2/14 2 () (O O) O O (O O) id γ γ id O O γ O O O γ η id id η I O O O O I γ O. O(n) n *5 γ η γ S M, N M N (M N)(n) ( ) M(k) Sk Ind S n S i1 S ik N(i 1 )
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1 1/14 * 1. Vassiliev Hopf P = k P k Kontsevich Bar-Natan P k (g,n) k=g 1+n, n>0, g 0 H 1 g ( S 1 H F(Com) ) ((g, n)) Sn. Com F Feynman ()S 1 H S n ()Kontsevich ( - - Lie ) 1 *2 () [LV12] Koszul 1.1 S F F. (1) (F )S M F[S n ] M = {M(n) n Z 0 } S f : M N S n f = {f n Hom Sn (M(n), N(n))} S S-Mod (2) 2 S M, N M N S (M N)(n) := i+j=n (3) 2 S M, N *3 M N S Ind Sm S i S j M(i) F N(j). (1.1) (M N)(n) := k 0 M(k) Sk N k (n). N k S (1.1) S k S k S n (4) S f : M M g : N N f g : M N M N (5) S I I(1) := F (), I(n) := 0 (n 1) S S-Mod *4 (S-Mod,, I). O = (O, γ, η) (S-Mod,, I) S O S γ : O O O () S η : I O () *1 yanagida@math.nagoya-u.ac.jp *2 operad *3 composite *4 [Mac98] monoidal category
2 2/14 2 () (O O) O O (O O) id γ γ id O O γ O O O γ η id id η I O O O O I γ O. O(n) n *5 γ η γ S M, N M N (M N)(n) ( ) M(k) Sk Ind S n S i1 S ik N(i 1 ) N(i k ) (1.2) k 0 2 i i k = n (i 1,..., i k ) γ γ(i 1,..., i k ) : O(k) O(i 1 ) O(i k ) O(i i k ) (1.3) i j (j = 1,..., k) (i i k ) May [May72] γ(i 1,..., i k ) (1.2) M N (µ; ν 1,..., ν k ) (0) S I (I, γ, η = id I ) (1) V End V End V (n) := Hom(V n, V ) γ f : V k V, f j : V ij V (j = 1,..., k) γ(i 1,..., i k )(f; f 1,..., f k ) := f(f 1 f k ) : V (i1+ +i k) V. (2) Com. S Com(0) := 0 n > 0 Com(n) := F() γ (1.3) id F (3) Assoc. S Assoc(n) := F[S n ]()γ (1.3) k = 2, i 1 = l, i 2 = m σ S l Assoc(l) τ S m Assoc(m) γ(l, m)(1; σ, τ) := σ τ, γ(l, m)((1, 2); σ, τ) := (1, 2) l,m (σ τ) σ τ S l+m Assoc(l + m) {1,..., l} σ {l + 1,..., l + m} τ (1, 2) l,m (1, 2) l,m := () ( ) 1,..., l, l + 1,..., l + m. l + 1,..., l + m, 1,..., l γ(i 1,..., i k )(σ; τ 1,..., τ k ) := σ i1,...,i k (τ 1 τ k ). (4) Lie Lie. S Lie(n) x 1,..., x n Lie x i 1 Lie γ Lie F = C Klyatchko [Kl74] Lie(n) Ind Sn Z n χ. χ Z n (Z n 1 n ). O, P α : O P S γ η *5 the space of n-ary operations
3 3/14. A O ( O ) O End A. S M V M(V ) M(V ) := n 0 M(n) Sn V n (M N)(V ) = M(N(V )) *6 S f : M N V S f(v ) : M(V ) N(V ) φ : V W S M(φ) : M(V ) M(W ) : A γ A : O(A) A 2 () (O O)(A) γ(a) O(A) O(O(A)) O(γA) O(A) I(A) η(a) O(A) γ A γ A γ A A O Com, Assoc, Lie Lie O = Assoc A O(A) = n 0 O(n) Sn A n n 0 A n = T (A) () γ A : T (A) A γ O(n) *7 [MSS02] [LV12] Koszul S-Mod M N := n M(n) Sn N n M N = n (M(n) C N n ) Sn N n S n S n M N := n (M(n) C N n ) Sn (S-Mod,, I). *8 C = (C,, ε) (S-Mod,, I) *9 S C S : C C C () S ε : C I () 2 () C C C C id 1 C C (C C) C id C (C C) I C C C O I ε id id ε (1) (O, γ, η) O O (2) V End c V End c V (n) := Hom(V, V n ) *6 [LV12, Proposition 5.1.3] *7 [Mac98] symmetric monoidal category *8 cooperad *9 comonoid
4 4/ Koszul [GiK94] Koszul [LV12, Chap. 7]. () (S ) M S M F(M) S η M : M F(M) (F(M), η M ) O S f : M O f = fη M F(M) S M S T n M T 0 M := I, T 1 M := I M, T n M := I (M T n 1 M),.... S i n : T n 1 M T n M i 1 : I I M 1 i n := id I (id M i n 1 ) i n T M T M := colim n T n M (1.4) T m M T n ()i 2 M T n 1 M T n M M T M j ([LV12, Theorem 5.5.1]). M (F(M), η M ) = ((T M, γ, η), j) γ γ m,n : T m M T n M T m+n M m = 0 I T N M = T n M γ 0,n := id TnM m > 0 T m M T n M T n M M (T m 1 M T n M) (id,id γm 1,n) T n M M T m+n 1 M i+j T m+n M. F(M) w id I(1) = F id I F(M) w(id) := 0, µ M(n) w(µ) := 1 w(µ; ν 1,..., ν n ) := w(µ) + w(ν 1 ) + + w(ν n ) F(M) (r) F(M) r F(M) (0) = F id, F(M) (1) = M. O I S O O/I {µ; ν 1,..., ν k } I γ(µ; ν 1,..., ν k ) I. S E S R F(E) (2) (E, R) * 10 O(E, R) := F(E)/(R) (R) R F(E) Com, Assoc, Lie S E E(2) = V, E(n) = 0 (n 2) F(E)(0) = 0, F(E)(1) = F, F(E)(2) = V F(E)(3) = V Ind S3 S 2 V 3V V. µ, ν V µ I ν, µ II ν, µ III ν F(E)(3) µ I ν(x, y, z) = µ(ν(x, y), z), µ II ν(x, y, z) = µ(ν(z, x), y), µ III ν(x, y, z) = µ(ν(y, z), x) *10 quadratic operad
5 5/14 S 3 F(E)(3) V = Fµ 1 E E 1 u I := µ I µ {u I, u II, u III } F(E 1 )(3) R Com := u I u II, u II u III O(E 1, R Com ) = Com R Lie := u I + u II + u III O(E 1, R Lie ) = Lie Assoc E 1 V = Fµ Fν 2 E E E 2 F(E 2 )(3) 12 u 1 := µ I µ, u 2 := ν II µ, u 3 := ν II ν, u 4 := µ III ν, u 5 := µ III µ, u 6 := ν I µ, u 7 := ν I ν, u 8 := µ II ν, u 9 := µ II µ, u 10 := ν III µ, u 11 := ν III ν, u 12 := µ I ν. R Assoc := u i u i+1 i = 1, 3, 5, 6, 9, 11 O(E 2, R Assoc ) = Assoc. S M * 11 F c (M) S C F c (M) C S C M F c (M) F c (M) S T M I (id) := id id ν M(n) (µ) := id µ + µ id n I M M I T 1 M T 1 M T 1 M T n = I (M T n 1 M) (µ; ν 1,..., ν k ) M T n 1 M (µ; ν 1,..., ν k ) := id (µ; ν 1,..., ν k ) + + (µ; ν 1,..., ν k ) + (i j (1.4) ) M T n 1 M id M M (T n 1 M T n 1 M) (M T n 1 M) T n 1 M j i T n M T n M. F c (M) F(M) F c (M) (r) r. S E S R F c (E) (2) (E, R) * 12 C(E, R) F c (E) C C F c (E) F c (E) (2) /R 0 C F c (E) C C(E, R) F c (E) C(E, R) O = O(E, R) Koszul O (E, R) * 13 O (E, R) := C(sE, s 2 R). s S * 14 S n K = p Z K p (sk) p := K p 1 S n sk S M (sm)(n) := s(m(n)) M(n) = M 0 (n) () S. O = O(E, R) (O ) O.. S := End sf 1 1 F S sf ( 1.1.2(1)) S c := End c sf *11 cofree cooperad *12 quadratic cooperad *13 [LV12] 180 anti-shriek *14 shift s suspension s
6 6/14 S S n S(n) = Hom F ((sf) n, sf) s 1 n sgn Sn, 1 n. (1) S M, N Hadamard M H N (M H N)(n) := M(n) N(n) S n S (2) O, P Hadamard S Hadamard O H P * 15 Hadamard (3) O * 16 Hadamard S H O C Hadamard S c H O O(E, R) Koszul * 17 O (E, R) O (E, R) := (S c H O ). (E, R) O ([GiK94, (2.1.11)]). () Com Lie, Assoc Assoc, Lie Com. ( 1.2.1) ([GiK94]). O = O(E, R) E(0) = 0 E(n) (O ) O. 1.3 () [GiK94] [LV12, 6.6] S S-Mod dg S dg S-Mod dg S n M(n) S n M(n) = p Z M p (n) ( 1) d : M p (n) M p 1 (n) dg S (M, d) dg S n H (M) ( S ) dg S dg S-Mod (dg S-Mod,, I) (O, γ, η) dg dg S (O, d) γ : O O O η : I O 0 dg S dg (C,, ε) B : { dg } { dg } dg * 18 dg O 0 dg ε : O I dg * 19 dg C 0 dg η : I C *15 S *16 operadic suspension O S so *17 [GiK94] quadratic dual operad *18 augmented dg operad *19 coaugmented dg cooperad
7 7/14 Ω : { dg } { dg } C = (C,, ε, η) dg * 20 C := coker(η : I C) Ω(C) := F(s 1 C) C d 2 : Fs 1 C F(s 1 C) Fs 1 C s (1) (Fs 1 Fs 1 ) (C (1) C) id τ id (Fs 1 C) (1) (Fs 1 C) F(s 1 C) (2) F(s 1 C) d 2 2 = 0 F(s 1 C, d 2 ) dg d 1 C d C Ω(C) d 1 d 2 dg Ω(C) := ( F(s 1 C), d 1 + d 2 ) Ω(C) [LV12, 6.5.2]. Ω B B : { dg } { dg } : Ω ([GiK94], [LV12, Prop ]). C = C(E, R) C * 21 p : Ω(C) C = O(s 1 E, s 2 R) Koszul p : H 0 (Ω(C)) C p.. p : Ω(O ) O O Koszul Koszul O B(P) dg. Com, Assoc, Lie Koszul () Koszul [LV12, Chap 8]. Koszul O Ω(O ) Koszul 1.4 O C, A, L Koszul O Koszul Koszul Ω(O ) dg *20 coaugmentation coideal *
8 8/14. Ω(O ) A O O O dg A dg Ω(O ) End A O A Ω(O ) O End A O. O = Com, Assoc, Lie O C, A, L half edge S T λ : S T T λ s 1, s 2 S λ(s 1 ) = λ(s 2 ) λ (1) Γ Flag(Γ) Flag(Γ) σ Flag(Γ) λ (Flag(Γ), σ, λ) (2) Flag(Γ) Γ flag( half-edge) (3) λ Γ Γ Vert(Γ) λ : Flag(Γ) Vert(Γ) v Vert(Γ) Leg(v) := λ 1 (v) Flag(Γ) Leg(v) v n(v) := Leg(v) v (4) σ 2 flag (a, b) Γ Edge(Γ) σ Γ Leg(Γ) n(γ) := Leg(Γ) flag Γ n(v) = 2 Edge(Γ) + n(γ). (2.1) v Vert(Γ) Γ = (Flag(Γ), σ, λ) 1 Γ flag [0, 1] v Vert(Γ) Leg(v) flag 0 [0, 1] Γ 2 flag 1 [0, 1]. (1) Γ Γ (2) Γ Γ 1 Flag(Γ) = {1,..., 9}, σ = (46)(57), λ = {1,..., 5} {6, 7, 8, 9} Γ * 22. (1) 2 Γ i = (Flag(Γ i ), σ i, λ i ) (i = 1, 2) φ : Γ 1 Γ 2 φ : Flag(Γ 1 ) Flag(Γ 2 ) φ σ 1 = σ 2 φ λ 1 (f) = λ 1 (f ) λ 2 (φ(f)) = λ 2 (φ(f )) (2) *22 [GeK98]
9 9/ Γ. (1) (g, n) Z 2 0 (2g 2) + n > 0 (2) Γ g : Vert(Γ) Z 0 (Γ, g) v Vert(Γ) (g(v), n(v)) (3) Γ g(γ) g(γ) := dim H 1 ( Γ ) + g(v). (2.2) v Vert(Γ) 2.2 [GeK98] Feynman S F V = {V ((g, n)) g, n Z 0 } V ((g, n)) S n (g, n) V ((g, n)) = 0 S V I n := I V ((g, I)) V ((g, I)) := [ V ((g, n))] Sn {1,..., n} I Γ = (Γ, g) S V V ((Γ)) V ((Γ)) := v Vert(Γ) V ((g(v), Flag(v))). S * 23 V colim Γ V (Γ) colim (Γ, g) e Edge(Γ) e Γ e g e : Γ e Z 0 g : Γ Z 0 * 24 (Γ e, g e ) * 25 Γ Γ e * 26 (g, n) Z 2 SG((g, n)) g(γ) = g (Γ, g) φ : {1,..., n} Leg(Γ) (Γ, g, φ) (Γ, g, φ) (Γ e, g, φ) S () M MV ((g, n)) := colim V ((G)) G Iso SG((g.n)) *23 endfunctor *24 e = {f, f } Flag(Γ) Γ λ : Flag(Γ) Vert(Γ) v = λ(f) v = λ(f ) 1 w g (w) := g(v) + g(v ), g e = g *25 g(γ e) = g(γ). *26 contraction
10 10/14 e v v w 2 Iso SG((g, n)) SG((g, n)) colim * 27 M [Mac98] * 28 (M, µ, η) µ : M 2 M η V : V M(V ) 1 n Γ 1 n V V ((Γ1 n)) MV. (1) C F : C C µ : F 2 F η : id C F (2) (F : C C, µ, η) (C, γ) C C γ : F (C) C γ F (γ) = γ µ C : F 2 (C) C γ η C = id C M. (1) MV (g, n) Γ [SG(g,n)] V ((Γ)) Aut(Γ) [SG(g, n)] Aut(Γ) (2) S V 2 i : V ((g, n)) V ((g, n )) V ((g + g, n + n 2)), ξ i,j : V ((g, n)) V ((g + 1, n 2)). Deligne-Mumford M g,n S F Deligne-Mumford () M((g, n)) := M g,n µ M SG((0, n)) n > 2. (2.2) (Γ, g, φ) SG((0, n)) n Γ H 1 (Γ) g(v) = 0 ( v Vert(Γ)) g : Vert(Γ) Z 0 S V 0 S n {V (n)} n> * 29 O O(n) S n S n+1 = S n, τ n := (0, 1,..., n) a O(m)b O(n) (a m b) = b 1 a a := τ m a O(m). Com, Assoc, Lie *27 F : D C colim x D F (x) C C {ι y : F (y) C} y D ι zf (f) = ι y y, z D f : y z *28 monad triple *29 cyclic operad
11 11/14. O 0 {O((0, n))}. O O () ().. Com cyc Com cyc Frobenius Com cyc Com cyc ((g, n)) = F() 3 Kontsevich Lie [K93, K94] Koszul v Leg(v) o v Γ ( ) Γ, {o v } v Vert(Γ) Γ half-edge (flag) Surf(Γ) ( ) Γ, {o v } v Vert(Γ) Vert, Edge underlying graph Γ Vert, Edge ( ) Γ, {o v } v Vert(Γ) Γ ( ) Γ, {o v } v Vert(Γ) Γ Γ Vert(Γ) e Edge(Γ) * (1) 2 underlying graph (2) Γ Aut(Γ) Γ e Edge(Γ) Γ/e e * 31 Γ Γ/e Vert(Γ/e) e 1 2 Vert(Γ/e) 1 Vert(Γ) Γ Γ/e (Γ e ) *30 : Vert(Γ) ord e Edge(Γ) or e {±1} (ord, {or e} e Edge(Γ) ) O Γ s, t O Γ s = (ord, {or e} e Edge(Γ) ), t = (ord, {or e } e Edge(Γ)) ord ord or e = or e ( e Edge(Γ)), ord = ord 1 f Edge(Γ), or f or f, ore = or e ( e f) s, t O Γ s t u O Γ, s, u u, t O Γ / Γ *31 underlying graph Γ = (F = Flag(Γ), σ, λ) e = {f 1, f 2 } F Γ/e flag F := F \ {f 1, f 2 }, σ F F λ v i := λ(f i ) Vert(Γ) λ := ( F λ Vert(Γ) Vert(Γ)/(v 1 v 2 ) ) e v v i v
12 12/ Leg(Γ) = 3 Γ Γ Γ Γ = Γ RG Edge(Γ) Γ RG RG Γ. RG (Γ) := Γ/e e Edge(Γ) (RG, ) (RG, ) () HRG HRG HRG 3.3 (RG, ) RG conn HRG conn HRG conn g 0 S 1 H ΩAssoc ((g, 0)). ΩAssoc 3.4 Kontsevich Lie Chevalley-Eilenberg Lie g (g) g det x; = 1 (x g) g Chevalley-Eilenberg d(x 1 x n ) := ( 1) i+j 1 [x i, x j ] x 1 x i x j x n 1 i<j n d : n (g) n 1 (g) C (g) := ( (g), d) H CE (g) Lie a p 1,..., p n, q 1,..., q n θ θ ( n i=1 (p iq i q i p i ) ) = 0 Lie a n a n a n+1 Lie a := lim a n ([K93, K94]). H CE (a ) Prim H CE (a ) Prim H CE k (a ) Prim H CE (sp ) k s>, 2g 2+s>0 H 2g+s 1+k (RG g,s ; Q). RG g,s Surf(Γ)( ) g s
13 13/ Riemann {p 1,..., p s } g Σ g,s 1 2g 2 + s > 0 Σ g,s := Σ g,s \ {p 1,..., p s } X f : X Σ g,s [f] (X, [f]) X * 32 (X, [f]) (Y, [g]) h : X Y g h : X Y Σ g,s f Teichmüller T g,s Finchel-Nielson T g,s (R R>0 ) 3g 3 (X, [f]) T g,s * 33 X * 34 * 35 X 2 1 Tg,s dec Tg,s dec Teichmüller T dec g,s T g,s (a 1,..., a s ) (R >0 ) 6g 6+3s (R >0 ) s Penner[P87] T dec g,s MC g,s Σ g,s () MC g,s T g,s g.(x, [f]) := (X, [gf]) T dec g,s MC g,s dec Tg,s T g,s M g,s := T g,s /MC g,s s g M dec g,s := T dec g,s /MC g,s * 36 T g,s M dec g,s M g,s (R >0 ) s : H (M dec g,s, Q) H (M g,s, Q). (3.1) Γ Σ(Γ) Σ(Γ) s g () Γ RGr g,s Γ RGr g,s Vert(Γ) Edge(Γ) = 2 2g s Γ Edge(Γ) R >0 Γ RGr g,s RGr met g,s RGr met g,s RGr met g,s RGr g,s Γ RGr g,s σ Γ σ Γ R Edge(Γ) / Aut(Γ) * ([P87]). RGr met g,s M dec g,s. M g,s ( (3.1) ) M g,s ([MSS02, Theorem 5.67]). H 6g 6+3s (RG g,s ) H (M g,s, Q) H (M g,s, Q) H (BMC g,s, Q) *32 marking makring *33 decoration *34 puncture *35 horocycle *36 M g,s M dec g,s s M g,s *37 : Γ 1 Γ σ Γ σ Γ
14 14/14 Γ 3 χ(γ) = 1 n Gr (n) Γ Gr (n) Edge(Γ) R >0 Gr (n) Gr (n) met orbifold 3n 3 ϵ Poincaré H c 3n 3 (Gr (n) met, ϵ) H (Gr (n) met, Q). (3.2) Γ Gr (n) G (n) Vert(Γ) * 38 G (n) = (G (n), ) () (3.2) G (n) Out(n) (Culler-Vogtmann [CV86]). G (n) BOut(n) H 2n 2 (G (n), ) H (BOut(n), Q). G Lie c [K93, K94] Q[p 1,..., p n ] n i=1 dp i dq i Lie c n c n c n+1 c c Chevalley-Eilenberg Prim H CE (c ) Prim H k (c ) Prim H k (sp ) n 2 H k (G (n) ). [CV86] Culler, M., Vogtmann, K., Moduli of graphs and automorphisms of free groups, Invent Math., 84 (1986), [GeK98] Getzler, E., Kapranov, M. M., Modular operads, Compositio Math., 110 (1998), no. 1, [GiK94] Ginzburg, V., Kapranov, M., Koszul duality for operads, Duke Math. J., 76 (1994), no. 1, [Kl74] Klyatchko, A. A., Lie elements in the tensor algebra, Siberian Math. J., 15 (1974), [K93] Kontsevich, M., Formal noncommutative symplectic geometry, in The Gelfand Mathematical Seminars, , , Birkhäuser Boston, Boston, MA, [K94] Kontsevich, M., Feynman diagrams and low-dimensional topology, in First European Congress of Mathematics, Vol. 2 (Paris, 1992), , Progr. Math., 120, Birkhäuser, Basel, [L] Loday, J., Cyclic homology, Grundlehren der mathematischen Wissenschaften, 301, 2nd ed., Springer [LV12] Loday, J., Vallette, B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, 346. Springer, Heidelberg, [Mac98] Mac Lane, S., Categories for the working mathematician, Second edition, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, (1998). [MSS02] Markl, M., Shnider, S., Stasheff, J., Operads in algebra, topology and physics, Mathematical Surveys and Monographs, 96. American Mathematical Society, Providence, RI, [May72] May, J. P., The geometry of iterated loop spaces, Lectures Notes in Mathematics, 271, Springer- Verlag, Berlin, [P87] Penner, R. C., The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys., 113 (1987), *38 RG
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