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
[1] S. Axelrod, I. M. Singer, Chern–Simons perturbation theory, in Proceedings of the XXth DGM Conference, CattoS., Rocha A. (eds.), pp. 3–45, World Scientific, Singapore, 1992 , II, J. Diff. Geom. 39 (1994), 173–213.
[2] D. Bar-Natan, Perturbative Aspects of the Chern-Simons Topological Quantum Field Theory, Ph.D. Thesis,
Princeton University, 1991.
[3] D. Bar-Natan, B. McKay, Graph Cohomology - An Overview and Some Computations, draft, 2001.
[4] R. Bott, Morse theory indomitable, Publ. Math. I.H.E.S., 68 (1988), 99–114.
[5] E. H. Brown, Jr, Twisted Tensor Products, I, Ann. of Math. 69, no. 1 (1959), 223–246.
[6] R. Budney, A family of embedding spaces, Geom. Topol. Monogr. 13 (2008), 41–83.
[7] D. Burghelea, R. Lashof, The homotopy type of the space of diffeomorphisms. I, Trans. Amer. Math. Soc. 196(1974), 1–36; II, Trans. Amer. Math. Soc. 196 (1974), 37–50.
[8] A. S. Cattaneo, P. Cotta-Ramusino, R. Longoni, Configuration spaces and Vassiliev classes in any dimension,Algebr. Geom. Topol. 2 (2002), 949–1000.
[9] J. Cerf, La stratification naturelle des espaces de fonctions differentiables reelles et le theoreme de la pseudo-
isotopie, Publ. Math. I.H.E.S. 39 (1970), 5–173.
[10] J. Francis, Smooth structures on PL 4-manifolds,
https://mathoverflow.net/questions/7892/smooth-structures-on-pl-4-manifolds
10![35, Lemma 3.1] π0Diff(D4, ∂) = π0Ω
5PL(4)/O(4) = 0
[11] K. Fukaya, Morse Homotopy and Chern–Simons Perturbation Theory, Comm. Math. Phys. 181 (1996), 37–90.
[12] M. Goussarov, Finite type invariants and n-equivalence of 3-manifolds, C. R. Acad. Sci. Paris Ser. I Math. 329
(1999), no. 6, 517–522.
[13] S. Garoufalidis, M. Goussarov, M. Polyak, Calculus of clovers and finite type invariants of 3-manifolds, Geom.
Topol. 5, no. 1 (2001), 75–108.
[14] S. Garoufalidis, T. Ohtsuki, On finite type 3-manifold invariants. III. Manifold weight systems, Topology 37
(1998), no. 2, 227–243.
[15] E. Guadagnini, M. Martellini, M. Mintchev, Perturbative aspects of the Chern-Simons field theory, Phys. Lett.B 227 (1989), 111–117.
[16] K. Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1–83.
[17] A. Hatcher, A proof of the Smale conjecture, Diff(S3) ≃ O(4), Ann. of Math. (2) 117 (1983), no. 3, 553–607.
[18] A. Hatcher, A 50-Year View of Diffeomorphism Groups, talk slides, 2012.
[19] , , 2003, .
[20] , , , , I, 2016, .
[21] M. Hutchings, Floer homology of families I, Algeb. Geom. Topol. 8 (2008), 435–492.
[22] K. Igusa, Twisting cochains and higher torsion, J. Homotopy Relat. Struct. 6 (2011), no. 2, 213–238.
[23] M. A. Kervaire, J. W. Milnor, Groups of homotopy spheres: I, Ann. of Math. 77 no. 3 (1963), 504–537.
[24] R. Kirby (Ed.), Problems in Low-Dimensional Topology, https://math.berkeley.edu/~kirby/
[25] M. Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol.II (Paris, 1992), Progr. Math. 120 (Birkhauser, Basel, 1994), 97–121.
[26] M. Kontsevich, Vassiliev’s knot invariants. I. M. Gelfand Seminar, 137–150, Adv. Soviet Math., 16, Part 2,Amer. Math. Soc., Providence, RI, 1993.
[27] G. Kuperberg, D. P. Thurston, Perturbative 3-manifold invariants by cut-and-paste topology,
arXiv:math/9912167.
[28] S. Kwasik, Low-dimensional concordances, Whitney towers and isotopies, Math. Proc. Camb. Phil. Soc. 102,
no. 1 (1987), 103–119.
[29] C. Lescop, Splitting formulae for the Kontsevich–Kuperberg–Thurston invariant of rational homology 3-spheres,
math.GT/0411431.
[30] I. Madsen, J. Milgram, The classifying spaces for surgery and cobordism of manifolds. Annals of MathematicsStudies, 92, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. xii+279 pp.
[31] T. Ohtsuki, A polynomial invariant of rational homology 3-spheres, Invent. Math. 123 (1996), no. 2, 241–257.
[32] T. Ohtsuki, Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramif. 5 (1996), 101–115.
[33] T. Ohtsuki, Quantum invariants, A study of knots, 3-manifolds, and their sets, Series on Knots and Everything,29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
[34] , , 4, 2015, .
[35] D. Randall, P. A. Schweitzer, On Foliations, Concordance Spaces, and the Smale Conjectures, In “DifferentialTopology, Foliations, and Group Actions”, Contemp. Math. 161 (1994), 235–258.
[36] K. Sakai, Nontrivalent graph cocycle and cohomology of the long knot space, Algebr. Geom. Topol. 8 (2008), no.3, 1499–1522.
[37] T. Shimizu, An invariant of rational homology 3-spheres via vector fields, Algebr. Geom. Topol. 16 (2016),
3073–3101.
[38] S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626.
[39] V. A. Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications, 23–69, Adv. Soviet Math.,
1, Amer. Math. Soc., Providence, RI, 1990.
[40] T. Watanabe, On Kontsevich’s characteristic classes for higher-dimensional sphere bundles II: Higher classes,
J. Topol. 2 (2009), 624–660.
[41] T. Watanabe, Higher order generalization of Fukaya’s Morse homotopy invariant of 3-manifolds I. Invariants
of homology 3-spheres, Asian J. Math. 22, no. 1 (2018), 111–180.
[42] T. Watanabe, Morse theory and Lescop’s equivariant propagator for 3-manifolds with b1 = 1 fibered over S1,preprint, arXiv:1403.8030.
[43] T. Watanabe, Kontsevich’s characteristic classes for Diff(S4), in preparation.
[44] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399.