KazuhiroIchihara Thinpositionforincompressiblesurfacesin3-manifolds

Thin position for incompressible surfaces in 3-manifolds

Kazuhiro Ichihara

Nihon University

College of Humanities and Sciences

Joint work with

M. Ozawa& J. Hyam Rubinstein The 2nd Pan Pacific International Conference

on Topology and Applications

Thin position argument:

Many good results, for knots & 3-manifolds.

M: a closed irreducible orientable 3-manifold

F: a separating closed orientable incompressible surface in M S: a strongly irreducible Heegaard surface inM

Aim of this talk:

PutF in a kind ofthin position with respect to S.

Recall: a Heegaard splitting

implies a singularfoliation of M by copies ofS.

St: the level surface (0< t <1)

H₁^t, H₂^t: the handlebodies obtained by splittingM along S_t (As t→0,1, H₁^t or H₂^t converges to a graph in M)

We can assume:

• No loop in F ∩S_t bounds a disk on S_t.

• For t small enough, F ∩H₁^t consists of a family of meridian disks.

1st step

Perform all possible boundary compressionsof F ∩H₂^t so that bands of St get pushed acrossF from M− to M+. In a sense,

the effect is to makeS_t∩M₋ thin and S_t∩M₊ thick.

Fix this copy of S_t as level one and denote it byS_t1

with the initialS as S_t0 (where F ∩H₁^t0 are meridian disks).

2nd step

Repeat the process for H₂^t1 bounded byS_t1, but this time interchanging the roles ofM+, M−

so that bands of St1 get pushed across F from M+ to M−.

This will give a new level surface S_t2

for which S_t2 ∩M− is thick and S_t2∩M₊ thin.

We iterate until eventually F meets a handlebody

corresponding toH₂^t in meridian disks only, fort close to 1.

Alternating thin position

Call the obtained surface F in an “alternating” thin position.

Note:

there are a finite number of critical levels ˆt, for 0<ˆt <1, so that at such a level there is a single saddle critical point.

We can find at least one thin surface which is incompressible . Theorem.

LetF, S, M₊,M₋ as above. Then either;

• there is a non-critical level t so that S_t∩M₊ is incompressible and S_t∩M− has compressing disks on both sides of S_t, or the same with M₊, M− interchanged.

• there is a critical level tˆso that S_t∩M₊ (resp. S_t∩M₋) is incompressible for t <ˆt (resp. t >ˆt) with t close to t,ˆ or the same with M₊, M− interchanged.

• there is a critical level tˆso that bothS_t∩M₊, S_t∩M−

ˆ ˆ

If we consider the Hempel distanceof a Heegaard surface, we obtain the following corollary.

Corollary 1.

Under the same settings as in Theorem, suppose thatS has Hempel distanceat least 4. Then only the third possibility in Theorem can occur.

Corollary 2.

LetM₊, M− be compact orientable irreducible 3-manifolds with incompressible boundary∂M₊ ∼=∂M−. Amongst all incompressible and∂-incompressible surfaces in M₊, M−, choose the onesA₊, A− with |∂A₊| ≥ |∂A−| which minimize

h=|χ(A₊)|+|χ(A−)|+ 2(2|∂A₊| − |∂A−| −1).

Thenh gives a lower bound for the absolute value of the Euler characteristic of a Heegaard surface of Hempel distance at least 4 in a closed 3-manifold obtained by gluingM+ and M−

along their boundary.