On Reeb graphs derived from Heegaard splittings with distance 2 g ( - - PowerPoint PPT Presentation

井戸 絢子 (奈良女子大学 D1)

On Reeb graphs derived from Heegaard splittings with distance 2g

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. ・ M = A ∪ B , ・ A ∩ B = ∂A = ∂B = P : Heegaard surface

S : a closed orientable surface of genus ≥ 2 Def.(curve complex) CS : 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.

S Cs

The distance between two vertices is the smallest number of edges in any path between the vertices in CS.

A∪PB : a genus g (≥2) Heegaard splitting of M

• Def. (Hempel distance)

D(A), D(B) : the subsets of the curve complex CP corresponding to curves on P bounding disks in A , B. d(P):= d(D(A), D(B)) : the Hempel distance of P d(P)=0

DA DB

d(P)=1 d(P)=2

DA DA DB DB

• Thm. [Cf. Scharlemann-Tomova, 2006]

P, Q : genus g Heegaard surfaces in M Suppose that Q is not isotopic to P, then d(P) 2 ≦ g. ( i.e. d(P)>2g ⇒ Q is isotopic to P ) 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
• f 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

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.

Question P, Q : genus g Heegaard surfaces in M Suppose d(P)=2g. Is Q isotopic to P ? Question P, Q : genus g Heegaard surfaces in M Suppose d(P)=2g. 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)=2g. And as an application, we give an alternative proof of the above fact.

M = ΣA∪P×(0,1)∪ΣB A∪PB : a Heegaard splitting of M ΣA, ΣB : spines of A, B

spine of handlebody

Ps=f (s) (s (0, 1))

∈

: level surface of f

• 1

We may suppose P×(0,1) → (0,1) can be extended to a smooth map f : M → [0,1] s.t. ・f (ΣA)=0, f (ΣB)=1 ・each level surface is isotopic to P. : a sweep-out of P

• 2. Main result

2-1. Sweep-out 1

M

ΣA ΣB Ps

P, Q: Heegaard surfaces in M f :a sweep out of P f |Q : Q → [0,1] x~y (x, y ∈ M) ⇔ ∃s [0,1]

∈ s.t. x, y are in the same comp. of (f|Q) (s).

Q/~ : Reeb graph

Reeb graph of Q

• 1

2-2. Reeb graph

• Lemma. [Tao Li, Cf. Johnson]

P, Q : Heegaard surfaces in M ΣA, ΣB : spines of A, B fix f : sweep out of P

Suppose that d(P) 2. Then ≧ Q can be isotoped so that (1) Q∩ΣA and Q∩ΣB consists of finitely many points ; (2) Q is transverse to each Ps, s∈（0,1）, except for finitely many critical levels s1,..., sn (0,1) ; ∈ (3) at each critical level si, Q is transverse to Psi,

except for a single saddle or circle tangency;

(4) at each regular level Ps, every component of Q∩Ps is an essential curve on Ps.

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.

Q :

circle tangency:

• Lemma. [Tao Li, Cf. Johnson]

∃si, sj (si<sj) s.t. ∀s (

∈ si, sj) : regular value, each comp. of Ps∩Q is essential on both Ps and

• Q. For any small ε>0, Psi-ε∩Q contains a curve bounding a meridian disk in A, and Psj+ε∩Q

contains a curve bounding a meridian disk in B.

• Thm. [Cf. Scharlemann-Tomova, 2006]

P, Q : genus g Heegaard surfaces in M Suppose that Q is not isotopic to P, then d(P) 2 ≦ g.

si sj

• 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.

Q

P, Q: Heegaard surfaces in M f :a sweep out of P f |Q : Q → [0,1] x~y (x, y ∈ M) ⇔ ∃s [0,1]

∈ s.t. x, y are in the same comp. of (f|Q) (s).

Q/~ : Reeb graph

Reeb graph of Q

• 1

2-2. Reeb graph

si+ε sjｰε

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.

d(D(A),c)=1

1 2 3 3 2g-1 2g-3 2g-3

v2 v1 v2g-2 v2g-1

2g-2 2g-1

Main result

si+ε sjｰε

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.

Outline of proof PX, PY : P∩X, P∩Y

QA, QB : Q∩A, Q∩B s P Q

A B X Y PX PY QA QB

M

Suppose Q is not isotopic to P and d(P)=4 1 1 2 3 3

• Remark. PX (resp. PY, QA, QB) is incompressible in X (resp. X, Y, A, B)
• Lem. [Rubinstein-Scharlemann 1999]

Suppose that PX, PY, QA, QB are incompressible in respectively X, Y , A, B. Then PX ∂–compresses to one of QA or QB , and PY ∂–compresses to the other.

P Q

A B X Y

M

∂-comp. disks

• 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. PX PY QA QB

P Q

A B X Y

M

P*=Q* P*, Q* : subsurfaces in P, Q s.t. P* and Q* are isotopic (χ(P*)= -2 ) The curves corresponding to points

• f these levels are disjoint on P.

P*= Q* 1 1 2 3 3

∴ d(P) 3 : a contradiction ≦

Thank you