On Reeb graphs derived from Heegaard splittings with distance 2g

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)

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.

S C

A∪^PB : 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(P)=0

D^A

D^B

d(P)=1

d(P)=2

DA

D^A

D^B

D^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.

( 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 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

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 = Σ^A

∪

∪

A

∪

spine of handlebody

Ps=f (s) ^-1 (s∈(0, 1)) : level surface of f 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.

: a sweep-out of P

2. Main result

2-1. Sweep-out

0

M

Σ

Σ

P

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

-1

2-2. Reeb graph

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 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 :

Lemma. [Tao Li, Cf. Johnson]

∃sⁱ, s^j (sⁱ<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^si-ε∩Q contains a curve bounding a meridian disk in A, and P^sj+ε∩Q contains a curve bounding a meridian disk in B.

Thm. [Cf. Scharlemann-Tomova, 2006]

sⁱ ess. curves on P and Q s^j

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

-1

2-2. Reeb graph

sⁱ+ε s^j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

v

v

v

v

2g-1

Main result

s

s

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

P^X,P^Y : P∩X, P∩Y Q^A, Q^B : Q∩A, Q∩B s

P

Q

A B

X Y

P^X P^Y

Q^A

Q^B

Suppose Q is not isotopic to P

M

and d(P)=4 1

1

2

3

3

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.

P

Q

A B

X Y

M

∂-comp. disks

P^X P^Y

Q^A

Q^B

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 of these levels are disjoint on P.

P*= Q*

1

1

2

3

3

∴ d(P) 3 : a contradiction ≦

Thank you