Kontsevich-Kuperberg-Thurston ( ) Kontsevich-Kuperberg-Thurston Kontsevich Chern-Simons 3 1 1989 E. Witten Chern-Simons 3 ( ) ([14]) Witten 3 Chern-Simons M. Kontsevich [5], S. Axerod I. M. Singer [2] propagator 2 2 propagator Kontsevich Kontsevich [5] Chern-Simons 3 G. Kuperberg D. Thurston Kontsevich LMO [7] 3 ([6]) Kontsevich-Kuperberg- Thurston V. A. Vassiiev ([12]) Vassiiev ( ) 3 [10] shimizu@urims.yoto-u.ac.jp 121
LMO n Jacobi n A n ( ) Kontsevich-Kuperberg-Thurston LP-surgery 3 D. Moussard ([9]) 2 Kontsevich-Kuperberg-Thurston Kontsevich-Kuperberg-Thurston z KKT 3 A( ) n A( ) degree n A n ( ) z KKT n z KKT n n z KKT n 2.1 Jacobi A n ( ) zn KKT A n ( ) degree n Jacobi diagram 2n 3n 3 simpe oop ( ) Jacobi diagram 3n 1,, 3n 2n 1,, 2n edge oriented abeed Jacobi diagram degree n edge oriented abeed Jacobi diagram E n E n = { degree n connected edge oriented abeed Jacobi diagram} Jacobi diagram 3 oriented Jacobi diagram degree n Jacobi diagram R AS, IHX 2 reation A n ( ) A n ( ) ={degree n oriented Jacobi diagrams}r/as, IHX. Γ E n orientation A n ( ) [Γ] 2.2 3 3 3 S 3 Y 122
AS, IHX ( ) 3 Y N( ; Y ) Y Y N( ; S 3 ) S 3 = R 3 { } S 3 ϕ :(N( ; Y ), ) (N( ; S 3 ), ) ϕ N( ; Y ) N( ; S 3 ) N( ; Y ) \ R 3 2 (Y \ ) 2 \ Δ={(x 1,x 2 ) x 1 x 2 } C 2 (Y ) Futon-MacPherson Δ= {(x, x) x Y \ } B A B(A, B) B A bow-up B(A, B) =(A \ B) Sν B ν B A B Sν B Y 2 2 bow-up B(Y 2, 2 ) bow-down q 1 : B(Y 2, 2 ) Y 2 C 2 (Y ):=B(B(Y 2, 2 ),q 1 1 ( (Y \ )) q 1 1 ((Y \ ) ) q 1 1 (Δ \ 2 )) C 2 (Y ) 6 ([8] ) q : C 2 (Y ) Y 2 bow-down C 2 (Y ) C 2 (Y )=q 1 ( (Y \ )) q 1 ((Y \ ) ) q 1 ( 2 ) q 1 (Δ \ 2 ). ( (a) (b) (c) (d) ) 123
2.3 C 2 (Y ) propagator 2 Y Kontsevich, Kuperberg, Thurston framing Watanabe Morse C 2 (Y ) 2 C 2 (Y ) \ q 1 (Δ \ 2 ) 2 q 1 ( (Y \ )) q 1 ( (Y \ )) = ST Y (Y \ ) ST Y ϕ = ST S 3 = S 2 S 2 anti-symmetric voume form ω S 2 2 C 2 (Y ) \ q 1 (Δ \ 2 ) 2 ω Y q 1 (Δ \ 2 ) 2 z KKT bow-up q 1 (Δ \ 2 )=Sν Δ\ 2 ν Δ = TY SνΔ\ 2 = ST(Y \ ) ST(Y \ ) (ω Y ) 2 C 2 (Y ) 2 2 C 2 (Y ) ( ) 2 propagator propagator (Y \ ) 2n \Δ={(x 1,,x 2n ) x i x j if i j} edge oriented abeed Jacobi diagram Γ E n i {1,, 3n} P i (Γ) : (Y \ ) 2n \Δ C 2 (Y ) s(γ; i),t(γ; i) i Γ P i (Γ)(x 1,,x 3n )= (x s(γ;i),x t(γ;i) ) q 1 (Δ \ 2 ) 2 framing (Kontsevich, Kuperberg, Thurston ) τ : T (Y \ ) = (Y \ ) R 3 framing τ N( ;Y )\ = τ R 3 N( ;S 3 )\. τ R 3 : T R 3 = R 3 R 3 τ ST(Y \ ) = (Y \ ) S 2 p(τ) :ST(Y \ ) = (Y \ ) S 2 S 2 S 2 voume form ω S 2 2 p(τ) ω S 2 Ω 2 (q 1 (Δ \ 2 )) framing ω Y p(τ) ω S 2 2.1. ω 0 (τ) =ω Y p(τ) ω S 2 Ω 2 ( C 2 (Y )). 124
Y 3 H 2 (C 2 (Y ); R) H 2 ( C 2 (Y ); R) C 2 (Y ) 2 ω(τ) ω(τ) C2 (Y ) = ω 0 (τ) 2.2 (Kuperberg, Thurston [6]). (1) z KKT (Y ; τ) = ( ) 3n Γ E n (Y \ ) 2n \Δ i=1 P i(γ) ω(τ) [Γ] A n ( ) τ (2) Y τ δ n A n ( ) z KKT (Y )=z KKT (Y ; τ)+ 1 4 σ(τ)δ n A n ( ) τ Y σ(τ) τ N( ; Y ) Y framing signature defect *1 (1) A n ( ) AS,IHX ( ) 2 ST(Y \ ) 3 (norma bunde ) Thom 2 a 1,,a 3n S 2 1 γ 1,,γ 3n Y \ γ i N( ;Y )\ = a i. a i R 3 a i γ =(γ 1,,γ 3n ) { γi (x) c γi := γ i (x) ST xy x Y \ ( γ 1 i (0)) }cosure ST(Y \ ) STY Y c γi c γi γ i ( ) S 2 3 c γi c γi 2.3. c(γ i )=c γi c γi ST(Y \ ) *2 3 *1 signature defect [8], [1] τ 3 Y framing τ signature defect σ(τ) Z Y bound 4 X TX TX C Y τ X 1st Pontrjagin X p 1 (τ : X) σ(τ) =p 1 (τ : X) 3SignX Hirzebruch X *2 Y 1 125
c(γ i ) Thom 1 2 2 c(γ i) ω Y *3 1 ω γi 2.4. ω 0 (γ i )=ω Y ω γi Ω 2 ( C 2 (Y )). C 2 (Y ) 2 ω(γ i ) ω(γ i ) C2 (Y ) = ω 0 (γ i ) 2.5 ([11]). γ *4 (1) z(y ; γ) = ( ) 3n Γ E n (Y \ ) 2n \Δ i=1 P i(γ) ω(γ i ) [Γ] A n ( ) γ i (2) γ Ĩ( γ) A n( ) z(y )= z(y ; γ) Ĩ( γ) A n( ) γ Y 2.6. (1) Ĩ( γ) Watanabe [13] Y 3 anomay term Watanabe Y 3 (2) γ i framing τ : T (Y \ ) (Y \ ) R 3 framing τ (Y \ ) R 3 a i τ a i Y \ τ a =(τ a 1,,τ a 3n ) Ĩ( a) δ n σ( ) *3 ω Y S 2 voume form ω S 2 {a i, a i } S 2 *4 (1) z(y ; γ) ω 0 (γ i ) 126
framing 3 framed cobordism Ĩ(τ a) = 1σ(τ)δ 4 n Ĩ signature defect γ i τ a i c(γ i )=p(τ) 1 ({a i, a i }) Ĩ(τ a) = 1σ(τ)δ 4 n 2.7 ([11]). Y z n (Y )=z KKT (Y ) 3 3.1 Watanabe Morse ( ) 3 ([13]) Watanabe 3 A( ) 3 2 Morse 3 ([3]) Watanabe A 1 ( ) Fuaya Morse ([4]) Watanabe n Watanabe modify n zn FW zn FW 3 A n ( ) zn FW Jacobi (broen graph) Modui zn FW zn KKT z n 2 (propagator) Morse f i propagator ω(f i ) C 2 (Y ) f i a 1,,a 3n S 2 1 f i : Y \ R Morse-Smae Morse f i N( ;Y )\ = q ai. q ai : R 3 R q ai (x) = x, a i R 3 a i, R 3 f i Crit(f i )={p i 1,,p i i,q 1,,q i i } ind(p i j)=2, ind(qj)=1 f i i Q Morse-Smae : H 2 (Y \ ; Q) H 1 (Y \ ; Q) [p i ]= [q i ] Y 3, g : H 1 (Y \ ) H 2 (Y \ ) g([q i ]) = g [p i ] 127
{Φ t f i : Y \ = Y \ } t R gradient-ie gradf i 1 ϕ :(Y \ ) (Y \ ) (0, ) (Y \ ) (Y \ ),ϕ(x, y, t) =(y, Φ t f i (x)) M (f i )=ϕ 1 (Δ) A q i q i D p i j p i j Watanabe propagator Poincaré *5 4 M(f i ):= 1 2 (M (f i )+M ( f i )), g (A q i D p i + D p i A q i ). *6 gradf 1,, gradf 3n,a 1,,a 3n ) zn FW (Y ; f)= Γ E n ( 3n i=1 P i (Γ) 1 M(f i ) 3.1 (Watanabe [13]). gradf = (gradf 1,, gradf 3n ) gradf I(gradf) A n ( ) z FW n Y [Γ] (Y )=zn FW (Y ; f) I(gradf) A n ( ) 3.2 z n Watanabe [13] Kontsevich Chern-Simons z n *5 propagator Poincaré *6 gradf 1,, gradf 3n,a 1,,a 3n 128
3.2 ([11]). z FW n (Y )= z n (Y ). Outine of proof. M(f i ) (Y \ ) 2 4 M(f i ) \ Δ C 2 (Y ) (C 2 (Y ), C 2 (Y )) 4 M C (f i ) 4 ST(Y \ ) 1 2 c(gradf i)+, g (A q i D p i + D p i A q i ) 2 A q i D p i D p i A q i C 2 (Y )\q 1 (Δ\ 2 ) f i q 1 ( (Y \ )) = ST Y (Y \ ) M C (f i ) (ST Y (Y \ )) = 1({a 2 i, a i } (Y \ )) M C (f i ) Poincaré ω(gradf i ) z n (Y ;gradf)=z n FW (Y ; f) Ĩ Ĩ(gradf)=I(grad f) [1] M. Atiyah. On framings of 3-manifods. Topoogy, 29(1):1 7, 1990. [2] S. Axerod and I. M. Singer. Chern-Simons perturbation theory. In Proceedings of the XXth Internationa Conference on Differentia Geometric Methods in Theoretica Physics, Vo. 1, 2 (New Yor, 1991), pages 3 45. Word Sci. Pub., River Edge, NJ, 1992. [3] K. Fuaya. Morse homotopy and Chern-Simons perturbation theory. Comm. Math. Phys., 181(1):37 90, 1996. [4] M. Futai. On Kontsevich s configuration space integra and invariants of 3-manifods. Master thesis, Univ. of Toyo, 2006. [5] M. Kontsevich. Feynman diagrams and ow-dimensiona topoogy. In First European Congress of Mathematics, Vo. II (Paris, 1992), voume 120 of Progr. Math., pages 97 121. Birhäuser, Base, 1994. [6] G. Kuperberg and D. P. Thurston. Perturbative 3-manifod invariants by cut-and-paste topoogy. ArXiv Mathematics e-prints, December 1999. [7] T. T. Q. Le, J. Muraami, and T. Ohtsui. On a universa perturbative invariant of 3- manifods. Topoogy, 37(3):539 574, 1998. [8] C. Lescop. On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rationa homoogy 3-spheres. ArXiv Mathematics e-prints, November 2004. [9] D. Moussard. Finite type invariants of rationa homoogy 3-spheres. Agebr. Geom. Topo., 12(4):2389 2428, 2012. [10] T. Ohtsui. Finite type invariants of integra homoogy 3-spheres. J. Knot Theory Ramifications, 5(1):101 115, 1996. 129
[11] T. Shimizu. An invariant of rationa homoogy 3-spheres via vector fieds. ArXiv e-prints, 2013. [12] V. A. Vassiiev. Cohomoogy of not spaces. In Theory of singuarities and its appications, voume 1 of Adv. Soviet Math., pages 23 69. Amer. Math. Soc., Providence, RI, 1990. [13] T. Watanabe. Higher order generaization of Fuaya s Morse homotopy invariant of 3- manifods I. Invariants of homoogy 3-spheres. ArXiv e-prints, February 2012. [14] E. Witten. Quantum fied theory and the Jones poynomia. In Braid group, not theory and statistica mechanics, voume 9 of Adv. Ser. Math. Phys., pages 239 329. Word Sci. Pub., Teanec, NJ, 1989. 130