On keen Heegaard splitting
(joint work with Ayako Ido and Yeonhee Jang)
Tsuyoshi Kobayashi
Nara Women’s Univ.
Topology and Geometry of Low-dimensional Manifolds
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Contents
Part 1
Part 2
Part 3
Part 4
Preliminaries
Historical Background and Main Result
Technical Background
Idea of proof of Main Result
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Part 1
Preliminaries
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(Hempel) distance of Heegaard splittings
C : compression-body if
it is obtained from S × [0, 1] ∪ (3-ball)
(S : closed orientable surface) by attaching“ 1-handles ”to
S × {1} ∪ ∂3-ball.
Notation:
∂− C := S × {0},
∂+ C := ∂C \ ∂− C,
The genus of ∂+ C is called the genus of the compression-body C.
A compression body C is called a handlebody if ∂− C = ∅.
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Heegaard splittings
Heegaard splittings
M : compact orientable 3-manifold
C1 ∪Σ C2 : (genus-g) Heegaard splitting of M (Σ: Heegaard surface) if
Ci : genus-g compression-body (⊂ M)
s.t.C1 ∪ C2 = M, and C1 ∩ C2 = ∂+ C1 = ∂+ C2 = Σ.
It is known that any M admits a Heegaard splitting.
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An alternate expression of H.S.
Σ = Σg : orientable surface of genus g ≥ 2.
MCG(Σ): Mapping class group of Σ: the group of automorphisms of Σ.
An alternate expression of the H.S. C1 ∪Σ C2 is:
C1 ∪Σ C2 = C1 ∪h C2 ,
where h : Σ(= ∂+ C1 ) → Σ(= ∂+ C2 ) ∈ MCG(Σ)
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Curve complex
The curve complex was introduced by Harvey and has been used to
prove many deep results about the mapping class group ( [Harv]).
[Harv] W. J. Harvey. Boundary structure of the modular group. In I. Kra and
B. Maskit, editors, Riemann Surfaces and Related Topics: Proceedings of the
1978 Stony Brook Conference, volume 97 of Ann. of Math. Stud. Princeton,
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Curve complex
The curve complex C(Σ) is a simplicial complex defined as follows:
0-simplex ↔ (isotopy class of) an essential simple closed curve on Σ,
two 0-simplices are joined by a 1-simplex if they can be realized by
disjoint curves.
n-simplex ↔ disjoint non-parallel n + 1 simple closed curves on Σ.
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Distance in the curve complex
For vertices a and b (∈ C(Σ)(0) ),
d(a, b) = dC(Σ) (a, b) := (the smallest number of 1-simplexes
in a path connecting a and b in C(Σ)).
A shortest path [a0 , a1 , a2 , . . . , an ] connecting two vertices a0 and an
is called a geodesic.
For sets A and B(⊂ C(Σ)(0) ),
d(A, B) = dC(Σ) (A, B) := min{d(a, b)|a ∈ A, b ∈ B}.
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Let:
M: compact orientable 3-manifold
C1 ∪Σ C2 : Heegaard splitting of M
C(Σ): curve complex of Σ
D(Ci ) (⊂ C(Σ)(0) ): disk complex of Ci ,
(i.e. a ∈ D(Ci ) ⇔ a bounds a disk in Ci .)
Definition (Distance of Heegaard splitting)
d(C1 ∪Σ C2 ) (:= dC(Σ) (D(C1 ), D(C2 ))
)
= min{d(a, b)|a ∈ D(C1 ), b ∈ D(C2 )}
John Hempel, 3-manifolds as viewed from the curve complex,
Topology 40 (2001) 631-657
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Note: h ∈ MCG(Σ) induces a bijection on C(Σ)(0) , and it naturally
extends to an isometry on C(Σ) denoted by h̃.
Theorem(Hempel)
∃g: Σ → Σ s.t. d(C1 ∪gn C2 ) → ∞ (n → ∞)
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Part 2
Historical Background and Main Result
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Historical background
Masur-Minsky: C(Σ) is δ-hyperbolic.
(Roughly speaking: C(Σ) is quasi-equivalent to hyperbolic space.)
[MM1] H. Masur and Y. Minsky. Geometry of the complex of curves I:
hyperbolicity, Invent. Math. 138 (1999), 103–149.
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Historical background
Note: h ∈ MCG(Σ) induces a bijection on C(Σ)(0) , and it naturally
extends to an isometry on C(Σ) denoted by h̃.
Expected:
If h ∈ MCG(Σ) is pseudo-Anosov,
then, h̃ behaves like a hyperbolic isometry
on a hyperbolic space.
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Historical background
This is justified by Abrams-Schleimer to show:
Theorem
Suppose h ∈ MCG(Σ): pseudo-Anosov with certain technical property.
Then: d(V1 ∪hn V2 ) has quasi-linear growth with respect to n.
A.Abrams, and S.Schleimer, Distance of Heegaard splittings,
Geometry & Topology, 9(2005), 95–119.
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Historical background
Summary
Geometry of C(Σ) up to quasi-equivalence (Coarse geometry of C(Σ))
is similar to hyperbolic geometry (and, this intuition works very well).
However,
Theorem (Birman-Menasco)
∃geodesic segment γ in C(Σ) which cannot be extended to a longer
geodesic segment.
cf. Any geodesic segment in Hn is infinitely extendable.
Birman-Menasco, The curve complex has dead ends, Geometry
Dedicata, Published online: 07 April, 2014.
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Historical background
This fact represents a difference between Coarse Geometry and (Fine)
Geometry of C(Σ).
Motivation
It may be interesting to study Fine Geometry of C(Σ) (like the paper of
Birman-Menasco) particularly from the viewpoint of Heegaard Theory.
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keen Heegaard splitting
Definition (keen Heegaard splitting)
C1 ∪Σ C2 is keen. ⇐⇒ d(C1 ∪Σ C2 ) is realized by unique pair of
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def.
elements of D(C1 ) and D(C2 ).
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Remarks
Let C1 ∪Σ C2 : a Heegaard splitting.
Let [a0 , a1 , . . . , an ]:
a path in C(Σ) realizing d(C1 ∪Σ C2 ) = n.
In this talk ai (∈ C(Σ)(0) ) often stands for geometric representative of ai .
Remark 1
If a0 (or an ) is separating in Σ, then C1 ∪Σ C2 is not keen.
[a′0 , a1 , . . . , an ] is also a path realizing d(C1 ∪Σ C2 ).
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Remarks
Remark 2
If ∃a∗ ∈ C(Σ)(0) s.t. |a∗ ∩| a0 | = 1, a∗ ∩ a1 = ∅, then C1 ∪Σ C2 is not keen.
[a′0 , a1 , . . . , an ] is also a path realizing d(C1 ∪Σ C2 ).
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Remarks
Remark 1.
If a0 (or an ) is separating in Σ, then C1 ∪Σ C2 is not keen.
Remark 2.
If ∃a∗ ∈ C(Σ)(0) s.t. |a∗ ∩| a0 | = 1, a∗ ∩ a1 = ∅, then C1 ∪Σ C2 is not keen.
Hence,
If C1 ∪Σ C2 is keen, then:
• a0 : non-separating
• a1 : non-separating
• a0 ∪ a1 : separating
This implies:
If g(Σ) = 2, then C1 ∪Σ C2 is not keen.
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Main Result
Theorem (Ido-Jang-K)
∀g ≥ 3, ∀n ≥ 2, ∃M: closed 3-manifold with a keen H.S. C1 ∪Σ C2
s.t. g(Σ) = g, d(C1 ∪Σ C2 ) = n.
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Technical Background
Part 3
Technical Background
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Technical Background: Subsurface projection
X: essential non-simple subsurface of Σ.
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Technical Background: Subsurface projection
X : essential non-simple subsurface of Σ,
(0)
2C(X) : the power set of C(X)(0) .
(0)
Subsurface projection πX : C(Σ)(0) → 2C(X) is defined by
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Technical Background: Subsurface projection
Lemma 1 (Masur-Minsky’00)
πX is 2-Lipschitz, hence,
∀[a0 , a1 , . . . , am ] a geodesic in C(Σ) such that ai ∩ X , ∅
(i = 0, 1, . . . , m),
diamC(X) (πX (a0 ) ∪ · · · ∪ πX (am )) ≤ 2m.
Lemma 2 (Masur-Minsky)
Let C be a compression-body. If every essential disk in C
intersects ∂X (+ technical conditions), then diamC(X) (πX (D(C))) < ∞.
Remark
Tao Li gave a sharp upper bound: diamC(X) (πX (D(C))) ≤ 12.
T. Li, Images of the disk complex , Geom. Dedicata. 158 (2012),
121-136.
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Technical Background: Subsurface projection
Lemma 3 (Masur-Minsky)
Let Σ be as above.
∃c > 0, s.t. for ∀h: Σ → Σ: pseudo-Anosov,
∀γ ∈ C(Σ)(0) , ∀n ∈ Z, dC(Σ) (h̃n (γ), γ) ≥ c|n|.
H. Masur and Y. Minsky. Geometry of the complex of curves I:
hyperbolicity, Invent. Math. 138 (1999), 103–149.
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Warmup 1: Extending geodesics
Proposition 1
Let [a0 , . . . , an−2 ], [an−2 , an−1 , an ] (n ≥ 4, even) be geodesics in C(Σ),
where an−2 : non-separating in Σ.
Suppose diamC(Xn−2 ) (πXn−2 (an−4 ) ∪ πXn−2 (an )) ≥ 4n,
where Xn−2 := Cl(Σ \ N(an−2 )).
Then [a0 , a1 , . . . , an ] is a geodesic in C(Σ).
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Outline of proof of Proposition
Suppose, for a contradiction, ∃[b0 , b1 , . . . , bm ] s.t. m < n, b0 = a0 ,
bm = an .
Claim
∃j s.t. bj = an−2
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Claim
∃j s.t. bj = an−2
proof. If not, we can apply Lemma 1.
Lemma 1
πX is 2-Lipschitz, hence, ∀[a0 , a1 , . . . , am ] a geodesic in C(Σ) such that
ai ∩ X , ∅ (i = 0, 1, . . . , m), diamC(X) (πX (a0 ) ∪ · · · ∪ πX (am )) ≤ 2m.
In fact: If bj ∩ Xn−2 = ∅, then bj = an−2 .
⇒ diamC(Xn−2 ) (πXn−2 ([b0 , b1 , . . . , bm ])) ≤ 2m < 2n.
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diamC(Xn−2 ) (πXn−2 ([b0 , b1 , . . . , bm ])) < 2n.
Contradicting the assumption diamC(Xn−2 ) (πXn−2 (an−4 ) ∪ πXn−2 (an )) > 4n.
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By Claim,
⇒ j = n − 2, m − j = 2
⇒ m = n: a contradiction.
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Existence of [an−2, an−1, an]
Let [a0 , a1 , . . . , an−2 ]: geodesic in C(Σ), with an−2 : non-separating in Σ.
The existence of a geodesic [an−2 , an−1 , an ] as in Proposition 1 is given
as follows.
Let a∗n−1 = an−3 , a∗n = an−4 .
Take h(∈ MCG(Xn−2 )) : a p.A.
By Lemma 3, ∃p(> 0)
s.t. diamC(Xn−2 ) (πXn−2 (an−4 ) ∪ h̃p (πXn−2 (an−4 ))) ≥ 4n.
Then, let an−1 = h̃p (an−3 ), an = h̃p (an−4 ).
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Warmup 2: Heegaard splitting with distance n
Ido-Jang-K., Heegaard splitting of distance exactly n, AGT 14 (2014),
1395-1411.
Theorem
∀n ≥ 2, ∀g ≥ 2, ∃C1 ∪Σ C2 : Heegaard splitting of a closed 3-mfd. M
s.t. g(Σ) = g, d(C1 ∪Σ C2 ) = n.
(For simplicity, we suppose n ≥ 4, even.)
Let Ci (i = 1, 2) be a genus g compression body with ∂− Ci ≃ Σg−1 .
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Let Σ = ∂+ C1 .
[a0 , a1 , . . . , an+2 ]: a geodesic in C(Σ) constructed in Warmup 1
with
We identify a0 with ∂D1 .
Take a homeomorphism f : ∂+ C1 → ∂+ C2 s.t. f (an+2 ) = ∂D2 .
By analyzing D(Ci ), we can show that d(C1 ∪f C2 ) = n.
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We attach a genus g − 1 handlebody.
H to close the boundary component ∂− C1 ,
in a “sufficiently complicated” manner.
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“sufficient complicated”?
Notation
h : ∂H → ∂− C1 : homeomorphism, X = the component of cl(Σ \ N(a1 ))
πX : C(Σ)(0) → 2C(X) : subsurface projection,
P : C(X)(0) → C(∂− C1 )(0) : natural projection.
(0)
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“sufficient complicated”?
By Lemma 2, we can show
diamC(∂− C1 ) (P ◦ πX (f −1 (D(C2 ))))
is bounded (in fact, ≤ 12 by T. Li.).
By Hempel, we may take h so that:
dC(∂− C1 ) (P ◦ πX (f −1 (D(C2 ))), h(D(H))) > 2n
(: sufficiently complicated h.)
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Note (C1 ∪h H) ∪f C2 is a Heegaard splitting.
Proposition 2
d((C1 ∪h H) ∪f C2 ) = n
Proof. Suppose, for a contradiction,
d((C1 ∪h H) ∪f C2 ) = m ≤ n − 1.
Take a ∈ D(C1 ∪h H), b ∈ D(C1 ), d(a, b) = m, and
[a = a0 , a1 , . . . , am = b] a geodesic.
By Lemma 1,
diamC(X) (πX (a) ∪ πX (b)) ≤ 2m ≤ 2(n − 1).
This implies:
diamC(∂− C1 ) (P(πX (a))∪P(πX (b))) ≤ 2(n−1).
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By using elementary cut and paste (outermost disk) arguments,
we can show:
d(P(πX (a)), h(D(H))) ≤ 2.
These imply:
d(P ◦ πX (f −1 (D(C2 ))), h(D(H))) ≤ 2(n − 1) + 2 = 2n.
: a contradiction.
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Idea of proof of Main Result
Part 4
Idea of proof of Main Result
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Idea of the proof of Main Result
Theorem (Ido-Jang-K)
∀g ≥ 3, ∀n ≥ 2, M: closed 3-mfd. with a keen H.S. C1 ∪Σ C2 s.t. g(Σ) = g, d(C1 ∪Σ C2 ) = n.
C1 ∪Σ C2 : a genus g Heegaard splitting of a closed 3- manifold M.
[a0 , a1 , . . . , an−2 ]: a geodesic in C(Σ) with a0 ∈ D(C1 ), an ∈ D(C2 ).
Suppose:
1.
2. ∀a∗ ∈ D(C1 ) \ {a0 }, a1 ∩ a∗ , ∅,
∀b∗ ∈ D(C2 ) \ {an }, an−1 ∩ b∗ , ∅.
3. ∃k(∈ {2, . . . , n − 2}) s.t. ak : non-sep. with
diamC(X1 ) (πX1 (a0 ) ∪ πX1 (ak )) ≥ 4n + 16,
diamC(Xn−1 ) (πXn−1 (ak ) ∪ πXn−1 (an )) ≥ 4n + 16.
Then, C1 ∪Σ C2 : keen.
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