On Reeb graphs derived from Heegaard splittings with distance 2 g 井戸 絢子 ( 奈良女子大学 D1) 20 / 12 / 2010 結び目の数学Ⅲ @ 日本大学

1. Preliminaries M : a closed orientable 3-manifold Def. A ∪ P B : a (genus g ) Heegaard splitting of M ⇔ A, B ⊂ M : genus g handlebodies s.t. ∪ B , ・ M = A ・ A ∩ B = ∂A = ∂B = P : Heegaard surface

S : a closed orientable surface of genus ≥ 2 Def. ( curve complex ) C S : the curve complex of S s.t. ・ the vertices correspond to isotopy classes of essential simple closed curves on S ・ the edges are drawn between vertices corresponding to disjoint curves. C s S The distance between two vertices is the smallest number of edges in any path between the vertices in C S.

A ∪ P B : a genus g (≥2) Heegaard splitting of M Def. ( Hempel distance ) D ( A ) , D ( B ) : the subsets of the curve complex C P corresponding to curves on P bounding disks in A , B . d ( P ):= d ( D ( A ) , D ( B )) : the Hempel distance of P D B D A d ( P )=0

D A D B d ( P )=1 D A D B d ( P )=2

Thm. [Cf. Scharlemann-Tomova, 2006] P, Q : genus g Heegaard surfaces in M ≦ g. Suppose that Q is not isotopic to P, then d ( P ) 2 ⇒ Q is isotopic to P ) ( i.e. d ( P )> 2g Rubinstein -Scharlemann gave the list of 3-mfds each admitting inequivalent genus 2 Heegaard splittings. Recently, Berge-Scharlemann pointed out there is a missing case in the list, and gave the complete list. Their result implies the following fact. Fact [Cf. Berge-Scharlemann, 2010] P, Q : genus 2 Heegaard surfaces in M Suppose d ( P )=4 . Then Q is isotopic to P. M.Scharlemann, M.Tomova, Alternate Heegaard genus bounds distance , Geom. Topol. 10(2006) 593--617 H. Rubinstein and M. Scharlemann, Genus two Heegaard splittings of ori-entable 3-manifolds , in Proceedings of the 1998 Kirbyfest, Geometry and TopologyMonographs 2 (1999) 489-553. J. Berge and M. Scharlemann, Multiple genus 2 Heegaard splittings: a missed case , ArXiv preprint arXiv:0910.3921

Question Question P, Q : genus g Heegaard surfaces in M P, Q : genus g Heegaard surfaces in M Suppose d ( P )=2 g. Is Q isotopic to P ? Suppose d ( P )=2 g. Is Q isotopic to P ? In this talk, we show that Q is isotopic to a canonical position w.r.t. P , if Q is not isotopic to P and d( P )=2 g. And as an application, we give an alternative proof of the above fact.

2. Main result spine of handlebody 2-1. Sweep-out A ∪ P B : a Heegaard splitting of M Σ A , Σ B : spines of A, B Σ B M = Σ A ∪ P ×(0,1) ∪ Σ B M P s 1 We may suppose P ×(0,1) → (0,1) can be extended to a smooth map f : M → [0,1] s.t. ・ f ( Σ A )=0, f ( Σ B )=1 ・ each level surface is isotopic to P. 0 : a sweep-out of P Σ A P s =f ( s ) ( s (0, 1)) : level surface of f -1 ∈

2-2. Reeb graph P, Q : Heegaard surfaces in M f :a sweep out of P f | Q : Q → [0,1] -1 ∈ M ) ⇔ ∃ s [0,1] ∈ s.t. x, y are in the same comp. of ( f | Q ) ( s ) . x~y ( x, y Q / ~ : Reeb graph Reeb graph of Q

Lemma. [Tao Li, Cf. Johnson] P, Q : Heegaard surfaces in M Σ A , Σ B : spines of A, B fix f : sweep out of P ≧ Suppose that d( P ) 2. Then Q can be isotoped so that (1) Q ∩ Σ A and Q ∩ Σ B consists of finitely many points ; (2) Q is transverse to each P s , s ∈ （ 0,1 ） , ∈ except for finitely many critical levels s 1 ,..., s n (0,1) ; (3) at each critical level s i , Q is transverse to P s i , except for a single saddle or circle tangency; (4) at each regular level P s , every component of Q ∩ P s is an essential curve on P s . circle tangency : Q : T.Li. Saddle tangencies and the distance of Heegaard splittings, Algebr. Geom. Topol. 7 (2007), 1119—1134 . J. Johnson, Flipping and stabilizing Heegaard splittings, preprint 2008),ArXiv:math.GT/08054422.

Lemma. [Tao Li, Cf. Johnson] ∃ s i , s j ( s i < s j ) s.t. ∀ s ( ∈ s i , s j ) : regular value, each comp. of P s ∩ Q is essential on both P s and Q . For any small ε >0, P s i - ε ∩ Q contains a curve bounding a meridian disk in A , and P s j + ε ∩ Q contains a curve bounding a meridian disk in B . Q s i s j ess. curves on P and Q By making use surfaces satisfying the above Lemmas, Tao Li and Johnson give an alternative proof of the following theorem. Thm. [Cf. Scharlemann-Tomova, 2006] P, Q : genus g Heegaard surfaces in M ≦ g. Suppose that Q is not isotopic to P, then d ( P ) 2

2-2. Reeb graph P, Q : Heegaard surfaces in M f :a sweep out of P f | Q : Q → [0,1] -1 ∈ M ) ⇔ ∃ s [0,1] ∈ s.t. x, y are in the same comp. of ( f | Q ) ( s ) . x~y ( x, y Q / ~ : Reeb graph Reeb graph of Q s i +ε s j ｰ ε

Main result P, Q : genus g Heegaard surfaces in M Suppose that Q is not isotopic to P and d ( P )= 2g , then Q can be isotoped so that the corresponding Reeb graph G is as follows. s j ｰ ε s i +ε 1 2g-1 2g-3 3 2g-2 2 v 1 v 2g-1 v 2g-2 v 2 3 2g-3 2g-1 d(D (A ), c )= 1

Application; an alternative proof of the following fact Fact [Cf. Berge-Scharlemann, 2010] P, Q : genus 2 Heegaard surfaces in M Suppose d ( P )=4 . Q is isotopic to P. Q Outline of proof M X Y Suppose Q is not isotopic to P Q A and d(P)=4 A 1 3 P B 2 P Y P X Q B 3 1 s P X , P Y : P ∩ X, P ∩ Y Q A , Q B : Q ∩ A, Q ∩ B

Remark. P X (resp. P Y , Q A , Q B ) is incompressible in X (resp. X , Y , A , B ) Lem. [Rubinstein-Scharlemann 1999] Suppose that P X , P Y , Q A , Q B are incompressible in respectively X, Y , A, B . Then P X ∂–compresses to one of Q A or Q B , and P Y ∂–compresses to the other. Q ∂-comp. disks M X Y Q A A P P Y P X B Q B H. Rubinstein and M. Scharlemann, Genus two Heegaard splittings of ori-entable 3-manifolds , in Proceedings of the 1998 Kirbyfest, Geometry and TopologyMonographs 2 (1999) 489-553.

Q M Y X 1 3 2 P* = Q* A 3 P 1 B P*= Q* The curves corresponding to points of these levels are disjoint on P. ∴ d(P) 3 : a contradiction ≦ P* , Q* : subsurfaces in P, Q s.t. P* and Q* are isotopic (χ (P*)= -2 )

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