Rational Homotopy and Intrinsic Formality of E n -operads Part II

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1 Rational Homotopy and Intrinsic Formality of E n -operads Part II Benoit Fresse Université de Lille 1 φ Les Diablerets, 8/23/ ( exp ˆL }{{} t 12, ) }{{} t 23 Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

2 Plan Part I: 0. The notion of an operad 1. The little n-discs operads 2. The (co)homology of the little discs operads 3. The rational homotopy theory of operads 4. The statement of the intrinsic formality theorem Part II: 1. The Drinfeld-Kohno Lie algebra operad 2. The realization of the (n 1)-Poisson cooperad 3. The obstruction theory proof of the intrinsic formality theorem Appendix: The fundamental groupoid of the little 2-discs operad Appendix: The rational homotopy theory interpretation of Drinfeld s associators Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

3 1. The Drinfeld-Kohno Lie algebra operad Definition: The rth Drinfeld-Kohno Lie algebra ˆp(r) is defined by the presentation: where: ˆp(r) = ˆL(t ij, 1 i j r)/ [t ij, t kl ], [t ij, t ik + t jk ] a generator tij such that t ij = t ji is assigned to each pair 1 i j r, the commutation relations [tij, tkl] 0 hold for all quadruple of pair-wise distinct indices i j k l, and the Yang-Baxter relations [t ij, t ik + t jk ] = 0. hold for all triple of pair-wise distinct indices i j k l. Remark: We have Û(ˆp(r)) = ˆT(t ij, 1 i j r)/ [t ij, t kl ], [t ij, t ik + t jk ] with [u, v] = uv ±vu. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

4 The monomials t i1 j 1 t ir j r Û(ˆp(r)) are usually represented by chord diagrams on r strands. For instance: t 12 t 12 t 36 t 24 = In this representation, the commutator and Yang-Baxter relations read:. i j k l i j k l = 0, i j k + i j k i j k i j k = 0. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

5 Observation: The Drinfeld-Kohno Lie algebras inherit the structure of an operad in the category of complete Lie algebras ˆp, and the associated algebras of chord diagrams form an operad in the category of Hopf algebras Ûˆp: the symmetric group Σr acts on ˆp(r) by permutation of strand indices; the composition operations ˆp(k) ˆp(l) i ˆp(k + l 1) Û(ˆp(k)) Û(ˆp(l)) i Û(ˆp(k + l 1)), are given by the following insertion operations (in the chord diagram picture): = Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

6 2. The realization of the (n 1)-Poisson cooperad Reminder: We have H (D 2 (r)) = H (F( D 2, r)) = S(ω ij, 1 i j r) (ω 2 ij, ω ijω jk + ω jk ω ki + ω ki ω ij ) where deg(ω ij ) = 1 and ω ij = ω ji for each pair i j. Theorem (Kohno): Let: C CE (ˆp(r)) = Chevalley-Eilenberg cochain complex = (S(Σ 1 ˆp(r) ), ), We have a quasi-isomorphism of commutative dg-algebras κ : C CE (ˆp(r)) H (F( D 2, r)) given by the following mapping: { κ(t ij ) = ω ij, for each pair i j, for each arity r N. κ(π ) = 0, when π has weight m > 1. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

7 Observation (Tamarkin): The dg-algebras C CE (ˆp(r)) inherit composition coproducts C CE (ˆp(k + l 1)) i C CE (ˆp(k) ˆp(l)) C CE (ˆp(k)) C CE (ˆp(l)) and form a Hopf dg-cooperad. Furthermore, the Kohno map defines a quasi-isomorphism of Hopf dg-cooperads C CE (ˆp) H (D 2 ). Observation: This result extends to all operads D n, for a graded version of the Drinfeld-Kohno Lie algebra operad ˆp n defined by the same presentation, with deg(t ij ) = n 2 and t ij = ( 1) n t ij as unique changes. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

8 Proposition: The Hopf cooperad C CE (ˆp n) defines a cofibrant resolution of the cohomology cooperad H (D n ) = Pois c n 1. Corollary: We have Pois c n 1(r) = Mor dg Com (C CE (ˆp n(r)), Ω ( ))) Pois c n 1(r) = MC (ˆp n (r)) where MC (ˆp n (r)) is the simplicial set of forms such that deg (γ) = 1 and: γ ˆp n (r) ˆ Ω ( ) δ(γ) + 1 [γ, γ] = 0. 2 Remark: In the case of the standard Drinfeld-Kohno Lie algebra operad ˆp = ˆp 2, we have: MC (ˆp(r)) B(G Ûˆp(r)) = B(exp ˆp(r)), where B := classifying space functor from groups to spaces. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

9 3. The obstruction theory proof of the intrinsic formality theorem Idea: Use the model G : dg Hopf Op c SimpOp op : Ω to transport the problem in the category of Hopf dg-cooperads. Goal: Let K be a Hopf dg-cooperad such that we have: a cohomology isomorphism H (K) Pois c n 1, for some n 3, an involutive isomorphism J : K K which mirrors the action of a hyperplane reflection on D n in the case 4 n. Pick: a fibrant resolution of this Hopf dg-cooperad K ResOp (K) =: Q, a cofibrant resolution of the Poisson cooperad R := Res Com (Pois c n 1) Pois c n 1, and prove the existence of a morphism of Hopf dg-cooperads: Pois c n 1 Res Com (Pois c n 1)? Res Op (K) K which induces H (K) Pois c n 1. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

10 Definition: Let Map dg Hopf Op c (R, Q) = {Hopf dg-cooperads maps φ : R Q } so that Map dg Hopf Op c (R, Q) 0 = Mor dg Hopf Op c (R, Q). Constructions: Take Res Op (K) := Tot(Res Op (K)), where R = Res Op (K) := operadic triple coresolution of K, and Res Com (Pois c n 1) := Res Com (Pois c n 1), where Q = Res Op (H) := cotriple resolution of H = Pois c n 1 in dg Com. Observation: We have Map dg Hopf Op c ( R, Tot(Q )) = Tot Diag(X ) where X = Map dg Hopf Op c (R, Q ). Theorem (Bousfield): The obstructions to the existence of φ Tot(X ) 0 φ Mor dg Hopf Op c (R, Q) lie in π s+1 π s (X ). Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

11 Theorem (BF, Willwacher): We have a sequence of reductions: π π (X ) H BiDer dg Hopf Op c (Res Com (Pois c n 1), Res Op (Poisc n 1)) H (BiDef dg Hopf Op c (Pois c n 1, Pois c n 1) H (BiDef dg Hopf Op c (Graph c n, Pois c n 1) H (Q GC 2 n) where: BiDef dg Hopf Op c (, ) is an analogue for Hopf dg-cooperads of the Gerstenhaber-Schack deformation bicomplex of bialgebras, Graph c n is an operad of graphs such that Graph c n Pois c n 1. GC 2 n is Kontsevich s complex of graphs with at least bivalent vertices. Observation: For n 3, we have H 1 (Q GC 2 n) = 0 when n 0(4), while we have H 1 (Q GC 2 n) J = 0 for all n, for an involution J inherited from Pois c n 1 H (D n ) H (K) and Graph c n Pois c n 1. Corollary: The obstructions to the existence of a map φ Mor dg Hopf Op c (R, Q) vanish. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

12 References The intrinsic formality of E n -operads, with Thomas Willwacher Preprint arxiv: Homotopy of Operads and Grothendieck-Teichmüller Groups Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

13 Homotopy of Operads and Grothendieck-Teichmüller Groups Part I. From operads to Grothendieck-Teichmüller groups 1. Introduction to the general theory of operads 2. Braids and E 2 -operads 3. Hopf algebras and the Malcev completion 4. The operadic definition of (pro-unipotent) Grothendieck-Teichmüller groups Part II. The applications of homotopy theory to operads 1. Introduction to general homotopy theory methods 2. Modules, algebras and the rational homotopy of spaces 3. The rational homotopy of operads 4. Applications of the rational homotopy to E n -operads Part III. The computation of homotopy automorphism spaces of operads 1. The applications of homotopy spectral sequences 2. The case of E n -operads Appendices. 1. The construction of free operads 2. The cotriple resolution of operads 3. Cofree cooperads and the bar and Koszul duality of operads Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

14 Thank you for your attention! Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

15 Appendix. The fundamental groupoid of the little 2-discs operad The homotopy equivalence D 2 (r) F( D 2, r) implies that the space D 2 (r) is an Eilenberg-MacLane space K(P r, 1), where P r = π 1 F( D 2, r) is the pure braid group on r strands. The idea is to use the fundamental groupoids π D 2 (r) in order to get a combinatorial model of the operad D 2. We have D 2 B(π D 2 ), where B := classifying space functor from groupoids to spaces. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

16 Construction: The fundamental groupoid π D 2 (r) has Ob π D 2 (r) = D 2 (r) as object set. The morphisms Mor π D2 (r)(a, b) are homotopy classes of paths γ : [0, 1] D 2 (r) going from γ(0) = a to γ(1) = b. The map Mor π D2 (r)(a, b) disc centers Mor π F( D 2,r)(a, b) identifies the morphism sets of this groupoid with cosets of the pure braid group P r inside the braid group B r. Thus, a morphism in this groupoid can be represented by a picture of the form: γ = 1 2 = a Mor π D 2 (2) 2 1 = b Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

17 The structure of the fundamental groupoid operad: The groupoids π D 2 (r) inherit a symmetric structure, as well as operadic composition products i : π D 2 (k) π D 2 (l) π D 2 (k + l 1), and hence form an operad in the category of groupoids. In the braid picture, the operadic composition products can be depicted as cabling operations: = Mor PaB(2) Mor PaB(2) Mor PaB(3) Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

18 Ideas: There is no need to consider the whole D 2 (r) as object set. The groupoids of parenthesized braids PaB(r) are full subgroupoids of the fundamental groupoid π D 2 (r) defined by appropriate subsets of little 2-discs configurations Ω(r) D 2 (r) as object sets. These object sets Ω(r) are preserved by the operadic composition structure of little 2-discs so that the collection of groupoids PaB(r), r N, forms a suboperad of π D 2. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

19 Construction: The sets Ω(r), r = 2, 3, 4,..., prescribing the origin and end-points of paths in PaB(r), consist of little 2-disc configurations of the following form: i j i j k i j k i j k l (the indices i, j,... run over all permutations of 1, 2,... ). These configurations represents the iterated operadic composites of the following element µ = 1 2 D 2 (2) and the sets Ω(r), r N, are the components of the suboperad of D 2 generated by this element. This operad is free. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

20 The operad PaB is generated by the following fundamental morphisms: the associator = µ 1 µ α = Mor PaB(3) = µ 2 µ and the braiding 1 2 = µ τ = Mor PaB(2) 2 1 = tµ where t = (12) Σ 2. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

21 In the morphism set of the operad PaB: the associator satisfy the pentagon equation and we have two hexagon equations combining associators and braidings. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

22 The first one reads: Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

23 and the second one reads: Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

24 Theorem (Mac Lane + Joyal-Street): An operad morphism φ : PaB P, where P is any operad in the category of categories, is uniquely determined by: an object m Ob P(2), which represents the image of the little 2-disc configuration µ D 2 (2) under the map φ : Ob PaB(2) Ob P(2), an isomorphism a Mor P(3) (m 1 m, m 2 m), which represents the image of the associator α Mor PaB(3) (µ 1 µ, µ 2 µ), and an isomorphism c Mor P(2) (m, tm), which represents the image of the braiding τ Mor PaB(2) (µ, tµ), such that a and c satisfy the analogue of the usual pentagon and hexagon relations of braided monoidal categories in Mor P. This result implies that PaB is generated by operations defining the structure of a braided monoidal category. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

25 Appendix: The rational homotopy theory interpretation of Drinfeld s associators Construction: Take the groups of group like elements G(Ûˆp(r)) = {u Û(ˆp(r)) ɛ(u) = 1, (u) = u ˆ u} in the complete Hopf algebras Û(ˆp(r)). Regard these groups as the morphism sets of groupoids CD Q (r) such that Ob CD Q (r) = pt. These groupoids CD Q (r) form an operad (in the category of groupoids) CD Q, with the composition products on morphisms Mor CD Q }{{ (k) } =G Ûˆp(k) Mor CD Q (l) }{{} =G Ûˆp(l) i Mor CD Q (k + l 1) }{{} =G Ûˆp(k+l 1) induced by the operadic composition of chord diagrams. Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26

26 Reminder: We have π D 2 PaB D 2 B(PaB) and Pois c 1(r) = MC (ˆp(r)) B(G Ûˆp(r)) = B(CD Q (r)). Definition: The set of Drinfeld s associators Ass Q is the set of operad morphisms φ : PaB CD Q whose extension to a Malcev completion of the operad of parenthesized braids PaB CD Q PaB Q ˆφ is an equivalence of groupoids arity-wise. Observation: Any such φ : PaB CD Q gives a rational weak-equivalence on classifying spaces: φ : B(PaB) Q B(CD Q ) Benoit Fresse (Université de Lille 1) Intrinsic Formality of E n-operads Les Diablerets, 8/23/ / 26