Di ( S HOMFLY Witten (i){(iii) Feynman (i) K 1;K 2: S 1! R 3 (i) S 1 S 1 R S 1 S 1! S 2 lk( K 1;K 2)! S 2 S 2: S 1 S 1! S 2 ( t 1;t2) = K 2 (t2) K 1 (t1) jK 2 (t2) K 1 (t1)j etc: (i)

日本数学会・2018年度秋季総合分科会（於:岡山大学）・トポロジー分科会・特別講演msjmeeting-2018sep-10i001

Diff(S4)

( )∗

1.

C∞ M Diff(M)

C∞ ([18, 20])

1.1. Diff(M)

M = Sd d ≤ 3

d = 1 Diff(S1) ≃ O(2) d = 2

Smale Diff(S2) ≃ O(3) ([38]) d = 3 Diff(S3) ≃ O(4)

(Smale ) Smale Hatcher

([17]) d ≥ 5 ([18]) d = 4

([24, Problem 4.34, 4.126, 4.141])

1.2 (4 Smale ). Diff(S4) ≃ O(5).

Smale topological version: TOP(S4) ≃ O(5) Randall Schweitzer

([35]) Diff(Dd, ∂)

Dd Diff(Sd) ≃ O(d+1)×Diff(Dd, ∂)

1.2 Diff(D4, ∂) ≃ ∗

1.3 ([43]). Diff(D4, ∂) 6≃ ∗ 4 Smale

4 Smale

Diff(D4, ∂) ≃ Ω5 PL(4)/O(4) [7] PL(n) piecewise linear (Rn, 0) →

(Rn, 0) [30, 20] 1.3

1.4. PL(4) 6≃ O(4).

(cf. [10]) §6

2. 3

3 [33, 34]

2.1. Chern-Simons

Witten 3 M L (M ;L)

Witten , [44]

([2, 15, 1, 25] etc.) S3

M 1 ∞ framed

( :17K05252, 26400089)∗ 690-8504 1060e-mail: [email protected]

Jones HOMFLY

Witten

(i)–(iii) Feynman

(i)

K1, K2 : S1 → R3 (i)

S1 × S1∫S1×S1 Φ

∗ωS2 lk(K1, K2)

ωS2 S2 Φ : S1 × S1 → S2 Φ(t1, t2) =K2(t2)−K1(t1)|K2(t2)−K1(t1)|

etc.

(i) (ii) (iii)

(1)

3

Chern–Simons ([27, 29])

2.2. !

Vassiliev

([39])

K Z

ZK = F0 ⊃ F1 ⊃ F2 ⊃ · · · Fk k

2 resolution:

ZK ZK f : ZK → Z

k f(Fk) = 0 f ( type k)

k Fk/Fk+1 Fk/Fk+1⊗Q k

(1) (ii) Kontsevich

([26])

3 Witten

3

Z 3

([31, 32])

Z 3 M C

CM = G0 ⊃ G1 ⊃ G2 ⊃ · · · Gk/Gk+1

Garoufalidis Jacobi 3

[14]

Goussarov

3 ([12, 16, 13])

2.1 (Goussarov, ). K1, K2 type k

degree k

Z 3 1

1 degree 2k type 3k

1: 3 3 Y

Borromean

Jacobi

Jacobi AS, IHX

3

3. BDiff(M)

Kontsevich

Chern–Simons [25]

3 2

3.1.

Emb(S1,Rd) S1 → Rd Emb(S1,Rd) C∞

Kontsevich S1 1 1,3

(1)-(ii), (iii) G odd(S1),G even(S1)

I :

H(G odd(S1))→ H∗(Emb(S1,Rd);R) if d is odd

H(G even(S1))→ H∗(Emb(S1,Rd);R) if d is even

G odd(S1),G even(S1) S1 1,3 Q

dΓ =∑

e±Γ/e

Kontsevich [25] I

1,3 Cattaneo–Cotta-Ramusino–Longoni

([8]) 4 ([36]) etc.

Cattaneo Vassiliev 2 resolution

Emb(S1,Rd)

d = 3 2 resolution S0 = −1, 1

d Sd−3 Sd−3

parametrize resolution

2 Sd−3 × · · · × Sd−3 → Emb(S1,Rd)

2 [25] ! Riemann moduli

Lie

3.2. BDiff(M)

BDiff(M) Diff(M)

B M-bundles/ ←→ [B,BDiff(M)]

[X, Y ] X → Y BDiff(M)

M-bundle πkBDiff(M) Sk M-bundle

standard πkBDiff(M) ≈ πk−1Diff(M)

(k ≥ 1)

Kontsevich 3 1

G even, G odd

I :

H(G odd)→ H∗(BDiff(Dd, ∂);R) if d is odd

H(G even)→ H∗(BDiff(Dd, ∂);R) if d is even

BDiff(Dd, ∂) fiber framing Dd-bundle

standard

Ω4SO4 → BDiff(D4, ∂)→ BDiff(D4, ∂)

πjSO4 j ≥ 4 πiBDiff(D4, ∂)⊗

R ≈ πiBDiff(D4, ∂)⊗ R (i ≥ 1) Kontsevich

I : H(G even)→ Hom(π∗BDiff(D4, ∂),R)

3 H(G odd), H(G even)

HY(G odd), HY(G even)

3.1 ([43]). d = 4 I HY(G even)

. [40] 2k 3 γ =∑

Γ µΓ Γ ∈

G even

. 2k 3 Γ

(D4, ∂)-bundle πΓ : EΓ → BΓ Sk (D4, ∂)-bundle

. γ 7→ I(γ) 7→ 〈I(γ),∑

Γ EΓ · Γ〉 = Cγ C 0

I|HY

HY(G even) Bar-Natan McKay

([3])3 1.3 3.1

k = ×12

1 2 3 4 5 6 7 8 9

0 1 ? 0 1 0 0 0 1

3.2. π2BDiff(D4, ∂)⊗Q 6= 0 [3] π5BDiff(D4, ∂)⊗

Q 6= 0, π9BDiff(D4, ∂)⊗Q 6= 03Bar-Natan McKay unpublished !

k = 2 ? [3]

4. ( )

Goussarov, Habiro

4.1. Hopf link Borromean rings

Rd Hopf link Sp∪Sq → Rd p+ q = d−1 4 d = 4

1 ≤ p ≤ q, p+ q = 3 (p, q) = (1, 2)

Rd Borromean rings Sp ∪ Sq ∪ Sr → Rd p + q + r = 2d − 35 d = 4 1 ≤ p ≤ q ≤ r ≤ 3, p + q + r = 5

(p, q, r) = (1, 2, 2), (1, 1, 3)

4.2.

d = 4 3 3 Γ D4

Hopf link 2 4

Hopf link S1 S2 S2, S1

Hopf link 3 3

Y G1 ∪G2 ∪ · · · ∪G2k

2k k Type I k Type II

Gi Vi ⊂ IntD4

~VG = (V1, . . . , V2k)

4.3.

Vi ∂Vi V ′i

D4 Ki D4×Ki Vi×Ki

Vi-bundle Vi → Ki

Vi Vi → Ki Emb(S1,Rd) 2 resolution

i K1 × · · · ×K2k

4 t ∈ R, x ∈ Rp, y ∈ Rq K1 = (t, x, y) | t2 + |x|2 = 1, |y| = 0, K2 = (t, x, y) |

(t− 1)2 + |y|2 = 1, |x| = 05 x ∈ Rd−p−1, y ∈ Rd−q−1, z ∈ Rd−r−1 K1 = (x, y, z) | |y|

2

4 + |z|2 = 1, x = 0,

K2 = (x, y, z) | |z|2

4 + |x|2 = 1, y = 0, K3 = (x, y, z) | |x|2

4 + |y|2 = 1, z = 0

Vi → Ki Vi Type I 3

Vi 4 disk 1-handle 1 2-handle 2

3 (1, 2, 2) Borromean rings

V ′i V ′

i Vi S0

bundle V ′i ∪ (−V )→ S0 V ′

i

Vi Type II 3 Vi 4 disk 1-handle 2

2-handle 1 (1, 1, 2) Borromean

rings 4 (1, 2, 2) = (1, 1, 2) + (0, 1, 0) (1, 2, 2)

Borromean rings (1, 1, 2) 1

(Vi, ∂)-bundle Vi → S1

V ′i Vi (D4, ∂)-bundle

πΓ : EΓ → BΓ = K1 × · · · ×K2k, Ki = S0 or S1

k- K1× · · ·×K2k → BDiff(D4, ∂)

4 3

5. Morse ( )

(D4, ∂)-bundle πΓ Kontsevich

Morse 6

5.1. Morse Chern–Simons

Morse 3

([11]) Kontsevich Morse

3 Morse f1, f2, f3

ξ1, ξ2, ξ3 3 ±ξ1,±ξ2,±ξ3

( ) 3

g

etc.

6 4 Morse

1. Morse !

pseudo-isotopy

2. d = 4k − 1 3 Lescop ([29] §4–§5)

([40]) d = 4 d = 4k − 1

Lescop

3

(pi, qi fi ) 3 M Z 3

M − ∞ Morse f1, . . . , f3k ξ1, . . . , ξ3k

∞ 3

Morse attach Γ

N(Γ) Γ 7 µk ∈ Q

Γ cΓ ∈ Q

ZMorsek (γ;M) =

∑

Γ

µcl(Γ)cΓ(N(Γ)− n(Γ)) + µk signW

M ([41]) 2k 3

n(Γ) ∂W = M 4 W rank 3

Γ

3 0 Kontsevich

(Chern–Simons ) framing

([37])

5.2. Morse

Morse 2 Morse

([4, 19] )

π : E =⋃

t∈B Ft → B B F -

bundle E,B Riemann Morse smooth ft : Ft → R

(t ∈ B) fiberwise Morse 8 ft π

critical locus ft ∇ft

B Morse h : B → R

η E η η lift9 ω = −∇ft− η

η 6= 0 η ω = 0 ∇ft = η = 0

ft critical locus η

η Morse C∗(η), t0 ∈ B Morse C∗(ft0)

Σ(ω) = y ∈ E | ωy = 0 Z C∗(ω)

C∗(ω) = C∗(η)⊗ C∗(ft0)

7N(Γ) well-defined !ξ1, . . . , ξ3k generic8 ft Morse birth-death

fiberwise Generalized Morse Function (GMF) fiberwise GMF

9 dπ(ηx) = ηπ(x) η E

η = ∇(h π)

twisted differential

ω

Cerf critical locus

bump function ρ : E → [0, 1] ωρ = −∇ft− ρη

critical locus ρ

handle-slide Z-path Ei,j = Ci(η)⊗

Cj(ft0) φr : Ei,j → Ei−r,j+r−1 ω p⊗ q

φr(p⊗ q) =∑

p′⊗q′

#MZ(p⊗ q, p′ ⊗ q′) p′ ⊗ q′

M Z(p⊗ q, p′ ⊗ q′) p⊗ q p′⊗ q′ Z-path moduli

φr K. -T. Chen ([42, 43])

∑

k+ℓ=r

k,ℓ≥0

±φkφℓ = 0

φ1 C∗(ft0) η

Morse ∂ = φ0

∂φr ± φr∂ +∑

k+ℓ=r

k,ℓ≥1

±φkφℓ = 0

φr C∗(ω) twisted differential ∂φ

E

cf. [22, 5, 21] r ≥ 2 φr = 0

5.3. : Z-graph

[11] M3 × I 1

∇ft §5.2

Z-path

Z-graph 1

B η bundle

Z-graph B

(D4, ∂)-bundle πΓ : EΓ → BΓ Z-graph

N(Γ)

ZMorsek (γ; πΓ) =

∑

Γ

µcl(Γ)cΓ(N(Γ)− n(Γ))

n(Γ) null-cobordism rank 4

ZMorsek (γ; πΓ)

Kontsevich

ZKk (γ) = 〈I(γ), ·〉 : πkBDiff(D4, ∂)⊗Q→ Q

ZKk (γ)(π

Γ) [37] ZKk (γ) bordism

πΓ well-defined

5.4. Morse :

ZMorsek (γ; πΓ)

Z-path handle-slide

∇ft t ∈ B Z-path

Z-graph ∇ft

Z-graph

Morse

0

Z-graph level

〈I(γ),∑

ΓEΓ ·Γ〉 =

∑Γ Z

Morsek (γ; πΓ) ·Γ =

∑Γ CµΓ ·Γ = Cγ

6.

Emb+(S3,R4), Emb+(D

4,R4)

S3 → R4, D4 → R4

Diff(D4, ∂)→ Emb+(D4,R4)→ Emb+(S

3,R4)

6.1. Emb+(S3,R4) 6≃ Emb+(D

4,R4).

SO4 π0Emb+(D4,R4) = 0

6.2 (4 Schoenflies ). π0Emb+(S3,R4) = 0

C (M) = Diff(M × I, ∂M × I ∪M × 0) pseudo-isotopy

g1, g2 ∈ Diff(M, ∂) g2g−11 pseudo-isotopy

pseudo-isotopic

Diff(Dd+1, ∂)→ C (Dd)→ Diff(Dd, ∂)

3 Smale Diff(D3, ∂) ≃ ∗ ([17])

6.3. C (D3) 6≃ ∗.

C (D1) ≃ ∗, C (D2) ≃ ∗ π0Diff(D5, ∂) ≈ Θ6 = 0 ([9], [23]),

π1Diff(D4, ∂)⊗Q 6= 0

6.4. π1C (D4)⊗Q 6= 0.

PL pseudo-isotopy π1CPL(D4) = 0 ([7]) Cerf

d ≥ 5 π0C (Dd) = 0 ([9]) d ≥ 5 Dd 2

pseudo-isotopic ⇒ isotopic

6.5 (Cerf 4 ). π0C (D4) = 0.

M4 j : C (M4)→ CTOP(M4)

π0(j(C (M4))) = 0, pseudo-isotopic ⇒ topologically isotopic

Kwasik ([28])

6.6 ([24, Problem 4.34], [6]). π0C (D3) ≈ π0Diff(D4, ∂) = 0 S4

isotopic 10

6.6 C (D4)→ Diff(D4, ∂)

π0Diff(D5, ∂) = 0 6.5

cf. 3.2

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