On distance of Heegaard splittings
and bridge decompositions
Yeonhee Jang
(joint work with Ayako Ido and Tsuyoshi Kobayashi)
Nara Women’s University
October 31, 2013
Nara International Seminar House
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Outline
.
. .1 Curve complex
.
. .2 (Hempel) distance of Heegaard splittings
.
. .3 Proof of Main Theorem
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Curve complex
F = F g, p : orientable surface (of genus g and with p punctures)
s.t. 3g + p − 4 > 0.
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Curve complex
F = F g, p : orientable surface (of genus g and with p punctures)
s.t. 3g + p − 4 > 0.
.
Curve complex
..
The curve complex C(F) is a simplicial complex defined as follows:
.
.
..
k-simplex ↔ k (isotopy classes of) essential simple closed curves on
F which are mutually disjoint.
.
.
0-simplex ↔ (isotopy class of) an essential simple closed curve on F,
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Curve complex
F = F g, p : orientable surface (of genus g and with p punctures)
s.t. 3g + p − 4 > 0.
.
Curve complex
..
The curve complex C(F) is a simplicial complex defined as follows:
.
.
..
k-simplex ↔ k (isotopy classes of) essential simple closed curves on
F which are mutually disjoint.
.
.
0-simplex ↔ (isotopy class of) an essential simple closed curve on F,
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Curve complex
When g = 1 and p = 0 or 1, the definition of the curve complex of
F = F g, p is slightly modified:
.
Curve complex
.
..
The curve complex C(F) is a simplicial complex defined as follows:
.
..
k-simplex ↔ k (isotopy classes of) essential simple closed curves on
F which mutually intersect in one point.
.
.
0-simplex ↔ (isotopy class of) an essential simple closed curve on F,
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Curve complex
.
..
.
.
.
Distance in the curve complex
.
..
For a, b ∈ C (0) (F),
d(a, b) = dC(F) (a, b) := (the smallest number of 1-simplexes in a path
connecting a and b in C(F)).
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Curve complex
.
Distance in the curve complex
.
..
For a, b ∈ C (0) (F),
d(a, b) = dC(F) (a, b) := (the smallest number of 1-simplexes in a path
connecting a and b in C(F)).
.
..
.
.
A shortest path [a0 , a1 , a2 , . . . , a n] connecting two 0-simplexes a0 and
a n is called a geodesic.
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Curve complex
.
Distance in the curve complex
.
..
For a, b ∈ C (0) (F),
d(a, b) = dC(F) (a, b) := (the smallest number of 1-simplexes in a path
connecting a and b in C(F)).
A shortest path [a0 , a1 , a2 , . . . , a n] connecting two 0-simplexes a0 and
a n is called a geodesic.
.
..
d( A, B) = dC(F) ( A, B) := min{d(a, b) | a ∈ A, b ∈ B}.
.
.
For A, B ⊂ C (0) (F),
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Curve complex
.
Distance in the curve complex
.
..
For a, b ∈ C (0) (F),
d(a, b) = dC(F) (a, b) := (the smallest number of 1-simplexes in a path
connecting a and b in C(F)).
A shortest path [a0 , a1 , a2 , . . . , a n] connecting two 0-simplexes a0 and
a n is called a geodesic.
d( A, B) = dC(F) ( A, B) := min{d(a, b) | a ∈ A, b ∈ B}.
.
Fact (Harvey ’81, Hempel ’01)
C(F) is connected (i.e., d(a, b) < ∞ ∀ a, b : essential s.c.c. on F).
.
..
.
..
.
.
.
..
.
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For A, B ⊂ C (0) (F),
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Curve complex
.
Distance in the curve complex
.
..
For a, b ∈ C (0) (F),
d(a, b) = dC(F) (a, b) := (the smallest number of 1-simplexes in a path
connecting a and b in C(F)).
A shortest path [a0 , a1 , a2 , . . . , a n] connecting two 0-simplexes a0 and
a n is called a geodesic.
d( A, B) = dC(F) ( A, B) := min{d(a, b) | a ∈ A, b ∈ B}.
Fact (Harvey ’81, Hempel ’01)
C(F) is connected (i.e., d(a, b) < ∞ ∀ a, b : essential s.c.c. on F).
In fact, d(a, b) ≤ 2 + 2 log2 ι(a, b),
where
ι(a, b) is the (geometric) intersection number of a and b.
.
.
.
..
..
.
.
.
..
.
.
For A, B ⊂ C (0) (F),
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Curve complex
β
ε
α
β
δ
α
δ
γ
ε
γ
.
Distance in the curve complex
..
.
.
.
..
.
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Curve complex
β
ε
α
β
δ
α
δ
γ
ε
γ
.
Distance in the curve complex
..
d(a, b) = 0 ⇔ a = b ( a and b are isotopic),
.
.
.
..
.
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Curve complex
β
ε
α
β
δ
α
δ
γ
ε
γ
.
Distance in the curve complex
..
d(a, b) = 0 ⇔ a = b ( a and b are isotopic),
.
.
..
.
.
d(a, b) = 1 ⇔ a , b and a ∩ b = ∅,
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Curve complex
β
ε
α
β
δ
α
δ
γ
ε
γ
.
Distance in the curve complex
..
d(a, b) = 0 ⇔ a = b ( a and b are isotopic),
.
d(a, b) = 1 ⇔ a , b and a ∩ b = ∅,
.
..
.
.
d(a, b) = 2 ⇔ a ∩ b , ∅ and ∃z: s.c.c. on F disjoint from a ∪ b,
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Curve complex
β
ε
α
β
δ
α
δ
γ
ε
γ
.
Distance in the curve complex
..
d(a, b) = 0 ⇔ a = b ( a and b are isotopic),
.
d(a, b) = 1 ⇔ a , b and a ∩ b = ∅,
d(a, b) = 2 ⇔ a ∩ b , ∅ and ∃z: s.c.c. on F disjoint from a ∪ b,
.
..
.
.
d(a, b) ≥ 3 ⇔ ( a ∩ b , ∅ and) @z: s.c.c. on F disjoint from a ∪ b
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Curve complex
β
ε
α
β
δ
α
δ
γ
ε
γ
.
Distance in the curve complex
..
d(a, b) = 0 ⇔ a = b ( a and b are isotopic),
.
d(a, b) = 1 ⇔ a , b and a ∩ b = ∅,
d(a, b) = 2 ⇔ a ∩ b , ∅ and ∃z: s.c.c. on F disjoint from a ∪ b,
.
.
.
..
d(a, b) ≥ 3 ⇔ ( a ∩ b , ∅ and) @z: s.c.c. on F disjoint from a ∪ b
⇔ F \ (a ∪ b) consists of (open) disks
or once-punctured disks.
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(Hempel) distance of Heegaard splittings
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(Hempel) distance of Heegaard splittings
.
Compression-bodies
V : compression-body if it is obtained from S × [0, 1] (S: closed
orientable surface) by attaching “2-handles” to S × {0}.
.
..
∂-V
∂+V
.
..
.
.
V
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(Hempel) distance of Heegaard splittings
.
Compression-bodies
V : compression-body if it is obtained from S × [0, 1] (S: closed
orientable surface) by attaching “2-handles” to S × {0}.
.
..
∂-V
∂+V
V
.
..
.
.
∂+ V := S × {1}, ∂− V := ∂V \ ∂+ V ,
the genus of ∂+ V is called the genus of the compression-body V .
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(Hempel) distance of Heegaard splittings
.
Compression-bodies
V : compression-body if it is obtained from S × [0, 1] (S: closed
orientable surface) by attaching “2-handles” to S × {0}.
.
..
∂-V
∂+V
V
.
..
A compression-body V is called a handlebody if ∂− V = ∅.
.
.
∂+ V := S × {1}, ∂− V := ∂V \ ∂+ V ,
the genus of ∂+ V is called the genus of the compression-body V .
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(Hempel) distance of Heegaard splittings
.
.
..
∂-V1
∂-V2
∂+V1
∂+V2
V1
g
.
.
(Generalized) Heegaard splittings
M : compact orientable 3-manifold
V1 ∪ F V2 : (genus- g) Heegaard splitting of M ( F: Heegaard surface)
if Vi : genus- g compression-body, V1 ∪ V2 = M and
.V1 ∩ V2 = ∂+ V1 = ∂+ V2 = F.
..
V2
g
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(Hempel) distance of Heegaard splittings
.
.
..
∂-V2
∂+V1
∂+V2
V1
g
V2
g
.
Theorem (due to Moise ’52)
..
Every compact orientable 3-manifold admits a Heegaard splitting of some
genus.
.
..
.
.
.
∂-V1
.
.
(Generalized) Heegaard splittings
M : compact orientable 3-manifold
V1 ∪ F V2 : (genus- g) Heegaard splitting of M ( F: Heegaard surface)
if Vi : genus- g compression-body, V1 ∪ V2 = M and
.V1 ∩ V2 = ∂+ V1 = ∂+ V2 = F.
..
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(Hempel) distance of Heegaard splittings
.
Distance of Heegaard sptlittings (Hempel ’01)
V1 ∪ F V2 : genus- g Heegaard splitting of M
C(F) : curve complex of F
D(Vi ) (i = 1, 2) : the maximal subcomplex of C(F) spanned by curves
(∈ C(F)) that bound disks in Vi
· · · disk complex of Vi
.
..
.
.
.
..
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(Hempel) distance of Heegaard splittings
.
.
..
.
.
Distance of Heegaard sptlittings (Hempel ’01)
V1 ∪ F V2 : genus- g Heegaard splitting of M
C(F) : curve complex of F
D(Vi ) (i = 1, 2) : the maximal subcomplex of C(F) spanned by curves
(∈ C(F)) that bound disks in Vi
· · · disk complex of Vi
{
d(V
∪
V
)
:=
d
(D(V
),
D(V
))
1 F 2
C(F)
1
2 .
.
..
C(F)
D(V1 )
V2
d(L,S)=1
d(
D(V2 )
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(Hempel) distance of Heegaard splittings
.
Known Results on Hempel distance
..
.
.
..
.
.
M = V1 ∪ F V2
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(Hempel) distance of Heegaard splittings
.
Known Results on Hempel distance
..
.
M = V1 ∪ F V2
.
..
.
.
d(V1 ∪ F V2 ) = 0 ⇔ M: reducible or F:“stabilized”,
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(Hempel) distance of Heegaard splittings
.
Known Results on Hempel distance
..
.
M = V1 ∪ F V2
d(V1 ∪ F V2 ) = 0 ⇔ M: reducible or F:“stabilized”,
(Hempel ’01, Hartshorn ’02)
∃S(⊂ M) : essential surface ⇒ d(V1 ∪ F V2 ) ≤ 2 g(S),
.
..
.
.
(Scharlemann-Tomova ’06)
∃F′ : “alternate” Heegaard surface of M ⇒ d(V1 ∪ F V2 ) ≤ 2g(F′ ),
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(Hempel) distance of Heegaard splittings
.
Known Results on Hempel distance
..
.
M = V1 ∪ F V2
d(V1 ∪ F V2 ) = 0 ⇔ M: reducible or F:“stabilized”,
(Hempel ’01, Hartshorn ’02)
∃S(⊂ M) : essential surface ⇒ d(V1 ∪ F V2 ) ≤ 2 g(S),
(Scharlemann-Tomova ’06)
∃F′ : “alternate” Heegaard surface of M ⇒ d(V1 ∪ F V2 ) ≤ 2g(F′ ),
“Casson-Gordon’s rectangle condition” ⇒ d(V1 ∪ F V2 ) ≥ 2,
.
..
.
.
(Hempel ’01, Berge, Scharlemann ’12) gave sufficient conditions for
d(V1 ∪ F V2 ) ≥ 3,
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(Hempel) distance of Heegaard splittings
.
Known Results on Hempel distance
..
.
M = V1 ∪ F V2
d(V1 ∪ F V2 ) = 0 ⇔ M: reducible or F:“stabilized”,
(Hempel ’01, Hartshorn ’02)
∃S(⊂ M) : essential surface ⇒ d(V1 ∪ F V2 ) ≤ 2 g(S),
(Scharlemann-Tomova ’06)
∃F′ : “alternate” Heegaard surface of M ⇒ d(V1 ∪ F V2 ) ≤ 2g(F′ ),
“Casson-Gordon’s rectangle condition” ⇒ d(V1 ∪ F V2 ) ≥ 2,
.
..
(Hempel ’01, Evans ’06, Minsky-Moriah-Schleimer ’07, Lustig-Moriah
’09, ...) gave Heegaard splittings with d(V1 ∪ F V2 ) ≥ n for ∀n.
.
.
(Hempel ’01, Berge, Scharlemann ’12) gave sufficient conditions for
d(V1 ∪ F V2 ) ≥ 3,
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(Hempel) distance of Heegaard splittings
.
Theorem 1 (Ido-J.-Kobayashi)
.
.
.
..
For any integers n ≥ 2 and g ≥ 2,
∃V1 ∪ F V2 : genus- g Heegaard splitting (of a closed 3-manifold) with
.d(V1 ∪ F V2 ) = n.
..
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(Hempel) distance of Heegaard splittings
.
.
..
.
.
.
Theorem 1 (Ido-J.-Kobayashi)
.
..
For any integers n ≥ 2 and g ≥ 2,
∃V1 ∪ F V2 : genus- g Heegaard splitting (of a closed 3-manifold) with
.d(V1 ∪ F V2 ) = n.
..
.
.
Remark
.
..
Examples of closed orientable 3-manifolds with genus- g Heegaard
splittings with n = 0 or 1 are known (∀(g, n) , (2, 1)).
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(Hempel) distance of Heegaard splittings
.
.
..
Theorem 1 was proved independently by [Qiu-Zou-Guo], and also by
[Yoshizawa] in case n is even.
.
.
.
Theorem 1 (Ido-J.-Kobayashi)
.
..
For any integers n ≥ 2 and g ≥ 2,
∃V1 ∪ F V2 : genus- g Heegaard splitting (of a closed 3-manifold) with
.d(V1 ∪ F V2 ) = n.
..
.
.
Remark
.
..
Examples of closed orientable 3-manifolds with genus- g Heegaard
splittings with n = 0 or 1 are known (∀(g, n) , (2, 1)).
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Proof of Theorem 1
.
Theorem 1 (Ido-J.-Kobayashi)
.
.
.
..
For any integers n ≥ 2 and g ≥ 2,
∃V1 ∪ F V2 : genus- g Heegaard splitting (of a closed 3-manifold) with
.d(V1 ∪ F V2 ) = n.
..
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Proof of Theorem 1
.
Theorem 1 (Ido-J.-Kobayashi)
.
.
.
..
For any integers n ≥ 2 and g ≥ 2,
∃V1 ∪ F V2 : genus- g Heegaard splitting (of a closed 3-manifold) with
.d(V1 ∪ F V2 ) = n.
..
(Step 1) Construct a geodesic of length (n + 2) in C(F).
(Key tool : Masur-Minsky’s subsurface projection)
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Proof of Theorem 1
.
Theorem 1 (Ido-J.-Kobayashi)
.
.
.
..
For any integers n ≥ 2 and g ≥ 2,
∃V1 ∪ F V2 : genus- g Heegaard splitting (of a closed 3-manifold) with
.d(V1 ∪ F V2 ) = n.
..
(Step 1) Construct a geodesic of length (n + 2) in C(F).
(Key tool : Masur-Minsky’s subsurface projection)
(Step 2) Glue two compression-bodies with “simple” disk complexes
(using the above geodesic).
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Proof of Theorem 1
.
Theorem 1 (Ido-J.-Kobayashi)
V1
.
.
.
..
For any integers n ≥ 2 and g ≥ 2,
∃V1 ∪ F V2 : genus- g Heegaard splitting (of a closed 3-manifold) with
.d(V1 ∪ F V2 ) = n.
..
(Step 1) Construct a geodesic of length (n + 2) in C(F).
(Key tool : Masur-Minsky’s subsurface projection)
(Step 2) Glue two compression-bodies with “simple” disk complexes
(using the above geodesic).
(Step 3) Attach handlebodies in a way “complicated enough” to the
boundary of the manifold.
V2
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Subsurface projection
F : surface, X : essential non-simple subsurface of F,
P(C (0) (X)) : the power set of C (0) (X).
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Subsurface projection
F : surface, X : essential non-simple subsurface of F,
P(C (0) (X)) : the power set of C (0) (X).
.
Subsurface projection π X : C (0) (F) → P(C (0) (X))
..
.
γ
F
X
γ X={a1,a2} πX(γ)={∂N(∂X a1),∂N(∂X a2)}
ignore
inessential
loops
γ
a1
a2
.
.
.
..
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Subsurface projection
F : surface, X : essential non-simple subsurface of F,
P(C (0) (X)) : the power set of C (0) (X).
.
Subsurface projection π X : C (0) (F) → P(C (0) (X))
..
.
γ1
γ
F
γ2
X
πX(γ)={γ1 ,γ2 }
.
..
.
.
γ
γ X
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Subsurface projection
π X : C (0) (F) → P(C (0) (X)) : subsurface projection
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Subsurface projection
.
π X : C (0) (F) → P(C (0) (X)) : subsurface projection
.
Lemma 1 (Masur-Minsky ’00)
.
..
Let [a0 , a1 , a2 , . . . , a m] is a geodesic in C(F) such that each ai intersects
.X. Then diamC(X) (π X (a0 ) ∪ π X (a m))) ≤ 2m.
..
.
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Subsurface projection
.
.
π X : C (0) (F) → P(C (0) (X)) : subsurface projection
.
Lemma 1 (Masur-Minsky ’00)
.
..
Let [a0 , a1 , a2 , . . . , a m] is a geodesic in C(F) such that each ai intersects
.X. Then diamC(X) (π X (a0 ) ∪ π X (a m))) ≤ 2m.
..
.
.
Lemma 2 (Li ’12)
.
..
Let V be a compression-body with ∂+ V = F. Assume that every essential
disk in V intersects ∂X and that V is not an I-bundle over X. Then
.diamC(X) (π X (D(V))) ≤ 12.
..
.
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Step 1: Path of length n ( n: even)
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Step 1: Path of length n ( n: even)
F : closed surface of (given) genus g(> 1)
α0 , α1 , α2 : essential s.c.c.s on F s.t.
α1
α0
α0 ∩ α2 =(1 point),
α1 : disjoint from α0 ∪ α2 .
α2
F
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Step 1: Path of length n ( n: even)
F : closed surface of (given) genus g(> 1)
α0 , α1 , α2 : essential s.c.c.s on F s.t.
α1
α0
α0 ∩ α2 =(1 point),
α1 : disjoint from α0 ∪ α2 .
α2
F
X2 := Cl(F \ N(α2 ))
f2 : F → F homeomorphism s.t.
f2 (N(α2 )) = N(α2 ),
diamC(X2 ) (π X2 (α0 ) ∪ π X2 ( f2 (α0 ))) ≥ 2n.
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Step 1: Path of length n ( n: even)
F : closed surface of (given) genus g(> 1)
α0 , α1 , α2 : essential s.c.c.s on F s.t.
α1
α0
α0 ∩ α2 =(1 point),
α2
α1 : disjoint from α0 ∪ α2 .
F
X2 := Cl(F \ N(α2 ))
f2 : F → F homeomorphism s.t.
f2 (N(α2 )) = N(α2 ),
diamC(X2 ) (π X2 (α0 ) ∪ π X2 ( f2 (α0 ))) ≥ 2n.
α3 := f2 (α1 ), α4 := f2 (α0 ).
f4
α0
α1
α2
α3
α4
5
α6
...
α n-1 αn
f2
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Step 1: Path of length n ( n: even)
F : closed surface of (given) genus g(> 1)
α0 , α1 , α2 : essential s.c.c.s on F s.t.
α1
α0
α0 ∩ α2 =(1 point),
α2
α1 : disjoint from α0 ∪ α2 .
F
Xi := Cl(F \ N(αi )) (i = 2, 4, . . . , n − 2)
fi : F → F homeomorphism s.t.
fi (N(αi )) = N(αi ),
diamC(Xi ) (π Xi (α0 ) ∪ π Xi ( fi (αi−2 ))) ≥ 2n.
αi+1 := fi (αi−1 ), αi+2 := fi (αi−2 ).
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
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Step 1: Path of length n ( n: even)
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
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Step 1: Path of length n ( n: even)
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
.
.
.
.
Lemma 3
..
.dC(F) (α0 , α k ) = k for ∀k = 2, 4, . . . , n.
..
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Step 1: Path of length n ( n: even)
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
.
.
.
.
Lemma 3
..
.dC(F) (α0 , α k ) = k for ∀k = 2, 4, . . . , n.
..
∵ Done if k = 2.
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Step 1: Path of length n ( n: even)
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
.
.
.
.
Lemma 3
..
.dC(F) (α0 , α k ) = k for ∀k = 2, 4, . . . , n.
..
∵ Done if k = 2.
So we assume that k > 2 and that dC(F) (α0 , α k−2 ) = k − 2.
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Step 1: Path of length n ( n: even)
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
.
.
Lemma 3
.
..
.dC(F) (α0 , α k ) = k for ∀k = 2, 4, . . . , n.
..
.
∵ Done if k = 2.
So we assume that k > 2 and that dC(F) (α0 , α k−2 ) = k − 2.
Assume that ∃ a path [β0 , β1 , . . . , β l ] s.t. β0 = α0 , β l = α k and l < k(≤ n)
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Step 1: Path of length n ( n: even)
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
.
.
Lemma 3
.
..
.dC(F) (α0 , α k ) = k for ∀k = 2, 4, . . . , n.
..
.
∵ Done if k = 2.
So we assume that k > 2 and that dC(F) (α0 , α k−2 ) = k − 2.
Assume that ∃ a path [β0 , β1 , . . . , β l ] s.t. β0 = α0 , β l = α k and l < k(≤ n)
{ ∃ j s.t. β j = α k−2
(by “diamC(X k−2 ) (π X k−2 (α0 ) ∪ π X k−2 (α k ))) ≥ 2n” and Lemma 1)
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Step 1: Path of length n ( n: even)
f4
α0
α1
α2
α3
α4
α5
α6
...
α n-1 αn
f2
.
.
Lemma 3
.
..
.dC(F) (α0 , α k ) = k for ∀k = 2, 4, . . . , n.
..
.
∵ Done if k = 2.
So we assume that k > 2 and that dC(F) (α0 , α k−2 ) = k − 2.
Assume that ∃ a path [β0 , β1 , . . . , β l ] s.t. β0 = α0 , β l = α k and l < k(≤ n)
{ ∃ j s.t. β j = α k−2
(by “diamC(X k−2 ) (π X k−2 (α0 ) ∪ π X k−2 (α k ))) ≥ 2n” and Lemma 1)
{ j ≥ k − 2 and l − j ≥ 2, a contradiction.
18 / 30
Extending geodesics
♣ Similarly, for any given two geodesics [a0 , a1 , . . . , a l ] and
[b0 , b1 , . . . , b m] in C(F) such that a l and b0 are non-separating on F, we
can find a self-homeomorphism h of F such that
diamC(F\al ) (π F\al (a0 ), π F\al ( f (b m))) > 2(l + m).
Then [a0 , a1 , . . . , a l = f (b0 ), f (b1 ), . . . , f (b m)] is a geodesic in C(F).
b0
a0
al
bm
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Extending geodesics
♣ Similarly, for any given two geodesics [a0 , a1 , . . . , a l ] and
[b0 , b1 , . . . , b m] in C(F) such that a l and b0 are non-separating on F, we
can find a self-homeomorphism h of F such that
diamC(F\al ) (π F\al (a0 ), π F\al ( f (b m))) > 2(l + m).
Then [a0 , a1 , . . . , a l = f (b0 ), f (b1 ), . . . , f (b m)] is a geodesic in C(F).
b0
a0
al
bm
20 / 30
Extending geodesics
♣ Similarly, for any given two geodesics [a0 , a1 , . . . , a l ] and
[b0 , b1 , . . . , b m] in C(F) such that a l and b0 are non-separating on F, we
can find a self-homeomorphism h of F such that
diamC(F\al ) (π F\al (a0 ), π F\al ( f (b m))) > 2(l + m).
Then [a0 , a1 , . . . , a l = f (b0 ), f (b1 ), . . . , f (b m)] is a geodesic in C(F).
b0
a0
al
bm
21 / 30
Step 2: Heegaard splitting with distance n ( n: even)
22 / 30
Step 2: Heegaard splitting with distance n ( n: even)
For i = 1, 2,
Vi := F g,0 × I ∪ (a 2-handle along D0 )
i
D0i
Vi
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Step 2: Heegaard splitting with distance n ( n: even)
For i = 1, 2,
Vi := F g,0 × I ∪ (a 2-handle along D0 )
i
D0i
Vi
.
Lemma 4
..
D0 is the only non-separating disk in Vi (up to isotopy),
.
i
∀ essential separating disk in Vi : disjoint from D0 .
i
.
.
.
..
22 / 30
Step 2: Heegaard splitting with distance n ( n: even)
For i = 1, 2,
Vi := F g,0 × I ∪ (a 2-handle along D0 )
i
D0i
Vi
.
Lemma 4
..
D0 is the only non-separating disk in Vi (up to isotopy),
.
∀ essential separating disk in Vi : disjoint from D0 .
i
.
..
.
Let [α0 , α1 , . . . , α n+2 ] be a geodesic (⊂ C(F)) obtained in Step 1, and
identify ∂+ Vi and F so that ∂ D0 = α0 and ∂ D0 = α n+2 .
1
2
By Lemmas 3 and 4, V1 ∪ F V2 is a Heegaard splitting with distance ≥ n.
D(V1)
α0
α1
α2 . . .
αn
.
i
αn+2
α n+1
D(V2)
22 / 30
Step 2: Heegaard splitting with distance n ( n: even)
On the other hand, we can find a loop α′ which bounds a separating disk
1
in V1 and disjoint from α0 ∪ α2 .
α0
D10
α2
V1
α1′
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Step 2: Heegaard splitting with distance n ( n: even)
On the other hand, we can find a loop α′ which bounds a separating disk
1
in V1 and disjoint from α0 ∪ α2 .
α0
D10
α2
V1
α1′
Similarly, we can find a loop α′ which bounds a separating disk in V2
n+1
and disjoint from α n ∪ α n+2 .
23 / 30
Step 2: Heegaard splitting with distance n ( n: even)
On the other hand, we can find a loop α′ which bounds a separating disk
1
in V1 and disjoint from α0 ∪ α2 .
α0
D10
α2
V1
α1′
Similarly, we can find a loop α′ which bounds a separating disk in V2
n+1
and disjoint from α n ∪ α n+2 .
{ Hempel distance of V1 ∪ F V2 is exactly n.
D(V1)
α0
α1
α′1
α2 . . .
αn α′n+1 αn+2
α n+1
D(V2)
23 / 30
Step 2: Heegaard splitting with distance n ( n: even)
On the other hand, we can find a loop α′ which bounds a separating disk
1
in V1 and disjoint from α0 ∪ α2 .
α0
D10
α2
V1
α1′
Similarly, we can find a loop α′ which bounds a separating disk in V2
n+1
and disjoint from α n ∪ α n+2 .
{ Hempel distance of V1 ∪ F V2 is exactly n.
D(V1)
α0
α1
α′1
α2 . . .
αn α′n+1 αn+2
α n+1
D(V2)
24 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
25 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
∂-V1
D1
V1
V2
F2
F1
Let D1 be the essential separating disk in V1 with ∂ D1 = α′ , then D1 cuts
1
V1 into a solid torus V 1 and the other component V 2 ( ∂− V1 × I).
1
1
25 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
∂-V1
D1
V1
V2
F2
F1
Let D1 be the essential separating disk in V1 with ∂ D1 = α′ , then D1 cuts
1
V1 into a solid torus V 1 and the other component V 2 ( ∂− V1 × I).
1
1
Let Fi (i = 1, 2) be the subsurface ∂+ V i ∩ F of F(= ∂+ V1 ).
1
25 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
∂-V1
D1
V1
V2
F2
F1
Let D1 be the essential separating disk in V1 with ∂ D1 = α′ , then D1 cuts
1
V1 into a solid torus V 1 and the other component V 2 ( ∂− V1 × I).
1
1
Let Fi (i = 1, 2) be the subsurface ∂+ V i ∩ F of F(= ∂+ V1 ).
1
Let π Fi : C (0) (F) → PC (0) (Fi ) : subsurface projection, and
let P : C (0) (F2 ) → C (0) (F2 ∪ D1 ) → C (0) (∂− V1 ) : natural projection.
25 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
∂-V1
D1
V1
V2
F2
F1
Let D1 be the essential separating disk in V1 with ∂ D1 = α′ , then D1 cuts
1
V1 into a solid torus V 1 and the other component V 2 ( ∂− V1 × I).
1
1
Let Fi (i = 1, 2) be the subsurface ∂+ V i ∩ F of F(= ∂+ V1 ).
1
Let π Fi : C (0) (F) → PC (0) (Fi ) : subsurface projection, and
let P : C (0) (F2 ) → C (0) (F2 ∪ D1 ) → C (0) (∂− V1 ) : natural projection.
By Lemma 2 (when n ≥ 3),
diamC(Fi ) (π Fi (D(V2 ))) ≤ 12,
and hence
diamC(∂− V1 ) (Pπ F2 (D(V2 ))) ≤ 12. · · · (1)
25 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
Let H be a handlebody of genus (g − 1) and choose a homeomorphism
h : ∂H → ∂− V1 so that
dC(∂− V1 ) (Pπ F2 (D(V2 )), h∗ (D(H))) ≥ 2n. · · · (2)
26 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
Let H be a handlebody of genus (g − 1) and choose a homeomorphism
h : ∂H → ∂− V1 so that
dC(∂− V1 ) (Pπ F2 (D(V2 )), h∗ (D(H))) ≥ 2n. · · · (2)
(This follows from (1) since dC(∂H) (D(∂H)), h n(D(∂H)) → ∞ as n → ∞
for h : pseudo-Anosov [Hempel ’01].)
PπF2D(V2) C( ∂-V1 )
D(H)
≧2n
hnD(H)
26 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
Let H be a handlebody of genus (g − 1) and choose a homeomorphism
h : ∂H → ∂− V1 so that
dC(∂− V1 ) (Pπ F2 (D(V2 )), h∗ (D(H))) ≥ 2n. · · · (2)
(This follows from (1) since dC(∂H) (D(∂H)), h n(D(∂H)) → ∞ as n → ∞
for h : pseudo-Anosov [Hempel ’01].)
PπF2D(V2) C( ∂-V1 )
D(H)
≧2n
hnD(H)
Let V ∗ := V1 ∪ h H.
1
26 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
Replace the gluing homeomorphism between ∂+ V1 and ∂+ V2 with another
one which is different from the original one only on F1 and satisfies
dC(F1 ) (π F1 (D(V2 )), ∂ D01 ) ≥ 2n. · · · (3)
∂-V1
D10
F1
D1
F2
27 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
Replace the gluing homeomorphism between ∂+ V1 and ∂+ V2 with another
one which is different from the original one only on F1 and satisfies
dC(F1 ) (π F1 (D(V2 )), ∂ D01 ) ≥ 2n. · · · (3)
∂-V1
D10
F1
D1
F2
Note
C( F1 )
πF1D(V2)
PπF2D(V2)
≧2n
≧2n
C( ∂-V1 )
h*D(H)
0
∂D 1
27 / 30
.
..
Step 3: Attaching handlebodies ( n ≥ 3)
.
d(V1∗ ∪ F V2 ) = n.
.
.
.
Proposition
..
28 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
.
d(V ∗ ∪ V ) = n.
1 F 2
.
..
Proof. Assume on the contrary that dC(F) (D(V ∗ ), D(V2 )) < n.
.
.
.
Proposition
..
1
28 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
.
d(V ∗ ∪ V ) = n.
1 F 2
.
..
Proof. Assume on the contrary that dC(F) (D(V ∗ ), D(V2 )) < n.
1
Then ∃a ∈ D(V ∗ ) \ D(V1 ), ∃b ∈ D(V2 ) s.t. dC(F) (a, b) < n.
.
.
.
Proposition
..
1
28 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
.
d(V ∗ ∪ V ) = n.
1 F 2
.
..
.
Proof. Assume on the contrary that dC(F) (D(V ∗ ), D(V2 )) < n.
1
Then ∃a ∈ D(V ∗ ) \ D(V1 ), ∃b ∈ D(V2 ) s.t. dC(F) (a, b) < n.
1
For any geodesic [a0 , a1 , . . . , a m] connecting a and b, every ai intersects
∂ D1 (and hence F1 and F2 ).
.
.
Proposition
..
28 / 30
.
Proposition
..
Step 3: Attaching handlebodies ( n ≥ 3)
.
.
d(V ∗ ∪ V ) = n.
1 F 2
.
..
.
Proof. Assume on the contrary that dC(F) (D(V ∗ ), D(V2 )) < n.
1
Then ∃a ∈ D(V ∗ ) \ D(V1 ), ∃b ∈ D(V2 ) s.t. dC(F) (a, b) < n.
1
For any geodesic [a0 , a1 , . . . , a m] connecting a and b, every ai intersects
∂ D1 (and hence F1 and F2 ). By Lemma 1, we have
diamC(F1 ) (π F1 (a) ∪ π F1 (b)) < 2n, · · · (4)
diamC(∂− V1 ) (Pπ X (a) ∪ Pπ X (b)) < 2n. · · · (5)
C( F1 )
πF1D(V2)
πF1(b)
＜2n
πF1(a)
PπF2D(V2)
PπF2(b)
≧2n
0
∂D 1
≧2n
C( ∂-V1 )
h*D(H)
＜2n
PπF2(a)
28 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
On the other hand, we can see by an “outermost disk argument” that
dC(F1 ) (π F1 (a), ∂ D01 ) = 0 or dC(∂− V1 ) (Pπ F2 (a), h∗ (D(H))) = 0.
πF1D(V2)
πF1(b)
C( F1 )
PπF2D(V2)
PπF2(b)
＜2n
πF1(a)
≧2n
0
∂D 1
≧2n
C( ∂-V1 )
h*D(H)
＜2n
PπF2(a)
29 / 30
Step 3: Attaching handlebodies ( n ≥ 3)
On the other hand, we can see by an “outermost disk argument” that
dC(F1 ) (π F1 (a), ∂ D01 ) = 0 or dC(∂− V1 ) (Pπ F2 (a), h∗ (D(H))) = 0.
πF1D(V2)
πF1(b)
C( F1 )
PπF2D(V2)
PπF2(b)
＜2n
πF1(a)
≧2n
0
∂D 1
≧2n
C( ∂-V1 )
h*D(H)
＜2n
PπF2(a)
This together with the inequalities (4) and (5) implies
dC(F1 ) (π F1 (D(V2 )), ∂ D01 ) < 2n, or
dC(∂− V1 ) (Pπ F2 (D(V2 )), h∗ (D(H))) < 2n,
which contradicts the inequality (2) or (3).
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Step 3: Attaching handlebodies ( n ≥ 3)
On the other hand, we can see by an “outermost disk argument” that
dC(F1 ) (π F1 (a), ∂ D01 ) = 0 or dC(∂− V1 ) (Pπ F2 (a), h∗ (D(H))) = 0.
C( F1 )
πF1D(V2)
πF1(b)
PπF2D(V2)
PπF2(b)
＜2n
πF1(a)
≧2n
0
∂D 1
≧2n
C( ∂-V1 )
h*D(H)
＜2n
PπF2(a)
This together with the inequalities (4) and (5) implies
dC(F1 ) (π F1 (D(V2 )), ∂ D01 ) < 2n, or
dC(∂− V1 ) (Pπ F2 (D(V2 )), h∗ (D(H))) < 2n,
which contradicts the inequality (2) or (3).
Similarly, we can fill ∂− V2 with a handlebody to obtain a Heegaard splitting
V ∗ ∪ F V ∗ with d(V ∗ ∪ F V ∗ ) = n, where each V ∗ is a handlebody.
1
2
1
2
i
29 / 30