H = H 1 (Jac(R); Z) Sp 1 H (Jac(R); Z) = Λ Z H, H (Jac(R); Z) = Λ Z H = Λ Z H Poincaré duality canonical ( ) canonical symplectic form foliation (2) F
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1 Sp-modules symplectic Sp-module low dimensional., 0 1, 2, 3, 4 (n),. foliation n. Sp-modules, intersection form H = H 1 (Σ; Z), µ : H H Z H rank 2g free module Q C Q foliation R Q, R H Q = H Q, H R = H R g n Sp Sp(2g, Q), Sp(2g, R). Sp(2g, Z)- Sp(2g, Q) 0 (0) Lie sp(2g, Q) S 2 H Q. Lie Lie 4 g = 1 sp(2, Q) sl(2, Q) sl(2, Q) S 2 H g=1 Q Q x 2, xy, y 2. R 1 (1) Sp Riemann R Jacobian 1
2 H = H 1 (Jac(R); Z) Sp 1 H (Jac(R); Z) = Λ Z H, H (Jac(R); Z) = Λ Z H = Λ Z H Poincaré duality canonical ( ) canonical symplectic form foliation (2) Foliation standard symplectic vector space H R 2n symplectic x 1,..., x n, y 1,..., y n coordinate standard symplectic form ω 0 = n dx i dy i i=1 H R = (R 2n, ω 0 ) standard symplectic vector space Hamilton foliation 1972 Gel fand-kalinin-fuchs 40 transverse symplectic foliation foliation. exotic Gel fand Kalinin arxiv first name Fuchs transversely symplectic foliation symplectic manifold Definition 1.1. (M 2n, ω) symplectic manifold, ω A 2 (M) 2-form dω = 0 ω n volume form. 2n symplectic form standard standard symplectic vector space symplectic manifold 2
3 symplectic ω n Darboux Theorem 1.2 (Darboux). symplectic manifold(m, ω) (R 2n, ω 0 ) symplectical symplectically diffeomorphism symplectomorphism contact structure contactomorphism symplectic symplectic Darboux local global transversely symplectic foliation F C - W codimension 2n transversely symplectic foliation 2n, leaf local normal product local global Gel fand-fuchs Chern-Simons Thurston W = α U α, ϕ α : U α R 2n standard symplectic space (R 2n, ω 0 ) submersion submersion submersion 1 leaf R 2n symplectic W f αβ = ϕ β ϕ 1 α : ϕ α (U α U β ) ϕ β (U α U β ) symplectomorphism U α ϕ 1 α (p) p ϕ α f αβ : symplectomorphism U β ϕ β Gel fand-fuchs-kalinin 3
4 n = 1, R Gel fand-kalinin-fuchs H 7 (W; R). GKF class exotic class. exotic, primary 1999 Metoki class H 9 (W; R) Bott Perchik 1974,5 Topology euler 100 Perchik transversely symplectic, n = 1 2 symplectic structure area preserving holonomy foliation Gel fand-kalinin-fuchs class Metoki class 0 Gel fand-kalinin-fuchs 40 transversely symplectic foliation Gel fand-fuchs theory, Chern-Simons theory GKF-class GF-theory Gel fand-fuchs 1969 Chern-Simons 1974 Annals Chern-Simons Thurston hyperbolization hyperbolic geometry 1982 Bulletin Haken hyperbolic structure 1976 pseudo-anosov Bulletin Bulletin Bulletin hyperbolic geometry Fathi-Poénaru-Laudenbach, 1988 Bulletin measured foliation measured lamination foliation lamination lamination , 100 Nielsen Nielsen Nielsen-Thurston 76 Thurston Nielsen 4
5 Nielsen ,3 Nielsen pseudo-anosov Thurston Nielsen Thurston Nielsen Riemann quasi-conformal map Teichmülar Gel fand-fuchs 1969 Chern-Simons 1974 Chern-Simons introduction Q Pontrjagin class Novikov Pontrjagin class piecewise linear formula 3 rational Pontrjagin class smooth tangent bundle connection Chern-Weil Pontrjagin form R Pontrjagin class formula Chern-Simons 2 Pontrjagin form exact form Chern-Simons p 1, first Pontrjagin class Gel fand-fuchs 5 Chern-Siomns foliation Godbillon-Vey foliation Gel fand-fuchs, Chern-Simons Godbillon Vey 70 1 de Rham well-defined de Rham cohomology 1971 Godbillon-Vey class, Roussarie PSL(2, R) locally homogeneous foliation Thurston Bulletin Noncobordant foliations of S foliation, Godbillon-Vey 1970 Gel fand-kalinin-fuchs class Thurston Thurston. Thurston monster Perelmann Thurston
6 cover S 1 Thurston, foliation, Gel fand-fuchs, Sp-module transverse foliation standard symplectic vector space H R symmetric algebra, S H R = R[x 1,..., x n, y 1,..., y n ] R x 1,..., y n derivation Der poly (S H R ) = {X = n ( i=1 f i ) + g i ; f i, g i R[x 1,..., y n ]} x i y i h S H R X Der poly (S H R ) X(h) = n i=1 ( ) h h f i + g i x i y i symplectic form symplectic form infinitesimal Der poly (S H R, ω 0 ) = {X Der poly(s H R ); L X ω 0 = 0} symplectic form Lie symplectic form infinitesimal symplectic form Lie symplectic symplectic Hamilton Proposition 1.3. L X ω 0 = 0 n ( F X = F ) x i y i y i x i F S H R /R = R[x 1,..., x n, y 1,..., y n ]/R i=1 modulo R F Hamiltonian function n = 1 4 derivation Corollary 1.4. Lie Der poly (S H R, ω 0 ) S 0 H R 6
7 Lie bracket Poisson bracket Der poly (S H R ) S 0 H R Der poly (S H R ) X, X F = X F = F X S 0 H R F S 0 H R X F = X X Der poly (S H R ) Y Der poly (S H R ), G = G Y S 0 H R [X, Y] Der poly (S H R ) S 0 H R, {F, G} Poisson bracket n ( F G {F, G} = F ) G x i y i y i x i i=1 Lie transversely symplectic foliation, x i y i linear derivation 1 2 subalgebra S 2 H R ( sp(2n, R)). Gel fand-fuchs foliation Gel fand-fuchs R 1 Gel fand-fuchs jet 3 H 3 2 foliation 3 (3) mapping class group. symplectic D. Johnson 1970 Torelli abel Johnson kernel Johnson genus g M g = π 0 Diff + (Σ g ) Sp g = 1 g 2 kernel 1 I g M g Sp(2g, Z) 1 Thurston Nielsen 1 Nielsen Nielsen pseudo-anosov Nielsen genus 2, Torelli pseudo-anosov Thurston 1 Thurston 76 genus 2 7
8 ( ) Thurston fill up transverse cut contractible separating separating Dehn-twist separating curve separating curve 2 separating curves Dehn-twist, Dehn-twist pseudo-anosov separating curves Dehn-twsit Torelli Nielsen Dehn-twist, Dehn-twist 2 GL(2, Z) Torelli Dennis Johnson ,8 1 Torelli abel torsion Q H 1 (I g, Q) = Λ 3 H Q /H Q, Sp H Q H Q u u ω 0 Λ 3 H Q Sp-map Sp-submodule Young GL 3 Sp Torelli Dennis Johnson Torelli 5, abel Johnson homomorphism τ 1 : I g Λ 3 H/H Z τ Johnson τ 1 abel Rohlin Birman-Craggs 2-torsion, homology cylinder 8
9 Johnson Malcév completion lower central series abel 2-step nilpotent group Dehn-Nielsen automorphims lower central series Johnson kernel ker τ 1, ker τ 1 = K g Johnson kernel g = 2 g 3 abel modulo torsion abel torsion rank Johnson filtration lower central series M g M g (1) = I g M g (2) = K g M g (3)... Malcév completion M g I g Sp(2g, Z), relative Malcév completion Deligne Hain Sp semi-simple part nilpotent part nipotent part + k=1 M g(k)/m g (k + 1) Lie Lie + k=1 M g(k)/m g (k + 1) Der ( L(π(Σ g ) ) Malcév completion derivation Lie disc completion Lie symplectic form relation symplectic form derivation Johnson τ 1 : M g ( ) 1 2 Λ3 H/H Sp(2g, Z) Johnson H Hain Λ 3 H/H π 1 PH 3 (Jac(R)) P primitive part primitive Λ 3 H/H Hain Deligne relative Malcév completion nilpotent part Hain 9
10 H (Λ 3 H/H) Sp-module Sp-module Sp part symplectic Sp- H (Λ 3 H/H) Sp H (M g ; Q), H (Λ 3 H/H) Sp image tautological algebra., Teichmüller moduli abel Λ 3 H/H Sp-module H, 1 Grassmann Sp-module (4) Kontsevich graph (co)homology. 1992,3. Kontsevich graph cohomology Lie commutative, associative, Lie c g, a g, l g c g Kontsevich graph cohomology stable, g g = 1, 2 unstable cohomology transversely symplectic foliation Gel fand-kalinin-fuchs a g unstable stable Kontsevich Riemann moduli. l g Kontsevich Kontsevich Lie l g, Riemann cohomology 2 5 (5) Gal( Q/Q) Johnson Lie ,6 Johnson 1994 expectation, τ image, Sp Hain Johnson 10
11 Johnson Hain Grothendieck, Deligne, Drinfeld, Grothendieck anabelian geometry, Soule element Johnson image genus 0 S 2 CP 1 3 rank 2 derivation Soule trace genus 2 genus 1 trace genus 2 genus genus 0, 1, 2 3 genus 0 Newton genus 1 genus 1 parabolic Euclidean, genus 1 parabolic hyperbolic elliptic genus 1 6 (6) Vassiliev finite type invariants. symplectic Kontsevich knot Morsification knot Morse critical point critical point configuration chord diagram linear chord diagram Vassiliev chord diagram circular chord diagram. S 1, 2k pairing chord diagram relation 4T-relation frame independence relation. relations Vassiliev finite type invariatnt generating function free algebra Bar-Natan, Vassiliev invariant HP generators 11
12 . k = 1, 2, 3, 4... google sequence, mathematics, number, chord diagram k linear chord diagram (2k 1)!! cyclic symmetry cyclic action plot cyclic invariant chord diagrams cyclic invariant linear chord diagrams cyclic invariant symmetric alternating representation, Young 90 Young symmetric product. alternating product. symmetry cyclic invariants chord diagrams 1,2,5,18,105,902,9749,... cyclic 0,1,2,17,88,897,9562,... chord diagrams 2000 Vassiliev knot cyclic invariant tensors unknown Vassiliev 3 M link L, (M 3, L) LMO (Le-Murakami J.-Ohtsuki) 3 finite type invariants LMO Kontsevich commutative algebra c g symmetric algebra S H Q, derivation 3 Kontsevich 3 2n + 1 M 2n+1 1 tangent bundle symplectic knot Vassiliev invariant generating function. 3 generating function, 3 relation 12
13 generating function Galfalidis commutative version S 3 H Q, abel S 4 H Q 1 bracket relation 3 generating function Galoufalidis-Nakamura Sp-module Stoimenow linear chord diagram parallel regular regular chord diagram Vassiliev invariant (2k 1)!! Zagier Zagier Stoimenow Dedekind η- function, modular form Zagier Vassiliev invariants and a strange identity related to the Dedekind eta-function Zagier : 1 parabolic symplectic form ideal Σ 1 Σ 1,1 symplectic form : parabolic hyperbolic Goldman Σ 1,1 1 H Lie Σ 1 ideal parabaolic? genus 2 genus 0 genus 1 tangential bese point, parabolic 13
(5 19 ) 10,,, 2 3 Mathematica,, 1 Gauss( ), ,8,, 1827 Gauss 50, Gauss 200 xyz R 3 Σ Σ, surface, S Σ Σ, Σ g 2
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