2008年度日本数学会秋期総合分科会, 2008.9.25
レンズ空間のデーン手術距離について
Surgical distance between lens spaces
市原一裕
Kazuhiro Ichihara
奈良教育大学 Nara University of Education 斎藤敏夫氏 (奈良女子大学) との共同研究
Joint work with Toshio Saito (Nara Women’s University)
§0. Back grounds
Classification of 3-manifolds Every closed orientable 3-manifold is;
• Reducible (containing essential 2-sphere),
• Toroidal (containing essential torus),
• Seifert fibered (foliated by circles), or
• Hyperbolic (admitting Riem.metric of curv.−1).
Conjectured by Thurston, (late ’70s) Established by Perelman (2002-03)
What’s the NEXT?
• Attack the remaining Open Problems.
(e.g., Virtually Haken Conjecture,
“Heegaard genus VS rank of π1” problem, etc. . .)
• Relate Geometric & Topological invariants.
(e.g., Volume conjecture (for knots), etc . . .)
• Study the Relationships between 3-manifolds.
(e.g., degree one map, Dehn surgery, etc . . .) (⇑ Today!)
Dehn Surgery
Let M be a closed orientable 3-manifold and K a knot in M.
Dehn surgery
1) Remove a neighborhood of K from M, 2) Gluing a solid torus back (along slope γ)
Solid torus 3-mfd;M Dehn surgery
(K, γ)
§1. Surgical distance M :=
orientation preserving homeomorphism type of connected closed orientable 3-manifolds
Notation
For [M], [M0] ∈ M, we consider
min
n
¯¯¯¯
¯¯¯¯
¯¯
[M] = [M0], [M1], · · · , [Mn] = [M0] ∈ M Mi+1 is obtained from Mi
by Dehn surgery on a knot.
We denote this value by d([M], [M0]).
Fact [Lickorish, Wallace]
d : M × M → Z+ is well-defined.
i.e., the graph of 3-manifolds is connected;
M
(K, slope)
§2. On hyperbolic knots
Notation For [M], [M0] ∈ M, we consider
min
n
¯¯¯¯
¯¯¯¯
¯¯
[M] = [M0], [M1], · · · , [Mn] = [M0] ∈ M Mi+1 is obtained from Mi
by Dehn surgery on a hyperbolic knot.
We denote this value as dH([M], [M0]).
Fact [Kawauchi]
d
H: M × M →
Z+is well-defined.
Fact [Kawauchi]
For [M], [M0] ∈ M, dH([M], [M0]) =
1 or 2 if d([M], [M0]) = 1 d([M], [M0]) otherwise
Problem 1.
When d([M], [M0]) 6= dH([M], [M0]) can occur?
Known Facts: (From S3)
• For some lens space L,
d([S3], [L]) = 1 & dH([S3], [L]) = 2.
In particular,
d([S3], [S2 × S1]) = 1 & dH([S3], [S2 × S1]) = 2.
d([S3], [RP 3]) = 1 & dH([S3], [RP 3]) = 2.
• P : Poincar´e homology sphere Σ(2, 3, 5).
3 3
Lens spaces
We here call a 3-manifold L a lens space if L is of Heegaard genus at most one
(i.e., constructed by gluing two solid tori).
Set L :=
orientation preserving homeomorphism type of lens spaces
Question: For which [L], [L0] ∈ L, dH([L], [L0]) = 1?
Theorem 1. [I.-Saito]
For ∀[L] ∈ L, ∃an infinite family [L1], [L2], · · · ∈ L such that dH([L], [Li]) = 1 for ∀i.
Theorem 2. [I.-Saito]
For ∀N > 0,
∃pair [L], [L0] ∈ L of types (p, q) and (p0, q0)
such that |p − p0| = N and dH([L], [L0]) = 1.
§3. On the set of lens spaces
Note that; d([L], [L0]) = 1 & dH([L], [L0]) ≤ 2.
Notation For [L], [L0] ∈ L, we consider
min
n
¯¯¯¯
¯¯¯¯
¯¯
[L] = [L0], [L1], · · · ,[Ln] = [L0]∈ L Li+1 is obtained from Li
by Dehn surgery on a hyperbolic knot.
We denote this value as dH([L], [L0])L. Problem 2.
Recall:
dH([L],[L0]) = 1 ⇒ dH([L], [L0])L = 1 by definition.
dH([L],[L0]) = 2 ⇒ dH([L], [L0])L ≥ 2 in general.
Question:
For which [L], [L0] ∈ L, dH([L],[L0]) = dH([L],[L0])L = 2 ?
Example. [I.-Saito]
dH(S3, S2 × S1)L = dH(S3, S2 × S1) = 2.
In fact,
dH(S3, L(64, 23)) = 1 & dH(L(64, 23), S2 × S1) = 1.
(using Yamada’s example)
