∃ α ∈ Ω 1 ( M ) s.t. . . 1 ξ = ker α . . 2 α ∧ dα > 0 . . Definition . An open book of ( M, ξ = ker α ) is called a supporting open book of 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” . F' F C ( F, ) ( F', ' t ) ○ C 6 / 21
. § 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 open book. Then, the monodromy of this open book has a positive . factorization. 7 / 21
. § 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 ) . . 8 / 21
. § 2 .Positive allowable Lefschetz fibrations (PALF) . . Definition . f : X 4 → D 2 is called a (positive) Lefschetz fibration (LF) if ∃{ b 1 , b 2 , . . . , b m } =: Crit( f ) ⊂ Int( D 2 ) s.t. . . 1 Crit( f ) is the set of critical values of f and for ∀ b i , ∃ ! p i ∈ f − 1 ( b i ) s.t. for ∀ p ∈ f − 1 ( b i ) \ { p i } f p : T p X → T f ( p ) D 2 : onto, d 2 f | f − 1 ( D 2 \ Crit( f )) is a fiber bundle over D 2 \ Crit( f ) . . . . 3 for ∀ p i (resp. ∀ b i ) ∃ ( z 1 , z 2 ) (resp. w ) : local cpx. coordinate of X (resp. D 2 ) s.t. w = f ( z 1 , z 2 ) = z 2 1 + z 2 2 . . 9 / 21
. § 2 .Positive allowable Lefschetz fibrations (PALF) . . Definition . LF f : X → D 2 is a positive allowable LF (PALF) if regular fiber of f is bounded and any vanishing cycle is homologically-nontrivial. . 10 / 21
. § 2 .Positive allowable Lefschetz fibrations (PALF) . . Definition . LF f : X → D 2 is a positive allowable LF (PALF) if regular fiber of f is bounded and any vanishing cycle is homologically-nontrivial. . 10 / 21
. § 2 .Positive allowable Lefschetz fibrations (PALF) . Handle decomposition of PALF D n PALF f : X → D 2 w/ fiber D n and m crit.pts. X ≈ ( D 2 × D n ) ∪ ( ∪ m i =1 H (2) ) i ≈ H (0) ∪ ( ∪ n j =1 H (1) i =1 H (2) ) ∪ ( ∪ m ) j i H ( k ) : k -handle 2 -handles attached to D 2 × D n along van. cycles w/ − 1 -framing. 11 / 21
. § 2 .Positive allowable Lefschetz fibrations (PALF) . First homology of the total space X C C C C n 1 2 n-1 D n { C 1 , C 2 , . . . , C n } : basis for H 1 ( D n ) H 1 ( X ) ∼ = H 1 ( D n ) / ⟨ γ 1 , γ 2 , . . . , γ m : van.cycles ⟩ [ γ i ] : ”homology class” of van. cycle γ i in H 1 ( D n ) [ γ i ] = ε i 1 C 1 + ε i 2 C 2 + · · · + ε in C n ( ε ij ∈ { 0 , 1 } , ∀ i ) To compute H 1 ( X ) , determine the SNF of (0 , 1) matrix A = ( ε ij ) . 12 / 21
. § 2 .Positive allowable Lefschetz fibrations (PALF) . First homology of the total space X C C C C n 1 2 n-1 D n { C 1 , C 2 , . . . , C n } : basis for H 1 ( D n ) H 1 ( X ) ∼ = H 1 ( D n ) / ⟨ γ 1 , γ 2 , . . . , γ m : van.cycles ⟩ [ γ i ] : ”homology class” of van. cycle γ i in H 1 ( D n ) [ γ i ] = ε i 1 C 1 + ε i 2 C 2 + · · · + ε in C n ( ε ij ∈ { 0 , 1 } , ∀ i ) To compute H 1 ( X ) , determine the SNF of (0 , 1) matrix A = ( ε ij ) . 12 / 21
. § 3 .Main Results . Suppose m ≥ n . X i 1 ,i 2 ,...,i n = ( D 2 × D n ) ∪ ( ∪ n k =1 H (2) i k ) : subhandlebody of X . Theorem (O.) . f : X → D 2 : PALF w/ fiber D n and m crit. pts. ∃ X i 1 ,i 2 ,...,i n ⊂ X s.t. H 1 ( X i 1 ,i 2 ,...,i n ) = 0 Then, ∂X :homology sphere ⇔ n = m . . Key Fact . 1 Y 4 = 0 -handle ∪ 2 -handles: 2 -handlebody . Q Y : intersection form of Y Then, Q Y = linking matrix determined by the diagram of Y . . . 2 If H 1 ( Y ) = 0 , ∂Y : homology sphere ⇔ Q Y : unimodular. . 13 / 21
. § 3 .Main Results . Suppose m ≥ n . X i 1 ,i 2 ,...,i n = ( D 2 × D n ) ∪ ( ∪ n k =1 H (2) i k ) : subhandlebody of X . Theorem (O.) . f : X → D 2 : PALF w/ fiber D n and m crit. pts. ∃ X i 1 ,i 2 ,...,i n ⊂ X s.t. H 1 ( X i 1 ,i 2 ,...,i n ) = 0 Then, ∂X :homology sphere ⇔ n = m . . Key Fact . 1 Y 4 = 0 -handle ∪ 2 -handles: 2 -handlebody . Q Y : intersection form of Y Then, Q Y = linking matrix determined by the diagram of Y . . . 2 If H 1 ( Y ) = 0 , ∂Y : homology sphere ⇔ Q Y : 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: ( ) t A − I m H 1 ( X ′ ) = 0 , ∂X = ∂X ′ and Q X ′ = A O = 1 if m = n ⇝ | detQ X ′ | > 1 if m > n ∴ ∂X : homology sphere ⇔ | detQ X ′ | = 1 ⇔ m = n □ 14 / 21
. § 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 → D 2 : PALF w/ fiber D n and m crit. pts. Then, ∂X :homology sphere ⇔ n = m and H 1 ( X ) = 0 . . . Corollary (O.) . Suppose n ∈ { 1 , 2 , 3 , 4 } . ( M, ξ ) : Stein fillable and supported planar open book ( D n , φ ) X : Stein filling induced by positive fact. φ = t γ 1 t γ 2 · · · t γ m Then, M : homology sphere ⇔ n = m and H 1 ( X ) = 0 . . 15 / 21
. § 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 → D 2 : PALF w/ fiber D n and m crit. pts. Then, ∂X :homology sphere ⇔ n = m and H 1 ( X ) = 0 . . . Corollary (O.) . Suppose n ∈ { 1 , 2 , 3 , 4 } . ( M, ξ ) : Stein fillable and supported planar open book ( D n , φ ) X : Stein filling induced by positive fact. φ = t γ 1 t γ 2 · · · t γ m Then, M : homology sphere ⇔ n = m and H 1 ( X ) = 0 . . 15 / 21
. § 3 . Main Results . Application : We consider PALF X n ( n > 2) whose van. cycles are as follows; b b b b b 1 2 3 n-3 n-2 c 16 / 21
To compute H 1 ( X n ) , determine the SNF of the following matrix: 1 1 0 0 0 0 0 1 1 · · · 0 0 0 0 0 1 0 0 0 . . ... . . . . . 0 0 0 1 1 0 0 0 0 · · · 0 1 1 0 0 0 0 0 1 ⇝ The SNF of this matrix is the identity matrix I n . ∴ H 1 ( X n ) = 0 By the theorem, ∂X n is a homology sphere. 17 / 21
. § 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 ̸≈ S 3 . ~ ~ -(n-2) -n all framing -1 ⇝ π 1 ( X n ) = 1 and H ∗ ( X n ) = 0 ( ∗ > 0) By Hurewicz theorem and Whitehead theorem, X n is contractible. 18 / 21
. § 3 . Main Results . ~ ~ -(n-2) 1 -n n-2 Casson invariant λ is Z -valued invariant of homology spheres. . Property of Casson invariant . λ ( S 3 ) = 0 ( surgery formula ) K ⊂ S 3 : knot, △ K ( t ) : Alexander poly. of K S 3 + 1 q K : 3-mfd obtained from 1 q surgery on K λ ( S 3 + 1 q K ) = q Then, 2 △ ′′ K (1) . . 19 / 21
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