Grothendieck-Teichmueller group, operads and graph complexes: a survey
aa r X i v : . [ m a t h . QA ] A p r GROTHENDIECK-TEICHM ¨ULLER GROUP,OPERADS AND GRAPH COMPLEXES: A SURVEY
SERGEI MERKULOV
Abstract.
This paper attempts to provide a more or less self-contained introduction into theory of theGrothendieck-Teichm¨uller group and Drinfeld associators using the theory of operads and graph complexes.
Contents . Introduction . Completions of groups, Lie algebras and algebras Completed filtered vector spaces, Lie algebras and algebras 3
Baker-Cambell-Hausdorff formula 4
Prounipotent completions 5
Quillen’s construction of the prounipotent completion 6
Examples of prounipotent completions 7
Profinite completions 9 . Monoidal categories and monoidal functors Monoidal categories (see e.g. [ES]) 9
Examples of monodial categories 10
Monoidal functors 12
Examples of monoidal functors 12
Closed symmetric monoidal categories 15 . Operads Basic axioms for operadic compositions 15
Axioms for a unit 16
Basic examples 17
Degree shift functor 21
Cosimplicial structures on operads 21
Lie algebras associated with operads 22 . Operad of parenthesized braids and d GT Geometric definition of the braid group 22
Algebraic definition of the (pure) braid group 24
Pure braids and semidirect products of free groups 24
Non- S operad of pure braids 25 An operad of parenthesized braids 26
D. Bar-Natan’s operad d P a B K Grothendieck-Teichm¨uller group d GT ( K ) 30 K. Furusho’s Theorem 32 . Infinitesimal braids,
GRT and associators Graded Lie algebra of PB n Operadic structure on Lie algebras of infinitesimal braids [T2] 34
Cosimplicial complex of the operad of infinitesimal braids 35
Operad of chord diagrams 35
Cosimplicial structure on c CD K D. Bar-Natan’s operad of parenthesized chord diagrams 36
Grothendieck-Teichm¨uller group
GRT and its Lie algebra 37 Associators 38 . Grothendieck-Teichm¨uller group, graph complexes and T.Willwacher theorems Operads of graphs 39 .2. M. Kontsevich graph complexes and T. Willwacher theorems 41
Oriented graph complexes 44
Sourced graphs 45
Multi-oriented graph complexes 45 . Some applications of the theory of Drinfeld’s associators,
GRT and graph complexes Universal quantizations of Lie bialgebras 47
Universal quantizations of Poisson structures 49
Solutions of the Kashiwara-Vergne conjecture 50
Formality theorem in the Goldman-Turaev theory 51
Cohomology of moduli spaces of algebraic curves and grt . Introduction
The Grothendieck-Teichmueller group GT (and its graded version GRT ) is one of the most interesting andmysterious objects in modern mathematics. This group together with its principal homogeneous space —the set of V. Drinfeld’s associators — plays a central role in many seemingly unrelated areas of mathematics:V. Drinfeld pioneered its applications of in the number theory and the theory of quasi-Hopf algebras [Dr2];P. Etingof and D. Kazhdan [EK, ES] used it to solve the Drinfeld quantization conjecture for Lie bialgebras;the formality theories of M. Kontsevich [K1, K2, K3] and D. Tamarkin [T1, T2] unravels the role of thegroup
GRT in deformation quantizations of Poisson structures; A. Alekseev and C. Torossian applied itto the solution of the Kashiwara-Vergne problem in the Lie theory [AT2]; the authors of [MW5] found itsinterpretation as a symmetry group of the involutive Lie bialgebra properad playing a key role in the stringtopology; A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef [AKKN1, AKKN2] proved its importance in theGoldman-Turaev theory of spaces of free homotopy loops in Rieman surfaces of genus g with n punctures;this group — more precisely its Lie algebra grt — plays a central role in the recent spectacular advances ofM. Chan, S. Galatius and S. Payne [CGP1] in the theory of the cohomology groups of moduli spaces M g of genus g algebraic curves. T. Willwacher [W1, W3] established a very important link between grt andcohomologies of some graph complexes which found many applications.Our main purpose is to give a more or less short and self-contained introduction into the theory of theGrothendieck-Teichm¨uller group and its applications through its operadic, properadic and graph complexesincarnations, and explain without proofs some of the results mentioned in the previous paragraph.The Grothendieck-Teichm¨uller theory first appeared in A. Grothendieck’s famous Esquisse d’un programme (1981) with the purpose to give a new geometric description of the universal Galois group
Gal ( Q ) of the fieldof rational numbers; A. Grothendieck’s main tool to approach this problem was the Deligne-Mumford modulistack M of algebraic curves of arbitrary genus with marked points and a very insightful observation thatthe (outer) automorphism group Out ( π geom ( M )) of the geometric fundamental group of this stack can bedescribed more or less explicitly as the set of elements of a free profinite group on two generators satisfyinga small number of equations (see [L] and [LS] for more details and references). A few years later (1989) V.Drinfeld introduced [Dr2] K -prounipotent Grothendieck-Teichm¨uller groups GT and GRT for an arbitraryfield K of characteristic zero while studying braided quasi-Hopf algebras and their universal deformations.The profinite version d GT of the first group contains Gal ( Q ) and is essentially Out ( π geom ( M )), where M is the of moduli spaces of genus zero curves with marked points.Drinfeld has written down explicitly [Dr2] systems of algebraic equations defining elements of both prounipo-tent groups GT and GRT . Thanks to the works of D. Bar-Natan [B-N] the algebro-geometric meanings ofthese two systems of equations are well-understood by now with the help of the theory of operads — GT is essentially the automorphism group of the topological operad d P a B of parenthesized braids while GRT isthe automorphism group of a much simpler graded analogue \ P a CD of the operad d P a B . The set of Drinfeldassociators can be identified with the set of isomorphisms d P a B → \ P a CD of operads. This part of the storyexplaining GT and GRT as automorphism groups of operads is covered in §§ resse has written a book [Fr] explaining this approach to GT and GRT in much more detail. In the secondpart, §§ GRT via the graphcomplexes (and compactified configuration spaces), and its applications to the deformation quantization the-ory of Poisson structures and Lie bialgebras, Lie theory, Goldman-Turaev theory, and the theory of modulispaces M g . We explain various graph complexes incarnations of the Lie algebra grt of GRT some whichappear quite mysterious at present. . Completions of groups, Lie algebras and algebras
A topological ring (in particular,a field) is a ring R which is also a topological space such that both the addition and the multiplicationoperations define continuous maps R × R → R .A topological vector space V is a vector space over a topological field K (say, R or C equipped with theirstandard topologies or with the discrete topology) which is endowed with a topology making the vectoraddition V × V → V and scalar multiplication K × V → V operations into continuous maps. One can definesimilarly topological K -(co)algebras , topological Lie algebras over K , topological Hopf algebras over K , etc.Topological vector spaces, rings and K -algebras are particular examples of uniform spaces in which it makesense to talk about Cauchy sequences and hence about their completeness. Any such a space V can becompleted in an essentially unique way with the help of equivalence classes of Cauchy sequences; V sits in itscompletion b V as a dense subset and the induced topology coincides with the original one. The completionof the tensor product b V ⊗ c W is denoted by b V b ⊗ c W (sometimes we abbreviate the latter notation to b V ⊗ c W when no confusion may arise).We shall be interested in this survey in a class of topological spaces, rings and (co)algebras whose topolo-gies are determined by filtrations. Their completions — often called complete filtered spaces, rings and(co)algebras — can be constructed in an explicit way as inverse limits. We shall explain the phenomenonfor algebras; the adoption of all the constructions below to Lie algebras, coalgebras, etc. is straightforward(see, e.g., [Ei]).Let A be a K -algebra equipped with a descending filtration by ideals,(1) A = m ⊃ m ⊃ m ⊃ . . . Hence we get a directed system of quotient rings and their morphisms, . . . −→ A/ m i +1 −→ A/ m i −→ . . . −→ A/ m −→ A/ m −→ . The associated completed filtered algebra is defined as the inverse limit, b A := lim ←− A/ m i := lim ←− A/ m i := ( a = ( a , a , . . . ) ∈ ∞ Y i =1 A/ m i | a j = a i mod m i for all j > i ) The algebra b A has an induced filtration by ideals, b m i := n a = ( a , a , . . . ) ∈ b A | a j = 0 for all j ≤ i o . It is clear that the quotient algebras A/ m i and b A/ b m i can be identified. The completed tensor product ofsuch completed algebras is defined by b A b ⊗ c A ′ := lim ←− i,j A/ m i ⊗ A ′ / m ′ i . The filtered algebra A and and its completion b A can be made into topological spaces by defining a basis ofopen neighborhoods of a point a in A or, respectively, in b A to be (cid:8) a + m i (cid:9) i ∈ N or, respectively , (cid:8) a + b m i (cid:9) i ∈ N . Such a topology on a filtered algebra is called the
Krull topology. It is not hard to see that b A is thecompletion of A as a topological algebra. Indeed, let { a i } i ≥ be a Cauchy sequence in A equipped with theKrull topology, that is, a sequence which satisfies the condition: for any open neighborhood U of zero in A here is a number N U such that for any i, j > N U one has a i − a j ∈ U . Equivalently, { a i } i ≥ is a Cauchysequence in A if and only if for any integer n there exists an integer N n such that a i − a j ∈ m i for all i, j > N n . Such a sequence always converges in b A to the point a whose n -th coordinate in Q n A/ m n is equal, bydefinition, to a N n mod m n . Reversely, any point in b A gives rise to a Cauchy sequence in A .We shall work below with filtrations generated by powers m i := I i of a fixed ideal I in A . The associatedcompletion b A is often called the I - adic completion of A , and the associated topology on A is called the I - adictopology . (i) Let A = K [ x , . . . , x n ] be a polynomial algebra over K , and let I be its maximal ideal. The I -adiccompletion of A , b A = K [[ x , . . . , x n ]] , is the algebra of formal power series over K .(ii) Let g be a positively graded Lie algebra, g = L ∞ i =1 g i , with g i being finite dimensional. It can beequipped with a descending filtration as in (1) given by the Lie ideals m i := L j ≥ i g j . The completionof g with respect to this filtration is then equal to b g = Y i ≥ g i and is called the degree completion of g (similarly one can define the degree completion of positivelygraded vector spaces, rings, algebras, etc).Consider next the universal enveloping algebra U ( b g ) of the completed graded Lie algebra b g . It is,by definition, the quotient of the tensor algebra ⊗ • b g by the ideal J generated by all elements of theform x ⊗ y − y ⊗ x − [ x, y ], x, y ∈ b g . The tensor algebra inherits a positive gradation from b g in thestandard way, ⊗ • b g = ∞ M i ∈ ( ⊗ • b g ) i with ( ⊗ • b g ) i := M i = i ... + inn ≥ ,i ,...,in ≥ b g i ⊗ . . . ⊗ b g i n . As [ b g i , b g j ] ⊂ b g i + j , the ideal J is homogeneous with respect to this gradation, i.e. it is generated bythe homogeneous elements. Therefore U ( b g ) comes equipped with an induced gradation and we candefine its graded completion b U ( b g ). To define the completed universal enveloping algebra b U ( b g ) we need only a descendingfiltration of g as in (1), a weaker structure than the positive gradation. However, in applications below sucha filtration always comes from a positive gradation on g as in the above Example . Use the Krull topology on b U ( b g ) to show that b U ( b g ) is a completed filtered Hopf algebra(with the coproduct, ∆ : b U ( b g ) → b U ( b g ) b ⊗ b U ( b g ), taking values in the completed tensor product). Recall that one can associate a group G := exp( g ) to a (com-pleted!) positively graded Lie algebra over a field K of characteristic zero, g = ∞ Y k =1 g k , with all graded components being of finite dimension. The group G coincides with g as a set (and theidentification map g → G is denoted by exp while its inverse G → g by log), and the group multiplication isdefined by the Campbell-Hausdorff formula:exp( x ) · exp( y ) := exp( bch ( x, y )) , here(2) bch ( x, y ) = log( e x e y ) = x + y + 12 [ x, y ] + 112 [ x, [ x, y ]] −
112 [ y, [ y, x ]] . . . , and e x := P ∞ k =0 x k k ! and log(1 + x ) := P ∞ k =1 ( − k x k k . This formal power series satisfies obviously theassociativity relation, bch ( x, bch ( y, z )) = bch ( bch ( x, y ) , z ) = log( e x e y e z ) , in the ring K hh x, y, z ii . Let lie n stand for the free Lie algebra over K generated by n letters x , . . . , x n and b lie n forits degree completion (with degrees of the generators x , . . . , x n set to be 1). Let ass n := K h x , . . . , x n i bethe free associative algebra generated by x , . . . , x n and c ass n = K hh x , . . . , x n ii its degree completion. Onecan make ass n and c ass n into Lie algebras by setting [ X, Y ] = XY − Y X . Then lie n ⊂ ass n and b lie n ⊂ c ass n are Lie subalgebras (in fact they are the smallest Lie subalgebras containing all the generators).One can make c ass n into a bialgebra (in fact, a Hopf algebra) by defining a coproduct,∆ : c ass n −→ c ass n b ⊗ c ass n , by first setting ∆( x i ) := 1 ⊗ x i + x i ⊗ c ass n by the rule∆( x i x i · · · x i k ) = ∆( x i )∆( x i ) · · · ∆( x i k ) . An element g ∈ c ass n is called grouplike if ∆( g ) = g ⊗ g . An element p ∈ c ass n is called primitive if∆( p ) = 1 ⊗ p + p ⊗
1. The following is true: • Let f be a formal power series from c ass n whose constant term is zero. Then e f ∈ d ass n is grouplikeif and only if f is primitive. • The set of primitive elements in c ass n is a Lie subalgebra of ( c ass n , [ , ]) which can be identified with b lie n . • The set of grouplike elements in c ass n forms a group with respect to the multiplication in c ass n whichcan be identified with exp( b lie n ).As a Hopf algebra, c ass n is precisely the completed universal enveloping algebra of b lie n . A linear algebraic group over a field K is, by definition, an algebraicgroup that is isomorphic to an algebraic subgroup of the group of invertible n × n matrices GL ( n, K ), thatis, to a subgroup defined by polynomial equations. Such a group G is called unipotent if its image under theembedding G → GL ( n, K ) lies in the set { g ∈ GL ( n, K ) | ( g − n = 0 } . An example of a unipotent group isthe group of all upper-triangular matrices in GL ( n, K ) with 1’s on the main diagonal.A group b G is called pro-unipotent if it is equal to the inverse limit, b G = lim ←− G i := ( ~g ∈ Y i ∈ I G i | f ij ( g j ) = g i ) , of some series of unipotent groups { G i } i ∈ I parameterized by a directed set ( I, ≤ ) and equipped with a systemof homomorphisms { f ij : G j → G i } i,j ∈ I such that i ≤ j satisfying the conditions : f ii = Id and f ik = f ij ◦ f jk for any i ≤ j ≤ k . The Lie algebra g of the prounipotentgroup G is defined as the inverse limit, b g = lim ←− g i of the Lie algebras g i of the groups G i . There exist mutually inverse maps exp : b g → b G and log : b G → b g . Such a system { G i , f ij } is called an inverse system of groups. Analogously one defines an inverse system of algebras,modules etc. .3.1. Definition. The prounipotent completion over a field K of an abstract group G is a prounipotentgroup b G ( K ) together with a homomorphism i : G → b G ( K ) satisfying the following universality property:if h : G → b H is a homomorphism of G into a prounipotent group b H over K , then there is a uniquehomomorphism f : b G ( K ) → b H such that h factors as the composition h : G i −→ b G ( K ) f −→ b H .Prounipotent completions are often called Malcev completions in the literature. One denotes sometimes theprounipotent completion over K of a group G simply by b G omitting thereby the reference to K . Let G be an abstract group, and K [ G ] := X g ∈ G λ g g | λ g ∈ K such that only finitely many λ g can be non-zero its group algebra over a field K . There is an algebra homomorphism, ε : K [ G ] −→ K : P g ∈ G λ g g −→ P g ∈ G λ g , called the augmentation . Let I K := Ker ε be the augmentation ideal, and let [K [ G ] be the I K -adic completion of K [ G ] (see § ). The ideal I K is generated by elements of the form g −
11, where 11 is the unit in G (oftendenoted by 1 by abuse of notation).The algebra K [ G ] can be maid into a Hopf algebra with the coproduct defined as follows∆ : K [ G ] −→ K [ G ] ⊗ K [ G ] P g ∈ G λ g g −→ P g ∈ G λ g g ⊗ g, This coproduct is continuous in the I K -adic topology on K [ G ] and hence induces by continuity a coproduct∆ : [K [ G ] −→ [K [ G ] b ⊗ [K [ G ]where b ⊗ is the completed tensor product. This gives [K [ G ] the structure of a complete filtered Hopf algebra.Consider the set of group-like elements,(3) b G := n x ∈ [K [ G ] : ∆( x ) = x ⊗ x and ε ( x ) = 1 o . This is a subgroup of the group of units in [K [ G ]. If char ( K ) = 0, then the K -powers of elements g ∈ G are well-defined in the ring [K [ G ], g λ = (11 − (11 − g )) λ := ∞ X i =0 λ ( λ − · · · ( λ − i + 1) i ! (11 − g ) i . In some cases the group b G is generated by such powers; the corresponding Lie algebra b g is generated byelements log g = log(11 − (11 − g )) := P ∞ i =1 (11 − g ) i . Note that the above formula for g λ implies,(4) hg λ h − = ( hgh − ) λ , and g λ g µ = g λ + µ for any g, h ∈ G and λ, µ ∈ K . The I K -adic topology on K [ G ] is Hausdorff as the intersection of all powers of I is the zero ideal. In this case one candefine the I -adic topology as the unique metric topology associated with the following metric on K [ G ]: d ( x, x ′ ) := || x − x ′ || , where, for any x ∈ K [ G ], its norm || x || is defined to be 2 − n with n being the largest natural number such that x ∈ I n . Thealgebra [K [ G ] can be identified with the metric completion of K [ G ]. .4.2. Theorem [Ha] . If I K /I K is a finite-dimensional vector space over K , then b G is a prounipotentcompletion of G . If G is finitely generated, then the condition of this Theorem is satisfied and the above explicit constructiongives us the prounipotent completion of G .It is often useful for practical computations to consider an increasing filtration of the group (3), . . . ⊆ b G l +1 ⊆ b G l ⊆ . . . ⊆ b G ⊆ b G, defined by b G l = b G ∩ (11 + c I K l ) . If the condition of Theorem is satisfied, then each group, b G l := b G/ b G l +1 is a unipotent group, and one can compute the prounipotent completion of G as the inverse limit b G = lim ←− b G l . char ( K ) = 0 . Consider the set of primitive elements , b g , of the completed group algebra [K [ G ] defined by b g := n x ∈ c I K | ∆( x ) = 11 ⊗ x + x ⊗ o . The bracket [ x, y ] = xy − yx makes b g into a Lie algebra. The c I K -adic topology of [K [ G ] induces a topology on b g making the latter into a complete topological Lie algebra. The logarithm and exponential functions giveus well-defined and mutually inverse homeomorphisms,log : 11 + c I K −→ c I K and exp : c I K −→
11 + c I K which restrict to the mutually inverse homeomorphisms,log : b G −→ b g and exp : b g −→ b G. Thus one can describe b G in terms of its Lie algebra b g which is sometimes a simpler object to define. Thelatter is the inverse limit of the Lie algebras of the prounipotent groups b G l defined above. (i) Let G be a group of finite type, that is, a quotient of the free group generated by a finite set { g , . . . , g n } by the normal subgroup generated by a finite number of relations { R i ( g , . . . , g n ) = 1 } i ∈ [ m ] , m ∈ N . Let K be a field of characteristic zero, and let b g be the quotient of the completed free Lie algebra b lie n generated by symbols { γ , . . . , γ n } by the ideal generated by the relations, { log R i ( e γ , . . . , e γ n ) = 0 } i ∈ [ m ] . Then the group b G ( K ) := exp b g gives us the prounipotent completion of G over K [Br].(ii) If F n is the free group in n letters, x , . . . , x n , then its prounipotent completion, b F n ( K ), over a field K of characteristic zero is equal to exp( d lie n ), where d lie n is the completed filtered Lie algebra over K generated by the following set, ( log x i = log(1 − (1 − x i )) = ∞ X k =1 k (1 − x i ) k ) i ∈ [ n ] . Put another way, F n ( K ) is generated by elements of the form x λi , λ ∈ K .The next examples are taken from [K]. iii) Let G = Z and let K be a field of characteristic zero. As Z is isomorphic to the free group generatedby a symbol t , its prounipotent completion b Z ( K ) can be computed using the construction in Example(i) above to give b Z ( K ) = exp( c lie ) ≃ K (as Abelian groups) . One can get the same result using Quillen’s construction as follows [K]. The group algebra K [ Z ] is thering of Laurent polynomials K [ t, t − ] with the augmentation ideal I K being a principal one generatedby t −
1. The completion [K [ Z ] is the formal power series K [[ T ]] with the inclusion K [ Z ] → [K [ Z ] givenby K [ t, t − ] −→ K [[ T ]]( t, t − ) −→ (1 + T, P ∞ k =1 ( − k T k ) . The ideal c I K is therefore the principal one generated by the symbol T (that is, the maximal ideal of K [[ T ]]) so that the augmentation map K [[ T ]] → K has the form T →
0. The coproduct is given by∆( T ) = 1 ⊗ T + T ⊗ T ⊗ T .If K has characteristic zero, then the set of group-like elements in [K [ Z ], i.e. the prounipotentcompletion of Z over K , is given by b Z ( K ) = { (1 + T ) α = exp( α log(1 + T )) ∈ K [[ T ]] | α ∈ K } ≃ K . Note that in this case b Z ( K ) l = b Z ( K ) ∩ (1 + c I K l ) = { } for l ≥ b Z ( K ) l = b Z ( K ) for all l .If K has characteristic p , then the element 1 + T has finite order in each group b Z ( K ) l = b Z ( K ) / b Z ( K ) ∩ (1 + c I K l +1 )as (1 + T ) p = 1 + T p ∈ c I K p . It follows [K] that the prounipotent completion of Z over K is equal to Z p , the Abelian group (infact, the ring) of p -adic integers.(iv) Let G = Z n = Z /n Z , n ≥
2. The associated group algebra over a field K is K [ Z n ] = K [ t ] / ( t n − K [ t ] by the principal ideal generated by ( t n − I K ⊂ K [ Z n ] is the principal one generated by t −
1. Note that t is a unit with t − = t n − .There is a factorization in K [ Z n ],( t − = t − t + 1= t n + t − t = t ( t n − + . . . + t + 2)( t − . If characteristic of K is prime to n or equal to zero, then t n − + . . . + t + 2 is also a unit in K [ Z n ] sothat I K = I K and hence \K [ Z n ] = lim ←− K [ Z n ] /I l K = K Hence the prounipotent completion of Z n in this case is the trivial group, c Z n ( K ) = { } (cf. Example(iii) above).If char ( K ) = p divides n , then one can assume that n = p d for some d ≥
1. As (1 − t ) p d =1 − t p d = 1 − t n = 0, one has I p d K = 0 so that c Z n ( K ) = Z n ( K ) and hence the prounipotent completionis given by the following set of group-like elements, c Z n ( K ) = { t i | ≤ i ≤ p d } ≃ Z p d (v) Examples (ii) and (iii) are special cases of a more general statement [K]: if G is a finitely generatedAbelian group, then b G ( K ) := (cid:26) G ⊗ Z K if char ( K ) = 0 G ⊗ Z Z p if char ( K ) = p. The prounipotent completion of the cartesian product of two groups is the cartesian product of their prounipotentcompletions. .6. Profinite completions. Let G be an abstract group and { I } a family of all normal subgroups I ⊂ G of finite index. This family can be made into a partially ordered set with respect to the inclusion. As I I ∈ { I } for any I , I ∈ { I } , this partially ordered set is directed so that { I } gives us an inverse systemof ideals and one can consider the inverse limit, e G := lim ←− G/I, which is called the profinite completion of G . For example, e Z = Q p Z p . . Monoidal categories and monoidal functors
For a category C its class of objects is denoted by Ob ( C ), and the set of morphisms from an object A to anobject B by Mor C ( A, B ). If Ob ( C ) is a set, then the category is called small . The composition of morphismsMor( A, B ) × Mor(
B, C ) → Mor(
A, C ) is denoted by ◦ . [ES] ). A category C is called a semigroup category if it is equippedwith a bifunctor · ⊗ C · C × C −→ C A, B ∈ Ob ( C ) −→ A ⊗ C B ∈ Ob ( C ) , and an isomorphism of trifunctors,Φ : ( · ⊗ C · ) ⊗ C · −→ · ⊗ C ( · ⊗ C · )Φ A,B,C : ( A ⊗ C B ) ⊗ C C −→ A ⊗ C ( B ⊗ C C )such that the diagram(5) (( A ⊗ C B ) ⊗ C C ) ⊗ C D Φ A ⊗ C B,C,D t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Φ A,B,C ⊗ Id * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ( A ⊗ C B ) ⊗ C ( C ⊗ C D ) Φ A,B,C ⊗ C D (cid:15) (cid:15) ( A ⊗ C ( B ⊗ C C )) ⊗ C D Φ A,B ⊗ C C,D (cid:15) (cid:15) A ⊗ C ( B ⊗ C ( C ⊗ C D )) A ⊗ C (( B ⊗ C C ) ⊗ C D ) Id ⊗ Φ B,C,D o o commutes for any A, B, C, D ∈ Ob ( C ).Commutativity of the diagram (5) is called the pentagon axiom . A remarkable fact is that the pentagonaxiom implies commutativity of all similar diagrams in the category S . More precisely, one has Let C be a semigroup category and consider an arbitrary col-lection A , . . . , A n of objects C . Given any two complete bracketings of the formal expression A ⊗ C ⊗ A ⊗ C . . . ⊗ C A n , then all isomorphisms from one bracketing to another composed of the associativity isomorphisms Φ and their inverses, are equal to each other. A semigroup category C is called monodial if it has an object 11 C (called a unit ) together with isomorphisms λ A : 11 C ⊗ C A −→ A, ρ A : A ⊗ C C −→ A such that the diagram ( A ⊗ C C ) ⊗ C B ρ A ⊗ Id ' ' PPPPPPPPPPP Φ A, C ,B / / A ⊗ C (11 C ⊗ C B ) Id ⊗ λ A w w ♥♥♥♥♥♥♥♥♥♥♥ A ⊗ C B commutes for any A, B ∈ Ob ( C ).A monoidal category is called strict if all the isomorphisms Φ, λ and ρ are identities. .1.2. Symmetric monoidal categories. A monoidal category is called symmetric if it is equipped withan isomorphism of bifunctors,(6) β : · ⊗ C · −→ · ⊗ op C · β A,B A ⊗ C B −→ B ⊗ C A such that(7) β B,A ◦ β A,B = Id and the diagram(8) A ⊗ C ( B ⊗ C C ) β A,B ⊗ C C / / ( B ⊗ C C ) ⊗ C A Φ B,C,A ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ ( A ⊗ C B ) ⊗ C C Φ A,B,C ❧❧❧❧❧❧❧❧❧❧❧❧❧ β A,B ⊗ Id ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ B ⊗ C ( C ⊗ C A )( B ⊗ C A ) ⊗ C C Φ B,A,C / / B ⊗ C ( A ⊗ C C ) Id ⊗ β A,C ❧❧❧❧❧❧❧❧❧❧❧❧❧ commutes for any A, B, C ∈ Ob ( C ).Commutativity of diagram (8) is called the hexagon axiom .There is a symmetric monoidal analogue of the above Mac Lane coherence theorem which says, that given anycomplete bracketings of the formal expressions A ⊗ C ⊗ A ⊗ C . . . ⊗ C A n and A σ (1) ⊗ C ⊗ A σ (2) ⊗ C . . . ⊗ C A σ ( n ) , σ ∈ S n , then all isomorphisms from one bracketing to another composed of the associativity isomorphismsΦ, symmetry isomorphisms β and their inverses, are equal to each other. A monoidal category is called braided if it is equipped with anisomorphism of functors (6) which makes the diagram (8) commutative while its inverse β − makes a versionof the diagram (8) in which the symbols β A,B ⊗ C C , β A,B and β A,C are replaced with β − B ⊗ C C,A , β − B,A and β − C,A respectively commutative as well.Mac Lane type coherence property of a braided monoidal category will be explained below in terms of thebraid groups. (i) Let
Vect K be the category whose objects are (complete filtered ) vector spaces over a field K andwhose morphisms, f : A −→ B , are continuous linear maps. This is a strict symmetric monoidalcategory with A ⊗ Vect K B := A ⊗ K B, the ordinary (completed) tensor product of vector spaces over K , Vect K := K and the symmetry morphism is given by β A,B ( A ⊗ K B ) = B ⊗ K A .(ii) Let Ass K be the category of (complete filtered topological) associative algebras (with unit) over afield K . This is a subcategory of Vect K . Moreover, Ass K is a strict symmetric monoidal categorywith respect to the structures induced from Vect K .(iii) Let Hopf K be the category of (complete filtered) Hopf algebras over a field K . This a subcategory of Ass K which is is strict symmetric monoidal with respect to the structures induced from Ass K .(iv) Let coAss K be the category of (complete filtered) associative coalgebras over a field K . This asubcategory of Vect K which is is strict symmetric monoidal with respect to the structures inducedfrom Vect K . The complete filtered versions of the categories listed below will be sometimes denoted by the same kernel symbol, butequipped with a wide hat, e.g., d Vect K , c Ass K etc. v) Let Lie K be the category whose objects are (complete filtered) Lie algebras over a field K and whosemorphisms, f : g −→ g , are (continuous) linear maps preserving Lie brackets, i.e. f ◦ [ · , · ] =[ f ( · ) , f ( · )]. This is a strict symmetric monoidal category with g ⊗ Lie K h := g ⊕ h , the direct sum of vector spaces over K , Lie K := 0 , and the symmetry morphism given by β g , h ( g ⊕ h ) = h ⊕ g .(i-v) ′ There are obvious dg , that is, differential graded , versions — dgVect K , dgAss K , dgHopf K , dgCoAss K , dgLie K — of all the above symmetric monoidal categories whose objects are dg vector spaces, dg associative algebras etc.We often use the following notation. Let V = ⊕ i ∈ Z V i be a Z -graded vector space, then for anyinteger k ∈ Z the symbol V [ k ] stands for the Z -graded vector space with V [ k ] i := V i + k and and s k for the associated isomorphism V → V [ k ]; for v ∈ V i one denotes | v | := i . For a pair of Z -gradedvector spaces V and V , the symbol Hom i ( V , V ) stands for the space of homogeneous linear maps ofdegree i , and Hom( V , V ) := L i ∈ Z Hom i ( V , V ); for example, s k ∈ Hom − k ( V, V [ k ]). Furthermore,we use the notation ⊙ n V for the n-fold symmetric product of the vector space V .(vi) Let Cat be the category of small categories (with functors as morphisms). This is a strict symmetricmonoidal category with Ob ( C ′ ⊗ Cat C ′′ ) := Ob ( C ′ ) × Ob ( C ′′ ) , and Mor C ′ ⊗ Cat C ′′ ( X ′ × X ′′ , Y ′ × Y ′′ ) = Mor C ′ ( X ′ , Y ′ ) × Mor C ′′ ( X ′′ , Y ′′ ) . Its full subcategory consisting of small groupoids (that is, small categories with every morphismbeing an isomorphism) is denoted by
CatG .(vii) Let
Cat ( Vect K ) be a category of small (complete filtered) K -linear categories, that is, a categoryof small categories which satisfy two conditions: (1) morphisms between any pair of objects forma (complete filtered) vector space, and (2) the composition of morphisms is a continuous bilinearmap. This is a subcategory of Cat but we make it into a symmetric monoidal category in a slightlydifferent way: Ob ( C ′ ⊗ Cat ( Vect K ) C ′′ ) := Ob ( C ′ ) × Ob ( C ′′ ) , and Mor C ′ ⊗ Cat ( Vect K ) C ′′ ( X ′ × X ′′ , Y ′ × Y ′′ ) = Mor C ′ ( X ′ , Y ′ ) ⊗ K Mor C ′′ ( X ′′ , Y ′′ ) . For future reference we denote by
Cat ( coAss K ) the (strict symmetric monoidal) subcategory of Cat ( Vect K ) such that the space of morphisms between any two objects is a complete filtered K -coalgebra, and the composition is compatible with the coalgebra structures.(viii) Let Set be the category of sets. This is a strict symmetric monoidal category with with I ⊗ Set J := I × J, the Cartesian product of sets I and J, Set := ∅ , and the symmetry morphism given by β A,B ( I × J ) = J × I .(ix) Let Group be the subcategory of
Set whose objects are groups and homomorphisms as morphisms.This is a strict symmetric monoidal category with respect to the the Catersian product; the unit isgiven by the trivial group 1.(x) Let
Top be the subcategory of
Set whose objects are Hausdorff spaces with compactly generatedtopology and continuous maps as morphisms. This is a strict symmetric monoidal category withrespect to the structures inherited from
Set . .3. Monoidal functors. Let ( C , ⊗ C , C ) and ( D , ⊗ D , D ) be monoidal categories. A functor F : C → D is called monoidal if it comes with a morphism 11 : 11 D → F (11 C ) and a natural transformation φ : F ( · ) ⊗ D F ( · ) −→ F ( · ⊗ C · ) φ A,B : F ( A ) ⊗ D F ( B ) −→ F ( A ⊗ C B )such that the following diagrams, ( F ( A ) ⊗ D F ( B )) ⊗ D F ( C ) Φ F ( A ) ,F ( B ) ,F ( C ) / / φ A,B ⊗ Id (cid:15) (cid:15) F ( A ) ⊗ D ( F ( B )) ⊗ D F ( C )) Id ⊗ φ B,C (cid:15) (cid:15) F ( A ⊗ C B ) ⊗ D F ( C ) φ A ⊗ C B,C (cid:15) (cid:15) F ( A ) ⊗ D F ( B ⊗ C C ) φ A,B ⊗ C C (cid:15) (cid:15) F (( A ⊗ C B ) ⊗ C C ) Φ A,B,C / / F ( A ⊗ C ( B ⊗ C C )) D ⊗ D F ( A ) ⊗ Id / / λ F ( A ) (cid:15) (cid:15) F (11 C ) ⊗ D F ( A ) φ C ,A (cid:15) (cid:15) F ( A ) F (11 C ⊗ C A )) F ( λ A ) o o F ( A ) ⊗ D D Id ⊗ / / ρ F ( A ) (cid:15) (cid:15) F ( A ) ⊗ D F (11 C ) φ A, C (cid:15) (cid:15) F ( A ) F ( A ⊗ C C )) F ( ρ A ) o o commute for any A, B, C ∈ Ob ( C ).A functor F between braided monoidal categories is called braided monoidal if it is monoidal and makes thediagram F ( A ) ⊗ D F ( B ) β / / φ A,B (cid:15) (cid:15) F ( B ) ⊗ D F ( A ) φ B,A (cid:15) (cid:15) F ( A ⊗ C B ) F ( β ) / / F ( B ⊗ C A ) commutative for any A, B ∈ Ob ( C ).A braided monoidal functor between symmetric monoidal categories is called symmetric monoidal . (i) The functor Lin K : Set −→ Vect K I −→ K [ I ] := the space of functions I → K is symmetric monodial.(ii) The (completed) universal enveloping functor, U : Lie K −→ Hopf K g −→ U ( g )is symmetric monoidal because U ( g ⊕ h ) = U ( g ) ⊗ K U ( h ).(iii) Consider a functor, ∆ K : CatG −→ Cat ( coAss K )which is identity on objects I ∈ Ob ( CatG ),∆ K ( I ) = I, and on morphisms it coincides with Lin K defined above in (i),∆ K (Mor CatG ( I, J )) = K [Mor CatG ( I, J )]where K [Mor Cat ( I, J )] is equipped with the unique coalgebra structure in which all elements f ofMor Cat ( I, J ) are group-like, ∆( f ) = f ⊗ K f . This functor is symmetric monoidal. Note that ∆ K ( S n ) = K [ S n ], the group algebra of S n = Mor CatG ([ n ] , [ n ]) equipped with its standard coalgebra structure. iv) Let ( g , [ , ]) be a Lie algebra, and CE • ( g ) its Chevalley-Eilenberg complex (also known as the barconstriction ) which is, by definition, a dg coalgebra CE • ( g ) := ⊙ • ( g [1]) = ∞ M n =0 ( g [1]) S n , with the coproduct ∆ : CE • ( g ) → CE • ( g ) ⊗ K CE • ( g ) given by∆ : s ( x ) ⊙ . . . ⊙ s ( x n ) := X [ n ]= I ⊔ I I ,I ≥ ( − σ ( I ,I ) x I ⊗ x I , where, for a naturally ordered subset I = { i , . . . , i k } of [ n ], we set x I := s ( x i ) ⊙ . . . ⊙ s ( x i k ) , and ( − σ ( I ,I ) is the sign of the permutation [ n ] → [ I ⊔ I ]. The differential, d : CE • ( g ) → CE • ( g ),is given by ds ( x ) ⊙ . . . ⊙ s ( x n ) := X ≤ i 1) = x . Let π ( X )( x , x ) stand forthe set of of all homotopy classes of paths in X from x to x . There is a natural composition map, π ( X )( x , x ) × π ( X )( x , x ) −→ π ( X )( x , x ) , which makes the set of points of X into a category, π ( X ), in which the set of morphisms from anobject x ∈ X to an object x ∈ X is identified with π ( X )( x , x ). Every such a morphisms isan isomorphism so that π ( X ) is in fact a groupoid called the fundamental groupoid of X . Thisconstruction gives us a monoidal functor,(9) π : Top −→ CatG X −→ π ( X ) . Note that for any subset A ⊂ X one can define a full subcategory of π ( X ) whose objects are pointsof A ; it is denoted by π ( X, A ).The set π ( X )( x , x ) =: π ( X, x ) is a group called the fundamental group of X base at x . If Top ∗ stands for the category of based topological spaces then π gives us a monoidal functor(10) π : Top ∗ −→ Group ( X, x ) −→ π ( X, x )We shall use below both these functors to construct operads in the categories CatG and Group outof operads in the categories in Top and Top ∗ respectively.(viii) Let G be a (discrete) group. For non-empty subsets A, B ⊂ G , let[ A, B ] = (cid:8) [ a, b ] := aba − b − | a ∈ A, b ∈ B (cid:9) stand for the subgroup of G generated by all commutators of elements of A with elements of B .Define inductively the descending central series of G as the following series of normal subgroups, G := G ⊇ G = [ G, G ] ⊇ G = [ G , G ] ⊇ . . . ⊇ G n = [ G n − , G ] ⊇ G n +1 = [ G n , G ] ⊇ . . . and let gr n ( G ) := G n /G n +1 be the associated n -th quotient. This is an Abelian group, i.e. a Z -module, so that it makes senseto define the following positively graded vector space over a field K , gr K ( G ) := ∞ M n =1 gr n ( G ) ⊗ Z K , he commutator map [ , ] : G × G −→ G ( a, b ) −→ aba − b − , induces on gr ( G ) K a linear skew-symmetric map,[ , ] : ∧ gr K ( G ) −→ gr K ( G )which satisfies the Jacobi identity. Hence this construction (which is natural in G ) gives us a functor b gr K : Group −→ Lie K G −→ b gr K ( G )which is obviously symmetric monoidal. Here b gr K ( G ) is the completion, Q ∞ n =1 gr n ( G ) ⊗ Z K , of thefiltered Lie algebra gr K ( G ). It is called the (completed) graded Lie algebra of G over K . Let f : G → H be a morphism of groups, and assume that G is residually nilpotent ,that is, its descending central series satisfies the condition ∞ \ i =1 G n = 1 . If the associated morphism of graded Lie algebras, gr Q ( f ) : gr Q ( G ) −→ gr Q ( H )is a monomorphism, then f is a monomorphism [CW]. A symmetric monoidal category C is called closed if forany object A ∈ Ob ( C ) the functor · ⊗ C A C −→ C B −→ B ⊗ C A has a right adjoint functor, Hom C ( A, · ) C −→ C B −→ Hom C ( A, B )which, by definition, is a functor satisfying the condition,Mor C ( B ⊗ C A, C ) ∼ = Mor C ( B, Hom C ( A, C )) , for any A, B, C ∈ Ob ( C ). One can show using Yoneda lemma that this condition implies(11) Hom C ( B ⊗ C A, C ) ∼ = Hom C ( B, Hom C ( A, C )) . The object Hom C ( A, B ) ∈ Ob ( C ) is called the internal hom of A and B .The categories Cat , Set and the category of finite-dimensional vector spaces are closed with Hom C ( A, B ) =Mor C ( A, B ). . Operads We shall mostly use in the subsequent sections the notion of operad at a rather elementary level — at thelevel of its basic properties which follow from its definition almost immediately. However, the definition itself(which is due to P. May [May]) is not, perhaps, very elementary, and we recommend a newcomer to payattention not only to the formal definition(s) of operad given below, but also to examples illustrating thatdefinition. Comprehensive expositions of the theory of operads can be found in the books [LV, MSS].From now on S stands for the groupoid of finite sets and their bijections. This is a subcategory of Set . An operad O in a symmetric monoidal category C is acollection of the following data [T2]:1) a functor O : S → C (which is often called an S - module ); ) for any finite sets I , J and an element i ∈ I a morphism(12) ◦ I,Ji : O ( I ) ⊗ C O ( J ) −→ O ( I \ { i } ⊔ J ) , natural in i , I , J (“the insertion of O ( J ) into the i -th slot of O ( I )”), such that the followingassociativity conditions hold:(a) for any finite sets I , J , K and any i , i ∈ I , i = i , the diagram O ( I ) ⊗ C O ( J ) ⊗ C O ( K ) ◦ I,Ji ⊗ Id / / Id ⊗ β O ( J ) O ( K ) (cid:15) (cid:15) O ( I \ { i } ⊔ J ) ⊗ C O ( K ) ◦ I \{ i }⊔ J,Ki / / O ( I \ { i , i } ⊔ J ⊔ K ) O ( I ) ⊗ C O ( K ) ⊗ C O ( J ) ◦ I,Ki ⊗ Id / / O ( I \ { i } ⊔ K ) ⊗ C O ( J ) ◦ I \{ i }⊔ K,Ji / / O ( I \ { i , i } ⊔ K ⊔ J ) commutes;(b) for any finite sets I , J , K and any i ∈ I and j ∈ J the diagram O ( I ) ⊗ C O ( J \ { j } ⊔ K ) ◦ I,J \{ j }⊔ Ki + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ O ( I ) ⊗ C O ( J ) ⊗ C O ( K ) Id ⊗◦ J,Kj ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ◦ I,Ji ⊗ Id + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ O ( I \ { i } ⊔ J \ { j } ⊔ K ) O ( I \ { i } ⊔ J ) ⊗ C O ( K ) ◦ I \{ i }⊔ J,Kj ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ commutes.Sometimes we abbreviate compositions ◦ I,Ji to ◦ i . An operad with unit is, by definition, an operad O equipped with a morphism, e : 11 C −→ O ( • ) , for any one-element set {•} which is natural in {•} and makes the following two compositions, O ( I ) −→ O ( I ) ⊗ C C Id ⊗ e −−−→ O ( I ) ⊗ C O ( • ) ◦ I, • i −−→ O ( I ) , O ( I ) −→ C ⊗ C O ( I ) e ⊗ Id −−−→ O ( • ) ⊗ C O ( I ) ◦ • ,I • −−→ O ( I ) , into identity maps for any set I and any i ∈ I . The skeleton of the groupoid S consists of the sets [ n ], n ≥ ∅ ). If thecategory C admits small colimits (and all categories we work with in this paper do have this property),then a functor O : S → C is equivalent to a collection of S n -modules, {O ( n ) := O ([ n ] } n ≥ , which is oftenabbreviated to an S -module in the category C . One can reformulate axioms of an operad in terms of the S -module {O ( n ) := O ([ n ] } n ≥ and the operadic compositions, ◦ i : O ( n ) ⊗ C O ( m ) −→ O ([ n ] \ i ⊔ [ m ]) = O ( n + m − , ∀ m, n ∈ N , i ∈ [ n ] , where one first identifies [ n ] \ i ⊔ [ m ] = { , , i − , i + 1 , . . . , n } ⊔ { ′ , ′ , . . . , m ′ } with the linearly ordered set, { , , . . . , i − , ′ , ′ , . . . , m ′ , i + 1 , . . . , n } , and then identifies the latter with [ n + m − 1] = { , , . . . , i − , i, i + 1 , . . . , m + n − } as linearly ordered sets.In this way one recovers the original definition of an operad given in [May]. We shall often use this freedombelow to describe an operad either as a functor O : S → C or simply as an S -module {O ( n ) ∈ Ob ( C ) } n ∈ N . .2.2. Non- S operads. As noted just above an operad O in a monoidal category C can be defined as acollection of objects, O = {O ( n ) ∈ Ob ( C ) } n ≥ , such that each objects O ( n ) carries a representation of S n ,and there are equivariant compositions, ◦ i : O ( n ) ⊗ C O ( m ) −→ O ( n + m − ∀ i ∈ [ n ] , which satisfy axioms (a) and (b) in § .If we forget in the definition above S n actions on O ( n ), n ≥ 1, and, correspondingly, omit the equivariancycondition on the compositions ◦ i , then we get a notion of a non- S operad . More precisely, a non- S operad in a (semigroup) category C is an operad O = {O ( n ) } in C such that the action of S n on each object O ( n )is trivial. Let O be an operad in a symmetric monoidal category C and F : C → D a symmetricmonoidal functor to some other category. Show that the data(i) the S -module structure, F O : S → D , given by the composition S O −→ C F −→ D , and(ii) the operadic “insertions”,¯ ◦ I,Ji : F O ( I ) ⊗ C F O ( J ) −→ F O ( I \ { i } ⊔ J ) , given by the compositions F O ( I ) ⊗ C F O ( J ) φ O ( I ) , O ( J ) −−−−−−−→ F ( O ( I ) ⊗ C O ( J )) F ( ◦ I,Ji ) −−−−−→ F ( O ( I \ { i } ⊔ J )) = F O ( I \ { i } ⊔ J ) , give us an operad F O in the symmetric monoidal category D . This fact is of an extreme importance inapplications — starting with a “geometric” operad in the category, say, of topological spaces, and applyingthe chain or homology functor one arrives to an operad in the category of vector spaces. This particularproperty of operads is another manifestation of the amazing unity of mathematics . . Introduce the notion of a morphism , ρ : O −→ O , of operads in a symmetric monoidalcategory C . . Introduce the notion of an ideal I , of an operad O in the category Vect K and constructthe quotient operad O / I . Here we define a few operads which will be used in some constructions below. Let C be a closed symmetric monoidal category. An arbitrary object A ∈ Ob ( C ) gives rise to an operad E nd A with the S -module, (cid:8) E nd A ( n ) := Hom C ( A ⊗ n , A ) (cid:9) and with the compositions ◦ [ n ] , [ m ] i given by the isomorphisms (use 11)),Hom C (cid:16) A ⊗ ( i − ⊗ C Hom C ( A ⊗ m , A ) ⊗ C A ⊗ ( n − i +1) , A (cid:17) ∼ = Hom C ( A ⊗ ( n + m +1) , A ) , i.e. literally by the substitution of the object Hom C ( A ⊗ m , A ) into the i -th input slot of Hom C ( A ⊗ n , A ).Let O be an operad in C . An O - algebra structure on an object A is, by definition, a morphism of operads, ρ : O −→ E nd A , which is also often called a representation of O in A . .3.2. Operad of parenthesized permutations. Consider an S -module in the category Set , P a : S −→ Set I −→ P a ( I ) := the set of all parenthesized permitations of I One can equivalently define P a ( I ) as- the set of all monomials built from elements of I using a non-commutative and non-associativeproduct (using each element of I once),- the set of all planar binary trees whose legs are labelled by elements of I (using each element once).For example, for I = { a, b, c, d, e } , (( ba )( e ( cd )) ≃ b a e c d ◦◦ ◦ ◦ ⑧⑧⑧ ❄❄❄✞✞✞ ✼✼✼ ✞✞✞ ✽✽✽✞✞✞ ✼✼✼ is an element of P a ( I ). The operadic compositions ◦ I,Ji = ◦ i are given by the substitution of a monomialfrom P a ( J ) into the i -th letter of a monomial P a ( I ), e.g. for I = { a, b, c, d, e } and J = { x, y, z } ,(( ba )( e ( cd )) ◦ c (( xy ) z ) = (( ba )( e ((( xy ) z ) d )) ∈ P a ( { I \ c } ⊔ J )or simply by grafting the root (which in our pictorial notation grows upward) of a planar binary tree from P a ( J ) into the i -labelled input leg of a tree from P a ( I ), b a e c d ◦◦ ◦ ◦ ⑧⑧⑧ ❄❄❄✞✞✞ ✼✼✼ ✞✞✞ ✽✽✽✞✞✞ ✼✼✼ ◦ c x y z ◦◦ ⑧⑧⑧ ❄❄❄❄✞✞✞ ✼✼✼ = b a e z dx y ◦◦ ◦ ◦◦◦ ⑧⑧⑧ ❄❄❄✞✞✞ ✼✼✼ ✞✞✞ ✽✽✽✞✞ ✼✼✼✞✞ ✼✼✼✆✆✆ ✼✼✼ . (i) Show that there is a one-to-one correspondence between Mor Set ( X × X, X ) and P a -algebra structures on X ∈ Ob ( Set ), that is, representations, ρ : P a → E nd X , of the operad P a in a set X .(ii) Describe representations of the operad Lin K ( P a ) in a vector space V , where Lin K is a monoidal functor Set → Vect K defined in § (i).(iii) Let O and O be operads in a symmetric monoidal category C . Show that the S -module defined by, O ⊗ C O : S −→ C I −→ O ( I ) ⊗ C O ( I ) , has an induced operadic structure. The resulting operad O ⊗ C O is called the tensor product of operads O and O . A graph with legs (or hairs) is a triple Γ = ( H (Γ) , ⊔ , τ ), where- H (Γ) is a finite set whose elements are called half-edges (or flags ),- ⊔ is a partition of H (Γ) into a disjoint union of subsets, H (Γ) = a v ∈ V (Γ) H ( v ) , parameterized by a set V (Γ) which is called the set of vertices of Γ; the subset H ( v ) ⊂ H (Γ) is calledthe set of half-edges attached to the vertex v ; its cardinality, H ( v ), is called the valency of v ,- τ : H (Γ) → H (Γ) is an involution, that is, a map satisfying the condition τ = Id . This map definesa new partition of H (Γ) into orbits ( h, τ ( h )) ⊂ H (Γ) which in general can have cardinality two( h = τ ( h )) or one ( h = τ ( h )). Orbits of cardinality 2 are called internal edges or simply edges ofthe graph Γ; the set of (internal) edges is denoted by E (Γ). Orbits of cardinality 1 are called legs (or hairs ); the set of legs is denoted by L ( G ). ote that L (Γ) = ∅ if and only the involution τ has no fixed points. In this case Γ = ( H (Γ , ⊔ , τ ) is calledsimply a graph . We shall study operads of such graphs in § geometric realization which is atopological space constructed as follows: (i) for each vertex v take H ( v ) copies of the interval [0 , 1] labelledby elements of H ( v ) and glue these intervals at the end-point 0, the result is a topological space (equippedwith the quotient topology from [0 , H ( v ) ) which is called the corolla of v ; (ii) then consider a union of allstars and, for each internal edge ( h, τ ( h )), identify the end-points 1 of the intervals [0 , 1] labelled by h and τ ( h ).An isomorphism of graphs with legs i : Γ → Γ is a bijection i : H (Γ ) → H (Γ ) which preserves partitions ⊔ and ⊔ (and hence induces a bijection of the sets of vertices V (Γ ) → V (Γ )) and commutes withinvolutions, i ◦ τ = τ ◦ i (and hence induces bijections E (Γ ) → E (Γ ) and L (Γ ) → L (Γ )). The associatedgroupoid of graphs and their isomorphisms is denoted by IsoGraph .A graph (with legs) is called connected (resp., simply connected) if its geometric realization is a connected(resp., simply connected) topological space. A connected and simply connected graph is called a tree . A rooted tree is a tree T with L ( T ) = ∅ and with one one of the legs, say r ∈ L ( T ), marked and called theroot . Elements in L ( T ) /r are called input legs or leaves of the rooted tree T ; the set of leaves is denoted by Leaf ( T ). Each vertex on a geometric realization of a rooted tree is connected to a root by a unique path;this defines a flow on the tree and hence a partition, H ( v ) = In ( v ) ⊔ r v . of the set of half-edges at each vertex v into a disjoint union of a unique element r v which lies on the pathfrom v to the root and the subset of the remaining half-edges, In ( v ), which are called input (half ) edges . Byabuse of language, the cardinality of In ( v ) is called the valency of a vertex v of a rooted tree T (which isequal to the valency of v in T viewed as an unrooted tree minus one). The corolla of a vertex in a rootedtree can be visualized as follows (with flow directed from bottom to the top) ◦ . . . ✾✾✾✾ ❍❍❍❍❍✆✆✆✆✈✈✈✈✈ Let T and T be rooted trees. An isomorphism of graphs, i : T → T , which sends the root of T intothe root of T is called an isomorphism of rooted trees . The associated groupoid of rooted trees and theirisomorphisms is denoted by IsoTree . Let IsoTree ( I ) be a category (in fact, a groupid) whose objects are rooted trees T equipped with an isomorphism l T : Leaf ( T ) → I (“labeling of input legs with elements of I ”) andmorphisms, f : T → T , are isomorphisms of rooted trees which respect the labelings, i.e. satisfy thecondition l T = l T ◦ f . An example of such a morphism is given by the following picture(13) v v v e e k l nm ◦◦ ◦ ✠✠✠✠✠✠ ✺✺✺✺✺✺✎✎✎✎✎✎ ✴✴✴✴✴✴ ✎✎✎✎✎✎ ✴✴✴✴✴✴ f −−−−−−−−−→ ( v ,v ,v → ( v ,v ,v e ,e → ( e ,e k,l,m,n ) → ( n,m,k,l ) v v v e e k l nm ◦◦ ◦ ✠✠✠✠✠✠ ✺✺✺✺✺✺✎✎✎✎✎✎ ✴✴✴✴✴✴ ✎✎✎✎✎✎ ✴✴✴✴✴✴ where V ( T ) = { v , v , v } , E ( T ) = { e , e } and Leaf ( T ) = { k, l, m, n } .Given any S -module E : S → C in a symmetric monoidal category C (with small colimits) there is anassociated free operad F ree hEi which has an obvious universal property. For simplicity, we shall explain theconstruction of F ree hEi in the case C = Vect K , the general case being essentially the same.Given an S -module, E : S → Vect K , to any I -labelled rooted tree T ∈ Ob ( IsoTree ( I )) with, say, n vertices,one can associate a vector space (“unordered tensor product over the set of vertices of T ”), T hEi = O v ∈ V ( T ) E ( In v ) := M p :[ n ] → V ( T ) E (cid:0) In p (1) (cid:1) ⊗ . . . ⊗ E (cid:0) In p ( n ) (cid:1) S n in a general symmetric monoidal category one has to take the equalizer over the set of isomorphisms[ n ] → V ( T )). It is not hard to see directly from this definition that the association T → T hEi defines afunctor from the category IsoTree ( I ) to the category Vect K , and hence it makes sense to consider its colimitcolim T ∈ IsoTree ( I ) T hEi , which exists in Vect K . As a vector space this colimit is (non-canonically) isomorphic to the direct sum,colim T ∈ IsoTree ( I ) T hEi ≃ M T ∈ Ob ( IsoTree ( I )) / ∼ T hEi where the summation runs over representatives T of isomorphism classes of I -labelled rooted trees from thecategory IsoTree ( I ). We use this colimit to define an S -module, F ree hEi : S −→ Vect K I −→ F ree hEi ( I ) := colim T ∈ IsoTree ( I ) T hEi , and notice that it has a natural structure of the operad with respect to graftings of trees and unorderedtensor products. This operad is called the free operad generated by the S -module E .Elements of F ree hEi ( I ) can be visualized as linear combinations of E -decorated I -labelled rooted trees . Onecan represent pictorially such a decorated tree as follows, e e e ◦◦ ◦ ✠✠✠✠✠✠ ✺✺✺✺✺✺✎✎✎✎✎✎ ✴✴✴✴✴✴ ✎✎✎✎✎✎ ✴✴✴✴✴✴ where e i are vectors from E ( In v i ), i = 1 , , 3. Such a representation is not unique; for example, isomorphism(13) leads to the identification e e e ◦◦ ◦ ✠✠✠✠✠✠ ✺✺✺✺✺✺✎✎✎✎✎✎ ✴✴✴✴✴✴ ✎✎✎✎✎✎ ✴✴✴✴✴✴ ≃ σ ( e ) e σ ( e ) ◦◦ ◦ ✠✠✠✠✠✠ ✺✺✺✺✺✺✎✎✎✎✎✎ ✴✴✴✴✴✴ ✎✎✎✎✎✎ ✴✴✴✴✴✴ where σ is the unique non-trivial automorphism of a two element set I (say I = { e , e } , { k, l } or { m, n } ,see (13)), and σ ( e i ) is the result of an action of this automorphism on the corresponding elements of E ( I ).Every element in F ree hEi ( I ) is an iterated operadic composition of (isomorphism classes of) the followingbasic decorated graphs, ◦ . . . ✾✾✾✾ ❍❍❍❍❍✆✆✆✆✈✈✈✈✈ | {z } I e ∈E ( I ) called the generating corollas . (i) Show that the operad of parenthesized permutations, P a , is the free operad in thecategory Set generated by the following S -module, E ( I ) := (cid:26) Bij ( I → [ I ]) if I = 20 otherwise . where Bij ( I → [ I ]) is the set of all bijections I → [ I ].(ii) Show that the operad Lin K ( P a ) (see § ii) is the free operad in the category Vect K generated bythe following S -module, E ( n ) := (cid:26) K [ S ] if n = 20 otherwise . .3.7. Operad of little disks. Let D a,λ := { z ∈ C | | z − a | ≤ λ } be the closed disk in C of radius λ and with center a ∈ C . The operad of little disks , D = {D ( n ) } n ≥ , is anoperad in the category, Top pc , of path-connected topological spaces given by the following data,- D (1) is the point (the unit in D );- D ( n ) for n ≥ n disjoint closed disks, { D a i ,λ i } i ∈ [ n ] , lying inside thestandard unital disk, D , .- the operadic composition, for any k ∈ [ n ], is given by ◦ k : D ( n ) × D ( m ) −→ D ( n + m − { D a i ,λ i } i ∈ [ n ] × { D b j ,ν i } i ∈ [ m ] −→ n { D a i ,λ i } i ∈ [ n ] \ k a { D λ k b j + a k ,λ k ν j } j ∈ [ m ] o . Put another way, we replace the closed k -th disk D a k ,λ k in a configuration from D ( n ) with the λ k -rescaled and a k -translated configuration from D ( m ). The homology monoidal functor (see § vi) sends the topological operad D into an operad, H • ( D ), in thecategory dgVect K . It was proven by F. Cohen that H • ( D ) is precisely the operad of Gerstenhaber algebras ,i.e. its representations H • ( D ) → E nd V in a graded vector space V are equivalent to a Gerstenhaber algebra structure on V .The fundamental groupoid monoidal functor π : Top ∗ → CatG (see § vii) sends D into an operad π ( D ) inthe category of groupoids. Let O be an operad in the category dgVect K . For any integer m one canuniquely associate to O an operad O{ m } with the property that there is a 1-1 correspondence, (cid:26) reprersentations of O in a graded vector space V (cid:27) ←→ (cid:26) reprersentations of O{ m } in a graded vector space V [ m ] (cid:27) . From this property one easily reads the structure of O{ m } = {O{ m } ( n ) } n ≥ as an S -module, O{ m } ( n ) = O ( n )[ m (1 − n )] ⊗ sgn ⊗ mn . where sgn n is the one-dimensional sign representation S n .Put another way, O{ m } is the tensor product of the operad O and the endomorphism operad E nd K [ − m ] ofthe 1-dimensional vector space concentrated in degree m . We shall often use below the following observation. Let O = {O ( n ) } n ≥ be an operad in a symmetric monoidal category C . Assume there isan element e ∈ O (2) satisfying the condition (14) e ◦ e = e ◦ e. Then the family of maps, (15) d i : O ( n ) −→ O ( n + 1) f −→ d i f := e ◦ f for i = 0 f ◦ i e for i ∈ [ n ] e ◦ f for i = n + 1 satisfies the equations (16) d j d i = d i d j − for any i < j, and hence makes the collection O = {O ( n ) } n ≥ into a pre-cosimiplicial object in the category C . A pictorial description of this composition can be found at http://en.wikipedia.org/wiki/Operad theory See http://en.wikipedia.org/wiki/Gerstenhaber algebra for the definition of this notion. roof. Using the second definition of operads in terms of decorated trees, we shall identify elements e ∈ O (2)and f ∈ O ( n ) with decorated corollas, e = ◦ e ❄❄⑧⑧ =: • ❄❄⑧⑧ , f = ◦ f n ... ✼✼✼ ❏❏❏❏✞✞✞tttt =: N n ... ✼✼✼ ❏❏❏❏✞✞✞tttt Then d d f = d • N n +1 ... ✱✱✱ ✾✾✾✒✒✒✆✆✆❄❄⑧⑧ = • • N n +2 ... ✱✱✱ ✾✾✾✒✒✒✆✆✆❄❄⑧⑧✒✒✒ ✱✱✱ On the other hand, d d f = d • N n +1 ... ✱✱✱ ✾✾✾✒✒✒✆✆✆❄❄⑧⑧ ! = • • N n +2 ... ✱✱✱ ✾✾✾✒✒✒✆✆✆❄❄⑧⑧❄❄⑧⑧ The condition (14) reads •• 21 3 ❄❄⑧⑧ ⑧⑧ ❄❄ = • • 21 3 ❄❄⑧⑧❄❄⑧⑧ so that we get d d f = d d f for any f ∈ O ( n ). Similar picturesestablish identities (16) for all other values of i ∈ [ n + 1] and j ∈ [ n + 2] with i < j . We leave it to the readerto draw the details. (cid:3) If O is an operad in the category Vect K , then under the conditions of the above Lemma one can associate to O a dg vector space, Simp • ( O ) := M n ≥ O ( n )[ − n ] , equipped with the differential d O ( n ) −→ O ( n + 1) f −→ df := P n +1 i =0 ( − i d i f, whose cohomology is denoted by H Simp • ( O ) and is called the cosimplicial cohomology of O . We shall seebelow that, for example, the cosimplicial cohomology of the operad of infinitesimal braids has much to dowith the main topic of these lectures, the Grothendieck-Teichm¨uller group. Let O = {O ( n ) } n ≥ be an operad in the category dgVect K with operadic compositions ◦ i : O ( n ) ⊗ O ( m ) → O ( m + n − ≤ i ≤ n . Consider a vector space O tot := M n ≥ O ( n )Then the map[ , ] : O tot ⊗ O tot −→ O tot ( a ∈ O ( n ) , b ∈ O ( m )) −→ [ a, b ] := P ni =1 a ◦ i b − ( − | a || b | P mi =1 b ◦ i a makes O tot into a dg Lie algebra [KM]. Moreover, the bracket descends to the space of coinvariants ( O tot ) S := L n ≥ O ( n ) S n making this vector spaces into a dg Lie algebra as well. If char ( K ) = 0, the space of coinvariants( O tot ) S is isomorphic via the symmetrization map to the space of S -invariants, ( O tot ) S := L n ≥ O ( n ) S n , sothat one has an induced dg Lie algebra structure on ( O tot ) S with Lie bracket given by the above formulafollowed with the symmetrization. . Operad of parenthesized braids and d GT Let C n ( C ) be the configuration space of pairwise distinctpoints in the plane C , C n ( C ) = { ( z , z , . . . , z n ) ∈ C | z i = z j for i = j } . The group S n acts on C n ( C ) by permuting the points, σ : ( z , z , . . . , z n ) −→ (cid:0) z σ (1) , z σ (2) , . . . , z σ ( n ) (cid:1) , σ ∈ S n . et C n ( C ) / S n stand for the associated set of orbits equipped with the quotient topology. Let p n = ( z , . . . , z n )be any point in C n ( C ) (say p n = { (1 , , . . . , n ) ∈ C } ) and let ¯ p n be its image under the projection C n ( C ) → C n ( C ) / S n . (i) The fundamental group B n := π ( C n ( C ) / S n , ¯ p n ) is called the group of braids .(ii) The fundamental group PB n := π ( C n ( C ) , p n )) is called the group of pure braids . It is often useful not to distinguish isomorphic groups. As topological spaces C n ( C ) and C n ( C ) / S n are path-connected, the groups π ( C n ( C ) , p n ) and π ( C n ( C ) / S n , ¯ p n ) are isomorphic to each fordifferent choices of base points. Therefore one should view B n and PB n as isomorphism classes, π ( C n ( C ) / S n )and, respectively, π ( C n ( C )), of the fundamental groups π ( C n ( C ) / S n , ¯ p n ) and, respectively, π ( C n ( C ) , p n );this point of view makes the choice of the base points irrelevant.The projection C n ( C ) → C n ( C ) / S n is a regular n !-sheeted covering with S n acting as covering transforma-tions. Thus the subgroup PB n ⊂ B n has index n !, and we have a short exact sequence of groups,1 −→ PB n −→ B n p −→ S n −→ . Let b be an element in B n and σ b := p ( b ) ∈ S n the associated permutation. One can visualize an element b ∈ B n as (an isotopy classes of) n disjoint continuous curves (strands), s i : [0 , −→ C × [0 , , i = 1 , , . . . , n, such that s i (0) = z i , s i (1) = z σ b ( i ) , and the composition [0 , s i −→ C × [0 , proj −→ [0 , b = 11 22 and b ′ = 11 22 33 ✵✵✵✵✻✻✻✻✻ ✸✸✸✸✸ ✷✷✷✷✷✷✷ represent braids b ∈ B , b ′ ∈ B with σ b = (12) and σ b ′ = (cid:18) (cid:19) , while b ′′ = 11 22 represents a pure braid from PB . Multiplication of braids is represented by the concatenation of strands, xy = yx. . .. . .. . . ∀ x, y ∈ B n For example, b ′′ = ( b ) . B is the free group on one generator b , i.e. there is an isomorphism of groups Z → B which sends n into b n . Similarly, PB is the free group on one generator b , i.e. its every element is ofthe form b n for some uniquely defined n ∈ Z . If K is a field of characteristic zero, then the prounipotentcompletions, b B ( K ) and c PB ( K ), of these groups coincide with { b µ | µ ∈ K } ≃ K = b F ( K ), the prounipotentcompletion of the free group in one generator. .2. Algebraic definition of the (pure) braid group. According to M. Artin, the braid group B n canbe identified with a group of finite type with generators, b i := 11 22 nn ... ... ii i +1 i +1 and (so called braid ) relations, b i b j = b j b i for | i − j | > ,b i b i +1 b i = b i +1 b i b i +1 . Hence the prounipotent completion b B n ( K ) of B n over a field K can be identified (see § i) with exp b b ,where b b is the quotient of the completed free Lie algebra b lie n on letters { γ i } ≤ i ≤ n with respect to the Lieideal generated by the following formal power series of Lie words,log( e γ i e γ j e − γ i e − γ j ) = 0 for | i − j | > , log( e γ i e γ i +1 e γ i e − γ i +1 e − γ i e − γ i +1 ) = 0 . Note that the permutation group S n is a group of finite type with generators σ i = ( i ( i + 1)) and relations σ i = 11 σ i σ j = σ j σ i for | i − j | > ,σ i σ i +1 σ i = σ i +1 σ i σ i +1 . Hence the projection p : B n → S n is given on generators by b i → σ i .According to W. Burau and A. A. Markov, the pure braid group PB n can be identified with a group of finitetype with generators, { x ij } ≤ i 1, the “forgetful” map f : C n +1 ( C ) −→ C n ( C )( z , . . . , z n +1 ) −→ ( z , . . . , z n )is a locally trivial fiber bundle [FV]. Its fiber is isomorphic to C \ { z , . . . , z n } whose first homotopy groupis the free group F n on n generators while the second homotopy group is trivial, π ( C \ { z , . . . , z n } , ∗ ) = 1.We can choose base points p n in C n ( C ), n ≥ 1, such that f ( p n +1 ) = p n . Hence the associated long exactsequence of homotopy groups reads,1 −→ π ( C n +1 ( C ) , p n +1 ) −→ π ( C n ( C ) , p n ) −→ F n −→ π ( C n +1 ( C ) , p n +1 ) −→ π ( C n ( C ) , p n ) −→ , n ≥ . s π ( C ( C ) , p ) = π ( C , p ) = 1, one obtains by induction that π ( C n ( C ) , p n ) = 1 for all n , and hence onegets an exact sequence of groups, 1 −→ F n −→ PB n +1 −→ PB n −→ . This exact sequence splits as the fiber bundle f : C n +1 ( C ) → C n ( C ) admits a cross-section, s : C n ( C ) −→ C n +1 ( C )( z , . . . , z n ) −→ ( z , . . . , z n , | z | + . . . + | z n | + 1) . Hence we proved the following For any n ≥ one has PB n +1 = F n ⋊ PB n = F n ⋊ ( F n − ⋊ . . . ⋊ ( F ⋊ ( F ⋊ F ))) Every element of PB can be uniquely represented as f ( b , b )( b b ) n for some n ∈ Z andsome element f ( x, y ) in the free group F on the generators x and y .Proof. The group PB is generated by x = b , x = b and x = b b b − , or, equivalently, by b , b and x x x = b b b b = ( b b ) = ( b b ) , where we used the braid relations b b b = b b b in B . As ( b b ) is central in PB and there are norelations between the generators b and b , the result follows. (cid:3) PB = F × Z .This corollary can be seen also from the fact that the fibration C ( C ) → C ( C ) is trivial.If K is a field of characteristic zero, then c PB ( K ) = c F ( K ) × K , with c F ( K ) generated by [ b ] λ , [ b ] λ , andthe factor K corresponding to [( b b ) ] λ , λ i ∈ K , i = 1 , , 3. On the contrary, b B ( K ) is a much simplergroup. If char ( K ) = 0 , then b B ( K ) ≃ b F ( K ) ≃ K . Proof. Set c := ( b b ) . As it is a central element in B , c λ commutes with any element in b B ( K ) for any λ ∈ K . Using the last equality in § (iii), we obtain b b b = b b b = c in b B ( K )so that b b = b − c , b − c b = c , and hence b = b = c in b B ( K ) . (cid:3) Let char ( K ) = 0 . The injection PB → B induces a surjection c PB ( K ) → b B ( K ) whosekernel is the prounipotent completion of the free group generated by b = c − / b and b := c − / b . S operad of pure braids. Consider a collection of groups, PB := { π ( C n ( C )) ≃ PB n } . Recall that π ( C n ( C )) stands for the isomorphism class of the fundamental group π ( C n ( C ) , p n ) defined fora particular choice of the base point p n ; as C n ( C ) is path connected, the class π ( C n ( C )) does not depend onsuch a choice. The collection PB can be made into a non- S operad with respect to the operadic composition, ◦ i : PB n × PB n −→ PB n + m − ∀ n, m ∈ N , i ∈ [ n ]( b ′ , b ′′ ) −→ b ′ ◦ i b ′′ which is defined by replacing the i -labelled strand in b ′ by the braid b ′′ made very thin. Perhaps, a morerigorous (and non-ambiguous) way to define this composition is to use the operad of little disks D definedin § and an elementary observation that, for every n ≥ 1, there is a continuous map c : D ( n ) −→ C n ( C ) hich associates to a configuration of little disks the configuration of their centers. This map is a homotopyequivalence so that π ( D ( n )) ≡ π ( C n ( C )) . Hence the operadic composition in D , ◦ i : D ( n ) × D ( m ) −→ D ( n ) ◦ i D ( m )induces an operadic composition in PB which can visualized precisely as explained in the beginning of thissubsection. Note that the (non-trivial) action of the permutation group S n on the topological space D ( n )induces a trivial action on the isomorphism class π ( C n ( C )) so that PB is a non- S operad indeed. PB . Let e be the identity element in the group PB (2) = PB . Then itsatisfies the condition e ◦ e = e ◦ e and hence gives rise to a pre-cosimplicial structure, d i : PB n −→ PB n +1 , i = 0 , , . . . , n + 1 , on the collection PB = { PB n } n ≥ given by the explicit formulae in (15). For example, one has in PB ,(17) d ( b ) = ( b ) , d ( b ) = ( b b ) , d ( b ) = ( b b ) , d ( b ) = ( b ) . In fact, this structure can be completed to a cosimplicial structure on PB by defining the operators s i : PB n +1 −→ PB n , i = 1 , , . . . , n + 1 , where s i erases the strand labelled by i . If we were able to make the operad of little disks D into anoperad in the category of based topological spaces, then, by applying the monoidal functor (10), we wouldhave obtained an operad π ( D ) in the category of groups. However this is impossible as there is no S n invariant configuration of little disks in D ( n ) for every n ≥ D gives us an operad in the category of groupoids, CatG ,with nontrivial action of permutations groups as endofunctors, σ : π ( D )( n ) → π ( D )( n ), σ ∈ S n , n ≥ π ( D )( n ) has too many objects (which arepoints in D ( n )). Let us construct by induction a “smallest possible” non-trivial suboperad, P a B , of theoperad π ( D ) such that P a B ( n ) is a full subcategory of π ( D )( n ) for each n ≥ n = 1 : Set P a B (1) to a full subcategory of π ( D )(1) which has only one object, the configuration D , ∈ π ( D )(1) (which is the unit in the operad D ). n = 2 : Let be any configuration of little disks in D (2), that is, a pair of disjoint disks inside the unitdisk labeled by 1 and 2, and set := σ ( ), where σ = (12) ∈ S . Set P a B (2) be the full subcategoryof π ( D )(2) whose only objects are and ; the S action on π ( D )(2) leaves the subcategory P a B (2)invariant. It is useful to identify Ob ( P a B (2)) with P a (2), the set of two planar binary corollas, = ◦ ⑧⑧ ❄❄ = ◦ ⑧⑧ ❄❄ n = 3 : Let ◦ i : D (2) × D (2) → D (3), i = { , } , be the operadic compositions in D ( n ). Set P a B (3) be thefull subcategory of π ( D )(3) whose only objects are p ◦ i q , where p , q ∈ { , } . It is clear that the set Ob ( P a B (3)) can be identified with P a (3), the set of parenthesized permutations of [3] = { , , } (see § ), or, equivalently, the set of [ ]-labelled planar binary trees, ◦ = ( ) , ◦ = ( ) , ◦ = ( ) , ◦ = ( ) , etc.n ≥ Ob ( P a B ) be the suboperad of Ob ( π ( D )) (in the category of sets) generated by the S -module E = { E ( n ) } , where E ( n ) := Ob ( P a B (1)) = D , for n = 1 ,Ob ( P a B (2)) = { , } for n = 2 , ∅ for n ≥ P a . et now P a B ( n ) be the full subcategory of the category π ( D ( n )). By construction, the collection of groupoids, P a B := {P a B ( n ) } n ≥ , is a suboperad of π ( D ). It was first introduced by D. Bar-Natan [B-N]. The idea to use the languageof operads in D. Bar-Natan’s approach to Drinfeld associators and the Grothendieck-Teichm¨uller group isdue to Tamarkin [T2] (see also the works of P. Severa and T. Willwacher [Se, SW]). This idea makes D.Bar-Natan’s story short and transparent.One can equivalently describe the operad P a B as functor, P a B : S −→ CatG I −→ P a B ( I ) = (cid:26) objects , Ob ( P a B ( I ))sets of morphisms , { Mor( A I , B I ) } A I ,B I ∈ Ob ( P a B ( I )) where P a B ( I ) is the category given by the following data:(i) the objects are parenthesized permutations of the set I , or, equivalently, planar I -labelled binarytrees equipped with the operadic structure explained in § ; put another way, the “object” partof the operadic structure is given by the functor P a : S −→ Set I −→ P a ( I ) =: Ob ( P a B ( I ))(ii) for any two objects A I and B I in Ob ( P a B ( I )), that is, for any two parenthesized permutations ofthe set I , the associated set of morphisms, Mor( A ( I ) , B ( I )), is defined to be the set of braids whosestrands connect the same elements of I (and hence have a non-ambiguous I -labelling). For example, β , := 12 21 ∈ Mor (12 , , , , := (11 2)(2 33) ∈ Mor ((12)3) , b ′ ◦ i b ′′ on morphisms is defined by replacing the i -labelled strand of thebraid b ′ with the braid b ′′ made thin. Show that the operad P a B is generated (via operadic and categorical compositions) bythe elements β , , 11 , , and their inverses.Find an explicit expression for the element ✽✽✽✽✽✽✽ ✭✭✭✭ ✸✸✸✸✸ ∈ Mor (1(23) , (23)1) in terms of the generators β •• and11 • , • , • . P a B . Let 11 be the identity element in Mor(12 , := 11 22 and, for any n ∈ N ≥ , define a family of functors, d i : P a B ( n ) −→ P a B ( n + 1) , i = 0 , , . . . , n, n + 1 , as follows:(i) on objects (i.e. on parenthesized permutations of [ n ]), one has d i : Ob ( P a B ( n )) −→ Ob ( P a B ( n + 1)) A n −→ d i A n := (12) ◦ A n for i = 0 A n ◦ i (12) for i ∈ [ n ](12) ◦ A n for i = n + 1 ii) on morphisms (i.e. on braids connecting the same elements in a pair of parenthesized permutations),one has d i : Mor( A n , B n ) −→ Mor( d i A n , d i B n ) f n −→ d i f n := , ◦ f n for i = 0 f n ◦ i , for i ∈ [ n ]11 , ◦ f n for i = n + 1Note that in the operad Ob ( P a B ), (12) ◦ (12) = (12) ◦ (12)so that the functors d i do not make the collection of groupoids {P a B ( n ) } n ≥ into a (pre)cosimplicial objectin the category of small categories. However these functors will be very useful for us below for the thefollowing reason. Let O be an operad in the category of small categories with same set of objects as in P a B . Let 11 , be the identity morphism from (12) to (12) in the category O (2). Then the formulae formallyidentical to the ones above give us a family of functors, d i : O ( n ) → O ( n + 1); moreover, any morphism ofoperads F : P a B → O respects these functors, i.e. d i ◦ F ( n ) = F ( n + 1) ◦ d i , for any n ∈ N and any i ∈ { , , . . . n + 1 } . The non- S operad PB also has a presimplicial structure given formally by the same for-mulae as above. Consider the elements d , , = 11 ((2(2 3)((3 4)4)) , d , , = (1((1 2)2) 3)(3 44) , d , , = ,d , , = , d , , = (1((1 2)(2 3)3) 44 , and check that they satisfy the pentagon equation (cf. (5))(18) d , , ◦ d , , = d , , ◦ d , , ◦ d , , Here ◦ denotes the categorical (rather than operadic) composition of morphisms.Following a long tradition (see, e.g., [Dr2, ES]), the elements d , , , d , , , d , , , d , , and d , , should have been denoted, respectively, by 11 , , , 11 , , , 11 , , , 11 , , and 11 , , so that the pentagonequation takes a familiar form (cf. equation (17.2) in the book [ES])11 , , ◦ , , = 11 , , ◦ , , ◦ , , . However, we shall mostly stick to the cosimplicial notation in this survey. One has in P a B (3), d β , = (2(3 3)2)11 , d β , = (1(2 2)1) 33 , d β , = (1 2) 33 (1 2) ❈❈❈❈❈❈ ❇❇❇❇❇❇❇ The permutation group S n acts on P a B ( n ) by relabelling of strands, for example,( · , , = 11 , , = (11 3)(3 22) , ( · , , = 11 , , = (33 1)(1 22) , ( · ( d β , ) = (1(3 3)1) 22 It is easy to see that the generators satisfy the following hexagon equation in P a B (3),(19) d β , = 11 , , ◦ d β , ◦ (cid:0) ( · − , , (cid:1) ◦ ( · d β , ◦ ( · , , . tudying analogously β − , = ( · β − , == (( · β , ) − = 12 21 ∈ Mor (12 , d β − , = 11 , , ◦ d β − , ◦ (cid:0) ( · − , , (cid:1) ◦ ( · d β − , ◦ ( · , , . for the generators β , and 11 , , of the operad P a B . Let O be an operad in the small category of groupoids with same set of objects as in P a B .There is a one-to-one correspondence between morphisms of operads, F : P a B −→ O , which are identical on objects, and elements F ( β , ) ∈ Mor O (12 , and F (11 , , ) ∈ Mor O ((12)3 , ,which satisfy the pentagon equation, (21) d F (11 , , ) ◦ d F (11 , , ) = d F (11 , , ) ◦ d F (11 , , ) ◦ d F (11 , , ) and the hexagon equations ,(22) d F ( β , ) = F (11 , , ) ◦ d F ( β , ) ◦ (cid:0) (23) · F (11 , , ) − (cid:1) ◦ (23) · d F ( β , ) ◦ (321) · F (11 , , ) , (23) d F ( β , ) − = F (11 , , ) ◦ d ( F ( β , ) − ) ◦ (cid:0) (23) · F (11 , , ) − (cid:1) ◦ (23) · d ( F ( β , ) − ) ◦ (321) · F (11 , , ) . Proof. As P a B is generated by β , and 11 , , , any morphism of operads F : P a B → O is uniquely determinedby its values on the generators which must satisfy the equations obtained by applying F to (18), (19) and(20).The reverse statement follows from the Mac Lane coherence theorem for braided monoidal categories, i.e.from the fact that all the relations between iterations (via operadic and categorial compositions) of thegenerators follow from (18), (19) and (20). (cid:3) d P a B K . Applying the functor ∆ K (see § (iii)) to the operad P a B we getan operad, P a B K := ∆ K ( P a B ) , in the symmetric monoidal category Cat ( coAss K ). For any finite set I the objects, A I , of the category P a B K ( I )are the same as objects of P a B , that is, the parenthesized permutations of the set I , while morphisms areformal K -linear combinations of braids,Mor P a B K ( A I , B I ) = (X i λ i f i | λ i ∈ C , f i ∈ Mor P a B ( A I , B I ) ) , which connect the same letters in the words A I and B I . The coproduct structure on morphisms is uniquelydefined by the condition that all f i are group-like.Every operad O in Cat ( coAss K ), in particular P a B K , comes equipped with a diagonal morphism of operads,(24) ∆ : O −→ O ⊗ O , which is given on objects A I ∈ Ob ( O ( I )) by ∆( A I ) = A I × A I , and on morphisms ∆ is given by the above coproduct in Mor O ( A I , B I ).Let us define one more operad in this category, P a K : S −→ Cat ( coAss K ) , whose objects are, by definition, identical to objects of the operad P a (and hence of P a B K ) and whosemorphisms spaces, Mor P a K ( A I , B I ), are identified with K equipped with the coproduct ∆(1) := 1 ⊗ P a K is the operad P a B K with “braiding” on morphisms forgotten. here is a augmentation morphism of operads (cf. § ), ε : P a B K −→ P a K , which is identical on objects, and is given on morphisms by the formula, ε : Mor P a B K ( A I , B I ) −→ Mor P a K ( A I , B I ) P i λ i f i −→ P i λ i The kernel, I K := Ker ε is an operadic ideal in P a B K . One can define its power, I m K , m ≥ 1, which is asuboperad of I K with the same objects as in I K but with morphisms given by the categorical compositionof at least m morphisms from I K . In this way we get a family of ideals, {I m K } m ≥ , of the operad P a B K , andhence an associated inverse system of operads in the category Cat ( coAss K ), . . . −→ P a B ( m ) K −→ P a B ( m − K −→ . . . −→ P a B (1) K −→ P a B (0) K , where P a B ( m ) K is the quotient operad P a B K / I m +1 K , m ≥ 0. The inverse limit, d P a B K := lim ←− P a B ( m ) K . is an operad in the category d Cat ( coAss K ). d GT ( K ) . The group of those automorphisms of the operad d P a B K which are identical on objects is called the Grothendieck-Teichm¨uller group and is denoted by d GT ( K ) orsimply by d GT or even GT . A similar group of automorphisms of P a B ( m ) K is denoted by GT ( m ) ( K ). The group d GT ( K ) coincides withthe inverse limit lim ←− GT ( m ) ( K ).Let us next characterize elements of d GT ( K ) as solutions of certain explicit algebraic equations. This char-acterization proves that the above operadic definition of d GT ( K ) coincides with the one given by V. Drinfeldin his original paper [Dr2]. [Dr2] . Let b F ( K ) be the pro-unipotent completion of the free group in two variables x and y . There is a one-to-one correspondence between elements of d GT ( K ) and pairs, (cid:16) f ∈ b F ( K ) , λ ∈ K ∗ (cid:17) , which satisfy the following equations, (25) f ( x, y ) = f ( y, x ) − , (26) f ( x , x ) x µ f ( x , x ) x µ f ( x , x ) x µ = 1 for x x x = 1 and µ := λ − , and (27) f ( x , x x ) f ( x x , x ) = f ( x , x ) f ( x x , x x ) f ( x , x ) in c PB ( K ) . The multiplication law in d GT ( K ) is given by, ( λ , f ) ( λ , f ) = (cid:0) λ λ , f ( f ( x, y ) x λ f ( x, y ) − , y λ ) f ( x, y ) (cid:1) Proof. By Theorem , every automorphism F of d P a B is uniquely determined by its values on thegenerators β , and 11 , , so that we have only to check that these values satisfy pentagon equation (21),two hexagon equations (22) and (23), and that F sends group-like elements in d P a B to group-like elementswith respect to the diagonal ∆. The latter condition is equivalent to saying that F ( β , ) and F (11 , , ) aregroup-like in Mor d P a B ( ) and Mor d P a B ( (12)3,1(23) ). Hence F ( β , ) ◦ β − , is grouplike in the coalgebra Mor d P a B ( ) Since we understand d P a B K as an operad in d Cat ( coAss K ), it is tacitly assumed that its every automorphism respects thediagonal ∆. nd F (11 , , ) ◦ − , , is grouplike in the coalgebra Mor d P a B ( (12)3,1(23) )which in turn imply (see §§ and ) that F ( β , ) ◦ β − , = [ β , ] µ ∈ d PB ( K )for some µ ∈ K with µ = − (as otherwise F would not be invertible), and F (11 , , ) ◦ − , , = f ( b , b )[( b b ) ] η ∈ d PB ( K )for some f ( x, y ) ∈ c F ( K ) and some η ∈ K . Here b and b stand for the renormalized generators (cf. § ), b := c − / b , b := c − / b , and b , b and c := ( b b ) are the grouplike elements of Mor d P a B ( (12)3,(12)3 ) obtained from the correspondingbraids in PB by attaching strands’ inputs and outputs to one and the same object (12)3, for example, b = (1(1 2)2) 33 = d ( β , ) , b := = 11 , , ◦ (2(2 3)3)11 ◦ − , , = 11 , , ◦ d ( β , ) ◦ − , , Using (4) we obtain d F ( β , ) = d ([ β , ] µ ◦ β , )= d ([ β , ] µ ) ◦ d ( β , )= 11 − , , ◦ [ b ] µ ◦ , , ◦ d β , so that F (11 , , ) ◦ d F ( β , ) = h f ( b , b )( b ) µ c η + µ i ◦ , , ◦ d β , . Denoting g = 11 , , ◦ d β , ◦ ( − , , = (11 2)(2 33) ◦ (2(3 3)2)11 ◦ = we obtain, using the results of § , g ◦ (cid:2) ( · f ( b , b ) (cid:3) − ◦ g − = f (cid:0) g ◦ ( · b ◦ g − , g ◦ ( · b ◦ g − (cid:1) − = f ( b , b − b − ) − so that the product of the first three terms on the r.h.s. of (22) now reads, F (11 , , ) ◦ d F ( β , ) ◦ (cid:0) ( · F (11 , , ) − (cid:1) = h f ( b , b )( b ) µ f ( b , b − b − ) − c µ i ◦ g. Next we have, g ◦ ( · d F ( β , ) ◦ g − = g ◦ ( · [ b ] µ ◦ g − ◦ g ◦ ( · d β , ◦ g − = h ( b − b − ) µ c µ i ◦ g ◦ ( · d β , ◦ g − , and hence F (11 , , ) ◦ d F ( β , ) ◦ (cid:0) ( · F (11 , , ) − (cid:1) ◦ ( · d F ( β , ) == h f ( b , b )( b ) µ f ( b , b − b − ) − ( b − b − ) µ c µ i ◦ h. where h := g ◦ ( · d β , = = d β , ◦ ( · − , , . s ( b b )( b )( b b ) − = b − b − and ( b b )( b )( b b ) − = b in b B ( K ), we have h ◦ ( · F (11 , , ) ◦ h − = [ f ( b , b − b − )] ◦ h ◦ ( · , , ◦ h − and hencer.h.s. of (22) = h f ( b , b )( b ) µ f ( b , b − b − ) − ( b − b − ) µ f ( b , b − b − ) c η + µ i ◦ h ◦ ( · , , . On the other hand, one has d b = b b b = ( b b ) b − = c / b − in b B ( K ), and hencel.h.s. of (22) = d F ( β , ) = [ c µ ( b − ) µ ] ◦ h ◦ ( · , , . Therefore, the first hexagon equation (22) is equivalent to the following equation in c PB ( K ), f ( b , b )( b ) µ f ( b , b − b − ) − ( b − b − ) µ f ( b , b − b − )( b ) µ c η = 1Hence η = 0 and the element f ( x, y ) ∈ c F ( K ) satisfies the equation(28) f ( x , x ) x µ f ( x , x ) − x µ f ( x , x ) x µ = 1for x x x = 1. Studying similarly the second hexagon equation (23), one obtains (using identities b − b b = b − b − , ( b b ) − ( b )( b b ) = b − b − and ( b b ) − ( b )( b b ) = b ) the following equation,( b ) − µ f ( b , b )( b ) − µ f ( b , b − b − ) − ( b − b − ) − µ f ( b , b − b − ) = 1which says that f ( x, y ) ∈ c F ( K ) satisfies(29) f ( x , x ) − x µ f ( x , x ) x µ f ( x , x ) − x µ = 1 . for x x x = 1.Equations (28) and (29) imply the first two claims, (25) and (26), of the Theorem.Note that relation between the generators x ij of PB n are invariant under their renormalizations, x ij → c λ ij x ij ,by powers of the central element c = ( b b · · · b n ) n . Therefore, when computing d i ( f ( b , b )), i = 0 , , , b and b with b and b . As one has in PB d ( b ) = x , d ( b ) = x x , d ( b ) = x x , d ( b ) = x , the l.h.s. of the “pentagon” equation (21) reads, d F (11 , , ) ◦ d F (11 , , ) = f ( x , x x ) f ( x x , x ) . Continuing in the same way we conclude that (21) is equivalent to the following equality in c PB ( K ), f ( x , x x ) f ( x x , x ) = f ( x , x ) f ( x x , x x ) f ( x , x ) . Taking inverses of both sides and using (25) we obtain from the latter equation the third claim, equation(27), of the Theorem.The final claim about the formula for the group multiplication in d GT ( K ) is left as an exercise. (cid:3) There is a surjection of groups, d GT ( K ) −→ K ∗ ( f, λ ) −→ λ. Its kernel is a subgroup of d GT ( K ) often denoted by d GT ( K ). A remarkable result of K. Furusho [Fu1] gives us a much shorter algebraiccharacterization of elements of the Grothendieck-Teichm¨uller group than the one given in the above Theorem : the pentagon equation (27) often implies in the hexagon equations (25) and (26). More precisely, onehas .8.1. Theorem [Fu1] . Let K be a field of characteristic and K be its algebraic closure. Suppose that anelement f = f ( x, y ) ∈ b F ( K ) satisfies Drinfeld’s pentagon equation (27). Then there exists an element λ ∈ K (which is unique up to a sign) such that the pair ( λ, f ) satisfies hexagon equations (25) and (26). Moreover,this λ is equal to ± p c ( f ) + 1 , where c ( f ) stands for the coefficient of XY in the formal power series f ( e X , e Y ) . Therefore, the group d GT ( K ) can be identified with the set of pairs ( f = f ( x, y ) ∈ b F ( K ) , ± p c ( f ) + 1) ∈ K \ f is a solution of Drinfeld’s pentagon equation with c ( f ) = − . . Infinitesimal braids, GRT and associators6.1. Graded Lie algebra of PB n . Let gr K be the functor associating to a discrete group thegraded Lie algebra obtained from the descending central series of that group (see § (viii) fordetails). It was proven in [Ko] that gr K ( PB n ) can be identified with the quotient of the free Liealgebra lie n ( n − / generated by the symbols { t ij = t ji } ≤ i There is an isomorphism of Lie algebras, t = K c ⊕ lie , where K c is the Abelian Lie algebra generated by c = t + t + t , and lie is the free Lie algebragenerated by t and t (or, equivalently, by t and t ) . .1.4. Exercise. As PB n is a group of finite type, its prounipotent completion c PB n ( K ) over afield K can be identified with the group exp( c pb n ( K )), where the complete Lie algebra c pb n ( K ) is thequotient of the free Lie algebra b lie n ( n − / on letters { γ ij } ≤ i Consider the maps of sets, f : J −→ I \ k ⊔ Jp −→ f ( p ) := p and g : I \ k ⊔ J −→ Ix −→ g ( x ) := ( i if x = i ∈ I \ kk if x = p ∈ J. with f being an injection, and Im f ◦ g being an subset of I of cardinality 1. Then the operadiccomposition ◦ I,Jk can be identified with the map f ∗ ⊕ g ∗ . Hence the Exercises just above imply that ◦ I,Jk is a Lie algebra homomorphism. We leave it to the reader to check that the homomorphisms ◦ I,Jk satisfy the axioms of a (non-unital) operad. (cid:3) The S -module b t of completed Lie algebras of infinitesimal braids is, of course, also an operad onthe category Lie K . Check that the operadic composition ◦ : t (3) ⊕ t (3) −→ t (5)([ t , t ] , [ t , t ]) −→ [ t , t ] ◦ [ t , t ]gives [ t , t ] ◦ [ t , t ] = [ t + t + t , t + t + t ] + [ t , t ] Let e := 0 be the zeroelement in t (2). It obviously satisfies the equation (14) and hence makes by Lemma § thecollection t = { t ( n ) } ≥ into the cosimplicial object in the category Lie K . The cohomology of theassociated cosimplicial dg Lie algebra, Simp • ( t ) = M n ≥ t n [2 − n ] , d = X i ( − i d i , was studied by Thomas Willwacher in his breakthrough paper [W1]. Its relation to theGrothendieck-Teichm¨uller group is discussed below. Applying the universal enveloping monoidal functor U : Lie K → Hopf K we get an operad CD K := U ( t ) = {CD K ( n ) := U ( t ( n )) } called in [B-N] the operad of chord diagrams because its elements, say, elements of CD K ( n ), can bepictorially presented as a collection of chords on n vertical strands (labelled by 1 , , . . . , n from leftto right) e.g. CD K (3) ∋ t t ←→ O O O O O O s an associative algebra CD K ( n ), n ≥ 2, is the quotient of the free non-commutative algebragenerated by symbols t ij = t ji , i = j i, j ∈ [ n ], modulo the ideal generated by the relations[ t ij , t kl ] = 0 if { i, j, k, l } = 4 and [ t ij , t ik + t kj ] = 0 if { i, j, k } = 3. The coproduct in CD K ( n ) isgiven on the generators as follows, ∆( t ij ) = 1 ⊗ t ij + t ij ⊗ . The operadic structure in CD K is completely determined by the one in t . For example, one has ◦ : CD K (3) ⊗ K CD K (3) −→ CD K (5)( t t , t t ) −→ ( t + t + t )( t + t + t ) t t . The I -adic completion of the associative algebra CD K ( n ) with respect to the maximal ideal I isdenoted by c CD K ( n ). The collection c CD K = { c CD K ( n ) } n ≥ is an operad in the category of completedfiltered Hopf algebras. For our purposes it is useful to view c CD K instead as an operad in the category Cat ( Coass K ) as follows: each competed filtered Hopf algebra c CD K ( n ) = b U ( t n ) is understood fromnow on as a category with one object • and with M or ( • , • ) := c CD K ( n ) equipped with its standardcoalgebra structure. c CD K . The element e := 1 in CD (2) satisfies equations (14) sothat formulae (15) make the collection c CD K = { b U ( t n ) } n ≥ into a cosimplicial object in the category dgAss K . Let t t ∈ b U ( t ). Check that d (cid:0) t t (cid:1) = t t , d (cid:0) t t (cid:1) = ( t + t ) t , d (cid:0) t t (cid:1) = ( t + t ) ( t + t ) . We have two operads inthe symmetric monoidal category Cat ( Coass K ), P a K defined in § and c CD K defined just above.Hence \ P a CD K := P a K ⊗ Cat ( Coass K ) c CD K is an operad in the category Cat ( Coass K ) called in [B-N] the operad of parenthesized chord diagrams .Elements of P a K can be pictorially represented as elements of P a B K but with braiding of strandsforgotten so that we represent morphisms in the category \ P a CD K ( n ) as linear combinations, withcoefficients in the algebra c CD K ( n ), of non-braided strands connecting identical symbols in a pair ofparenthesized permutations of [ n ], for example H , := t · 11 22 ∈ Mor \ P a CD K (2) (12 , , X , = 1 · 21 21 ✔✔✔✔✔✔✔✔✯✯✯✯✯✯✯✯ ∈ Mor \ P a CD K (2) (12 , , , := (11 2)(2 33) ∈ Mor \ P a CD K (3) ((12)3) , Show that the operad \ P a CD K is generated by H , , X , and 11 , , . Let 11 be the identity element in Mor \ P a CD K (2) (12 , , := 11 22 Show that, for any n ∈ N , the family of functors, d i : \ P a CD K ( n ) −→ \ P a CD K ( n + 1) , i = 0 , , . . . , n, n + 1 , efined on objects and morphisms of the category \ P a CD K ( n ) by formulae identical to the ones in § , makes the collection of categories { \ P a CD K ( n ) } n ≥ a (pre)cosimplicial object in the category Cat ( Coass K ). Check that d H , = 11 , ◦ H , = t 11 (2(2 3)3) , d H , = H , ◦ , = ( t + t ) (1(1 2)2) 33 ,d H , = H , ◦ , = ( t + t ) 11 (2(2 3)3) , d H , = 11 , ◦ H , = t (1(1 2)2) 33 . Let Φ = Φ( t , t )11 , , ∈ Mor \ P a CD K (3) ((12)3) , t , t ) ∈ K hh t , t ii ⊂ b U ( t ). Check that d Φ = 11 , ◦ Φ = Φ( t , t ) d , , , d Φ = Φ ◦ , = Φ( t + t , t ) d , , d Φ = Φ ◦ , = Φ( t + t , t + t ) d , , ,d Φ = Φ ◦ , = Φ( t , t + t ) d , , , d Φ = 11 , ◦ Φ = Φ( t , t ) d , , . GRT and its Lie algebra. The group of auto-morphisms of the operad \ P a CD K which preserve elements H , and X , is called the gradedGrothendieck-Tecihm¨uller group and is denoted by GRT [Dr2, B-N, T2]. There is a one-to-one correspondence between elements of GRT and grouplikeelements, Φ( x, y ) = e φ ( x,y ) , of the completed universal enveloping (Hopf ) algebra b U ( lie ) ≃ K hh x, y ii of the free Lie algebra on generators x and y which satisfy the following equations (30) Φ( x, y ) = Φ − ( y, x ) , (31) Φ( t , t )Φ( t , t ) − Φ( t , t ) = 1 in b U ( t ) , (32) Φ( t + t , t )Φ( t , t + t ) = Φ( t , t )Φ( t + t , t + t )Φ( t , t ) in b U ( t ) . Proof. Any automorphism F of the operad \ P a CD K is uniquely determined by its values on thegenerators H , , X , and 11 , , . These values are not arbitrary, they must respect the relationsbetween the generators. Mac Lane coherence theorem for symmetric monoidal categories impliesthat all the relations between the generators follow from the following two ones:Pentagon : d , , ◦ d , , = d , , ◦ d , , ◦ d , , Hexagon : d X , = 11 , , ◦ d X , ◦ (cid:16) (23) · − , , (cid:17) ◦ (23) · d X , ◦ (321) · , , . As an element of GRT preserves, by definition, the generators H , and X , , we conclude thatsuch an element F is uniquely determined by its value on 11 , , , F (11 , , ) ∈ Mor \ P a CD K (3) ((12)3 , ≃ b U ( t ) , which must satisfy the equations,(33) d F (11 , , ) ◦ d F (11 , , ) = d F (11 , , ) ◦ d F (11 , , ) ◦ d F (11 , , ) nd(34) d X , = F (11 , , ) ◦ d X , ◦ (cid:0) (23) · F (11 , , ) − (cid:1) ◦ (23) · d X , ◦ (321) · F (11 , , ) . By Corollary § b U ( t ) = K [[ c ]] × K hh t , t ii so that F (11 , , ) = f ( c )Φ( t , t )for some invertible formal power series, f ( c ) = a + a c + a c + . . . , a = 0in c = t + t + t , and some invertible formal power series Φ ∈ b U ( lie ). As 11 , , is grouplikein the coalgebra Mor \ P a CD K (3) ((12)3 , e φ forsome φ ∈ b lie .The hexagon equation (34) implies,Φ( t , t )Φ( t , t ) − Φ( t , t ) = 1 , f ( c ) = 1 . The hexagon equation is S -equivariant. Applying to that equation permutation (32), or, equiva-lently, applying this permutation to the above equation for Φ, we get,Φ( t , t )Φ( t , t ) − Φ( t , t ) = 1and hence Φ( t , t )Φ( t , t ) = 1 . This proves claims (30) and (31). The last claim follows immediately from the pentagon equationfor F (11 , , ) and the definition of the functors d i . We leave the details to the reader. (cid:3) The group GRT is prounipotent. Hence it is of the form exp( grt ) where the Lie algebra grt of GRT can be identified with elements φ ( x, y ) ∈ b lie which satisfy the equations(35) φ ( t + t , t ) + φ ( t , t + t ) = φ ( t , t ) + φ ( t + t , t + t ) + φ ( t , t )in b t , and(36) φ ( x, y ) = − φ ( y, x )(37) φ ( x, y ) + φ ( y, − x − y ) + φ ( − x − y, x ) . Show that φ ( x, y ) = [ x, y ] satisfies equations (35) and (36), but not (37).K. Furusho [Fu1] has proven that grt can be identified with elements φ ( x, y ) ∈ b lie which satisfythe pentagon equation (35) and do not contain [ x, y ] as a summand. Put another, in this casehexagon equations (36) and (37) follow from (35). Let Simp • ( t ) be the cosimplicial complex of the operad of infinitesimal braids(see ( )). Then H ( Simp • ( t )) = grt ⊕ K where the direct summand K is generated by [ t , t ] ∈ t . This is a reformulation of Proposition E.1 in [W1]. It gives a second proof of K. Furusho’s resultdiscussed just above. We have operads d P a B K and \ P a CD K in the category Cat ( coAss K ). An isomor-phism of operads (if it exists) A : d P a B K −→ \ P a CD K is called an associator [Dr2, B-N]. It is not hard to show the following .8.1. Theorem. There is a one-to-one correspondence between the set of associators and group-like elements, Φ( x, y ) = e φ ( x,y ) , of the completed filtered Hopf algebra b U ( lie ) of the free Lie algebraon generators x and y which satisfy the following equations (38) Φ( x, y ) = Φ − ( y, x ) , (39) Φ( t , t ) e − t Φ( t , t ) − e − t Φ( t , t ) e − ( t + t ) = 1 in U ( b t ) , (40) Φ( t + t , t )Φ( t , t + t ) = Φ( t , t )Φ( t + t , t + t )Φ( t , t ) in b U ( t ) . It is much harder to show that for any field K of characteristic zero the set of associators, As K := { d P a B K → \ P a CD K } is non-empty, see [Dr2] for the proof. The equations defining associators are algebraic, but the onlytwo explicit solutions we know involve transcendental methods in their constructions. The first associator was constructedby V. Drinfeld with the help of the monodromy of the flat Knizhnik-Zamolodchikov connection andis known nowadays as the Knizhnik-Zamolodchikov associator Φ KZ ( x, y ); this is a formal powerseries in two non-commutative formal variable x and y of the form,Φ KZ ( x, y ) = 1 + X m,k ,...,km − ∈ N ≥ km ∈ N ≥ ( − m ζ ( k , . . . , k m ) x k m − y · · · x k − y + regularized terms , where ζ ( k , . . . , k m ) is the multiple zeta value, the real number defined by the following convergingsum, ζ ( k , . . . , k m ) = X 1. There is an explicit iterativeconstruction of the regularized terms in the above formula in terms of multiple zeta values so that all coefficients of Φ KZ are rational linear combinations of that values. The second explicit associator Φ AT ( x, y )was constructed by A. Alekseev and C. Torossian in [AT1] using Fulton-MacPherson’s compactifiedconfiguration spaces of points in the complex plane and the integration theory of singular differentialforms on semialgebraic chains. It was proved to be a Drinfeld associator in [SW]. K. Furusho founda method in [Fu2] of computing coefficients of the formal power series Φ AT ( x, y ) in terms of rationallinear combinations of iterated integrals of M. Kontsevich weight differential forms associated to Liegraphs (see also a nice paper [BPP] on the systematic computation of weights of the M. Kontsevichformality map). . Grothendieck-Teichm¨uller group, graph complexes and T.Willwacher theorems7.1. Operads of graphs. In this section we consider only graphs Γ without hairs (see § )which are called from now on simply graphs . Recall that the set of vertices of Γ is denoted by V (Γ)and the set of edges by E (Γ). A graph Γ is called directed if each edge e = ( h, τ ( h )) ∈ E (Γ) comesequipped with a choice of a direction, that is, with a total ordering of its set { h, τ ( h ) } of half-edges.Here are a few examples of directed graphs • • / / , •• (cid:15) (cid:15) • • h h ◗◗◗◗◗ •• ♠♠♠♠♠ , • • " " •• b b , • • " " • • < < . et G n,l be the set of directed graphs Γ with n vertices and l edges such that some bijections V (Γ) → [ n ] and E (Γ) → [ l ] are fixed, i.e. every edge and every vertex of Γ is marked. Thepermutation group S l acts on G n,l by relabeling the edges so that, for each any integer d , it makessense to consider a collection of S n -modules, DG ra d = DG ra d ( n ) := Y l ≥ K h G n,l i ⊗ S l sgn ⊗| d − | l [ l ( d − n ≥ with the group S n acting on DG ra d ( n ) by relabeling the vertices. This is a Z -graded vector spaceobtained by assigning to each edge of a generating graph from G n,l the homological degree 1 − d .Note that if d is even, then each graph Γ ∈ DG ra d ( n ) is assumed to come equipped with a choice ofordering of edges up to an even permutation (an odd permutation acts as the multiplication by − • • " " • • < < ∈ DG ra d ∈ Z (2) vanishes identically as it admits an automorphismwhich changed the ordering of edges by an odd permutation, i.e. it is equal to minus itself.This S -module is an operad with respect to the following operadic composition, ◦ i : DG ra d ( n ) × DG ra d ( m ) −→ DG ra d ( m + n − , ∀ i ∈ [ n ](Γ , Γ ) −→ Γ ◦ i Γ , where Γ ◦ i Γ is defined by substituting the graph Γ into the i -labeled vertex of Γ and taking asum over all possible re-attachments of dangling edges (attached before to v i ) to the vertices of Γ .For example, • • " " •• b b ◦ •• (cid:15) (cid:15) = 12 3 •• (cid:15) (cid:15) • • " " •• b b + 12 3 •• (cid:15) (cid:15) • • " " •• b b + 12 3 •• (cid:15) (cid:15) • • ) ) ❘❘❘❘❘ •• u u ❧❧❧❧❧ + 12 3 •• (cid:15) (cid:15) • • i i ❘❘❘❘❘ •• ❧❧❧❧❧ Note that for d ∈ Z one has to choose of an ordering of edges in each graph summand of thecomposition Γ ◦ i Γ , and there is a canonical way to do by, roughly speaking, placing all the edgesof Γ in front of the edges of Γ .One can define an “undirected version” of the operad above by noticing that the group S l ⋉ ( S ) l acts on set G n,l of directed labelled graphs by relabeling the edges and reversing the directions ofthe edges. Hence for each fixed integer d ∈ Z , one can consider a collection of S n -modules, G ra d ( n ) := Y l ≥ K h G n,l i ⊗ S l ⋉ ( S ) l sgn | d | l ⊗ sgn ⊗ l | d − | [ l ( d − S n acts by relabeling the vertices. For d even elements G ra d ( n ) can be understoodas undirected graphs, e.g. • • ∈ G ra d ∈ Z (2) , ≡ , 12 3 ••• • ◗◗◗◗◗ •• ♠♠♠♠♠ ∈ G ra d ∈ Z (3)while for d odd one has identifications of the type • • " " •• b b = − • • " " • • < < in G ra d ∈ Z +1 (2) . Show that graphs in G ra d ∈ Z which have multiple edges vanish identically. The integer parameter d has sometimes a clear geometric meaning. With operadsof graphs DG ra d and G ra d for d ≥ R d , the most prominent of which is the Kontsevichformality theory [K2] for d = 2. Other examples of d = 2 field theories have been studied in,e.g., [Me2, Sh, W5]; for an example of a highly non-trivial d = 3 de Rham field theory we refer to[MW5]. It seems that to get non-trivial (in the sense that GRT plays a classifying role, see § d ≥ multi-oriented graphswhich are discussed below. .2. M. Kontsevich graph complexes and T. Willwacher theorems. As discussed in § above, for any operad O = {O ( n ) } n ≥ in the category of graded vector spaces the linear map[ , ] : O tot ⊗ O tot −→ O tot ( a ∈ P ( n ) , b ∈ P ( m )) −→ [ a, b ] := P ni =1 a ◦ i b − ( − | a || b | P mi =1 b ◦ i a ∈ P ( m + n − O tot := Q n ≥ P ( n ) into a Lie algebra [KM]; moreover, these bracketsinduce a Lie algebra structure on the subspace of invariants O tot S := Q n ≥ O ( n ) S n . For any d ∈ Z one can consider a degree shifted operad O{ d } (see § ) and conclude that for any operad O andany integer d the associated graded vector space O{ d } tot S ≃ Y n ≥ O ( n ) ⊗ S n sgn ⊗| d | n [ d (1 − n )]is canonically a Lie algebra . In particular, the graded vector spaces (the “directed full graphcomplex” and, respectively, the “full graph complex”) dfGC d := Y n ≥ DG ra d ( n ) ⊗ S n sgn ⊗| d | n [ d (1 − n )]and fGC d := Y n ≥ G ra d ( n ) ⊗ S n sgn ⊗| d | n [ d (1 − n )]are Lie algebras with Lie brackets given by the substitution of a graph into a vertex of anothergraph as explained above. Note that the homological degree of graph Γ from dfGC d or fGC d is givenby | Γ | = d ( V (Γ) − 1) + (1 − d ) E (Γ) . A generator Γ ∈ dfGC d can be understood as a directed graph (with no labeling of vertices oredges) equipped with an orientation or which is, by definition, a unital vector in the following1-dimensional Euclidian space R Γ := (cid:26) ∧ E (Γ) K [ E (Γ)] if d ∈ Z ∧ V (Γ) R [ V (Γ)] if d ∈ Z + 1Every directed graph has precisely two possible orientations, or and or opp , and one identifies(Γ , or ) = − (Γ , or opp ). We often abbreviate (Γ , or ) to Γ.A generator Γ ∈ fGC d can be understood as an undirected graph (with no labeling of vertices oredges) equipped with an orientation or which for d even is the same as defined just above, whilefor d odd it is defined as a unital vector in the 1-dimensional Euclidian vector space R Γ := ∧ V (Γ) R [ V (Γ)] ⊗ O e =( h,τ ( h )) ∈ E (Γ) ∧ R [ h, τ ( h )]where each edge is interpreted as the orbit consisting of two half-edges under the involution τ (see § ). Put another way, for d even an orientation or is a choice of ordering of edges (up to aneven permutation), while for d odd or is a choice of ordering of vertices (up to an even permutation)and a choice of direction on each edge (up to a flip and multiplication by − or and or opp , and one identifies (Γ , or ) = − (Γ , or opp ). L ie { − d } → O , where L ie is the operad of Lie algebras. This observation gives us a very useful approach to the Kontsevich graph complexes and theirribbon graphs generalizations (see [W1, MW2]) which we skip in this survey. .2.1. Exercise. Show that the graph γ := • • ∈ fGC d or γ := • • / / ∈ dfGC is a Maurer-Cartan element, i.e. it has degree 1 and satisfies [ γ , γ ] = 0Hence Lie algebras fGC d and, respectively, dfGC d come equipped with a compatible differential δ Γ := [Γ , γ ] = X v ∈ V (Γ) (cid:16) δ ′ v Γ − ( − | Γ | δ ′′ v Γ (cid:17) where δ ′ v splits the vertex v of Γ into two vertices δ ′ v • ✹✹✹✹✡✡✡✡ ✡✡✡✡ ✹✹✹✹ = X H ( v )= I ′⊔ I ′′ I ′ , I ′′≥ | {z } I ′ half-edges I ′′ half-edges z }| { • • ✹✹✹✹✡✡✡✡ ✡✡✡✡✹✹✹✹ and resp. δ ′ v • o o / / (cid:26) (cid:26) ✹✹✹✹ (cid:4) (cid:4) ✡✡✡✡ (cid:4) (cid:4) ✡✡✡✡ Z Z ✹✹✹✹ := X H ( v )= I ′⊔ I ′′ I ′ , I ′′≥ | {z } I ′ half-edges I ′′ half-edges z }| { • • / / o o / / (cid:26) (cid:26) ✹✹✹✹ (cid:4) (cid:4) ✡✡✡✡ (cid:4) (cid:4) ✡✡✡✡ Z Z ✹✹✹✹ connected by a (directed) edge and redistributes the attached half-edges to v (if any) along the twonew vertices in all possible ways, while δ ′′ v attaches to v a new univalent vertex δ ′ v • ✹✹✹✹✡✡✡✡ ✡✡✡✡ ✹✹✹✹ = 2 •• ❬❬❬❬❁❁❁❁✡✡✡✡ ✘✘✘✘ ✹✹✹✹ ✈✈✈✈ and resp. δ ′ v • o o / / (cid:26) (cid:26) ✹✹✹✹ (cid:4) (cid:4) ✡✡✡✡ (cid:4) (cid:4) ✡✡✡✡ Z Z ✹✹✹✹ := •• o o - - ❬❬❬❬ (cid:29) (cid:29) ❁❁❁❁ (cid:4) (cid:4) ✡✡✡✡ (cid:12) (cid:12) ✘✘✘✘ (cid:26) (cid:26) ✹✹✹✹ : : ✈✈✈✈ + •• o o - - ❬❬❬❬ (cid:29) (cid:29) ❁❁❁❁ (cid:4) (cid:4) ✡✡✡✡ (cid:12) (cid:12) ✘✘✘✘ (cid:26) (cid:26) ✹✹✹✹ z z ✈✈✈✈ Note that if the vertex v has at least one half-edge, then the two terms with univalent vertices inthe sum δ ′ v Γ cancel out with the two terms in δ ′′ v Γ so that in most cases one can ignore in the aboveformula for δ the second term δ ′′ v and assume in the first term δ ′ v that the summation runs overdecompositions H ( v ) = I ′ ⊔ I ′′ with I ′ ≥ I ′′ ≥ 1. For example, the graph γ is a cycle, δγ = [ γ , γ ] = 0, but the associated cohomology class is trivial, γ = − δ ( • ).The dg Lie algebra ( fGC d , δ ) was introduced by M.Kontsevich in [K1] in his attempt to calculateobstructions to universal quantizations of Poisson structures.Both dg Lie algebras contain dg Lie subalgebras, fcGC d and dfcGC d , generated by connected graphs.There is an isomorphism of complexes, fGC d = ⊙ •≥ ( fcGC d [ − d ]) [ d ] , dfGC d = ⊙ •≥ ( dfcGC d [ − d ]) [ d ]so that at the cohomology level H • ( fGC d ) = ⊙ •≥ ( H • ( fcGC d )[ − d ]) [ d ] , H • ( dfGC d ) = ⊙ •≥ ( H • ( dfcGC d )[ − d ]) [ d ]Hence there is no loss of important information when working with connected graphs only.Moreover, there is a monomorphism of graph complexes(41) fcGC d −→ dfcGC d which sends an undirected graph Γ into a sum of graphs obtained by interpreting each edge asthe sum of edges in both directions. It was proven in [W1](Appendix K) that this map is a quasi-isomorphism . Hence there is no need to study the complex dfGC d by itself so that we continue inthis subsection discussing only the complex fcGC d . That complex splits as a direct sum fcGC d = fcGC ≤ d ⊕ fcGC d ⊕ GC (cid:8) d where fcGC ≤ is spanned by connected graphs having at least one vertex of valency ≤ fcGC d isspanned by connected graphs with no vertices of valency ≤ dfcGC d contains two very important subcomplexes of oriented graphs and of sourced graphs whichare of great significance in applications (see below). , and GC (cid:8) d is generated by graphs with each vertex of valency ≥ 3. It was proven in [W1] that thecomplex fcGC ≤ d is acyclic while H • ( fcGC d ) = M j ≥ j ≡ d +1 mod 4 K [ d − j ] , where the summand K [ d − j ] is generated by the polytope with j vertices, that is, a connected graphwith j bivalent vertices, e.g. • •• ☞☞☞ ✷✷✷ for j = 3 (in fact this particular graph vanishes identically for d even as in this case it admits an automorphism reversing its orientation).The complex GC (cid:8) d contains a subcomplex GC d spanned by graphs with no loops , that is, edgesattached to one and the same vertex, and the inclusion is a quasi-isomorphism [W1], H • ( GC d ) = H • ( GC (cid:8) d ) . One of the major results in [W1] is the following There is an isomorphism of Lie algebras, H ( GC ) = grt , where grt is the Lie algebra of the Grothendieck-Teichm¨uller group GRT (see § H • < ( GC ) = 0 . There is an explicit construction of infinitely many cohomology classes { [ w n +1 ] } n ≥ in H ( GC ),more precisely, of their cycle representatives { w n +1 } n ≥ in GC . The first two classes can be givenexplicitly as follows w = ••• • rrrr☞☞☞☞☞☞☞☞ ✷✷✷✷✷✷✷✷▲▲▲▲ w = ••• •• • ❦❦❦❦ ✻✻✻✻✟✟✟✟❙❙❙❙ ❑❑❑❑❑ ✕✕✕✕✕✮✮✮✮✮ sssss + 52 ••• •• • ★★★★ ♠♠♠♠☎☎☎☎✔✔✔✔✔✔✔✔ ❋❋❋❋❋❋❋❋ ❑❑❑❑❑ ✕✕✕✕✕✮✮✮✮✮ sssss For higher n there is an explicit transcendental formula for the cycles w n +1 given in [RW] whichpresents each cycle as a linear combination of graphs with 2 n + 2 vertices and 4 n + 2 edges(42) w n +1 = X Γ ∈ G n +2 , n +2 c Γ Γ = λ n +1 •• • • ... ••• • ❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧❦❦❦❦❦❦ ❙❙❙❙❙❙ ✰✰✰✰✰✰✓✓✓✓✓✓❦❦❦❦❦❦❙❙❙❙❙❙✰✰✰✰✰✰✓✓✓✓✓✓ + . . . where the weights c Γ are given by explicit converging integrals over the configuration space of 2 n different points in C \{ , } (see (23) in [RW]). In particular, the coefficients λ n +1 of the wheel-typesummands in w n +1 are equal to zeta values ζ (2 n +1)(2 πi ) (2 n +1) up to a non-zero rational factor, cf. [G]. (Infact, from the presence in w n +1 of such wheel-graphs one can conclude almost immediately thatthese cycles are not boundaries.) The pro-nilpotent Lie algebra grt is isomorphicas a Z ≥ graded Lie algebra to the degree completion of the free Lie algebra generated by formalvariables of degrees n + 1 , n ≥ . It has been proved by F. Brown in [B] that grt does contain indeed the above mentioned free Liealgebra.Thus the graph cohomology classes [ w n +1 ] might generate the whole grt . enote by GC ≥ d the dg Lie subalgebra of fcGC d spanned by graphs with valency of each vertex ≥ H • ( fcGC d ) = H • ( GC ≥ d ) = H • ( GC d ) + M j ≥ j ≡ d +1 mod 4 K [ d − j ] . There are other differentials on the Kontsevich graph complexes GC d which have been studied in[MW1, KWZ1]. A graphs Γ from the dg Lie algebra dfGC d is called oriented if it contains no wheels , that is, directed paths of edges forming a closed circle. The subspace fGC ord ⊂ dfGC d spanned by oriented graphs is a dg Lie subalgebra. For example, • •• / / E E ☞☞☞☞☞☞ Y Y ✷✷✷✷✷✷ ∈ fGC ord while • •• / / (cid:5) (cid:5) ☞☞☞☞☞☞ Y Y ✷✷✷✷✷✷ fGC ord . Let dfcGC ord the subcomplex of fGC ord spanned by connected graphs, and GC or d ⊂ dfcGC ord thesubcomplex spanned by graphs which each vertex at least bivalent and with no bivalent vertices ofthe form • / / / / . Then [W3] H • ( fcGC ord ) = H • ( GC or d )so that there no loss of important information when working with GC or d solely. [W3] . H • ( GC or d +1 ) = H • ( GC ≥ d ) . Hence one has a remarkable isomorphism of Lie algebras, H ( GC or3 ) = grt . Moreover it followsfrom this theorem that H i ( GC or3 ) = 0 for i ≤ − H − ( GC or3 ) is a 1-dimensional space generatedby the graph •• B B •• \ \ . This theorem tells us that in order to have some non-trivial de Rham fieldtheory in dimension 3 (rather than two), one has to work with oriented graphs rather than withsimply directed (or even undirected) graphs. We refer to [MW5] for an example of such a de Rhamfield theory providing us with an explicit universal formula for deformation quantization of Liebialgebras.The original argument in [W2] does not give us an explicit relation between the cohomology groupsin the above Theorem. Let ^ GC or d +1 and ^ GC ≥ d be the graph complexes dual to GC or d +1 and GC ≥ d respectively. Then one also has H • ( ^ GC or d +1 ) = H • ( ^ GC ≥ d ). An explicit construction sending ahomology class in H • ( ^ GC ≥ d ) to a cohomology class in H • ( ^ GC or d +1 ) has been found in [Z1]. It canbe used to find an explicit 3-dimensional oriented incarnation w or ∈ H ( GC or3 ) of, for example,the tetrahedron cohomology class w ∈ H ( GC ) discussed above; w or is a linear combination oforiented graphs with 7 vertices and 9 edges.In fact, the complex GC or3 controls the deformation theory of the properad of Lie bialgebras so that GRT is essentially a symmetry group of that properad [MW4]. In string topology and the theoryof moduli spaces of algebraic curves it is important to work with involutive Lie bialgebras, andtheir deformation theory is controlled by the following deformation of the complex GC or3 . GC or . Consider a Lie algebra ( GC or3 [[ ~ ]] , [ , ]), where GC or3 [[ ~ ]] is the(completed) topological vector space spanned by formal power series in a formal parameter ~ ofhomological degree 2, and [ , ] are the Lie brackets obtained from the standard ones in GC or d by the K [[ ~ ]]-linearity. It was shown in [CMW] that the formal power series γ ~ := ∞ X k =1 ~ k − ... •• •• e e •• B B •• \ \ | {z } k edges s a Maurer-Cartan element in the Lie algebra ( fGC or [[ ~ ]] , [ , ]) and hence makes the latterinto a differential Lie algebra with the differential δ ~ Γ := [Γ , γ ~ ] . It was proven in [CMW] that H ( GC or [[ ~ ]] , δ ~ ) ≃ H ( GC or , δ ) ≃ grt as Lie algebras. Moreover, H i ( GC or [[ ~ ]] , δ ~ ) = 0 for all i ≤ − H − ( GC or [[ ~ ]] , δ ~ ) is a 1-dimensional vector space class generated by the formal powerseries P ∞ k =2 ( k − ~ k − ... •• •• e e •• B B •• \ \ | {z } k edges . A graphs Γ from the dg Lie algebra of connected oriented graphs dfcGC d is called sourced if it contains at least one vertex v of valency ≥ v = ≥ z }| { ... • c c ●●●●●● Z Z ✹✹✹✹ D D ✡✡✡✡ ; ; ✇✇✇✇✇✇ Such vertex is called a source . The subspace sfGC d ⊂ dfcGC d spanned by sourced graphs is a dg Liesubalgebra. Consider a smaller subspace sGC d ⊂ sfGC d spanned by sourced graphs with all verticesat least bivalent and with no bivalent vertices of the form • / / / / . It was proven by in [Z1] that H • ( sfGC d +1 ) = H • ( sGC d +1 ) = H • ( GC ≥ d )so that one gets one more incarnation of the Grothendieck-Teichm¨uller Lie algebra in dimension 3 grt = H ( sGC )in terms of sourced graphs. Similarly one can work with directed graphs with at least one target vertex of the form | {z } ≥ ... • ; ; ✇✇✇✇✇✇ D D ✡✡✡✡ Z Z ✹✹✹✹ c c ●●●●●● . The Grothendieck-Teichm¨uller Lie algebra appears nat-urally as a single entity in dimension d = 2 as the cohomology group H ( GC ) of the “ordinary”graph complex. In dimension 3 it reappears as the cohomology group H ( GC or3 ) = H ( sGC ) of the oriented or sourced graph complex, i.e. we have a sequence of isomorphism of Lie algebras, grt = H ( GC ) = H ( GC or3 ) = H ( sGC ) . What kind of graph complex GC ? d gives us an incarnation of grt in dimension d = 4 , , . . . , grt = H ( GC ) = H ( GC or3 ) = H ( sGC ) = H ( GC ?4 ) = H ( GC ?5 ) = . . . ?The answer to that question was found by Marko ˇZivkovi´c in [Z1, Z2] who introduced and stud-ied multi-oriented sourced graph complexes. Their algebro-geometric interpretation was found in[Me4] — they control quantizations and deformation theory of strong homotopy (even or odd) Liebialgebra structures, in particular of Poisson structures, on infinite-dimensional spaces spaces withbranes .Let us go back for a moment to the Lie algebra of directed connected graphs dfcGC d and define,for any integer k ∈ N ≥ , its extension d k fcGC d as a Z -graded vector space spanned by directedgraphs Γ with each edge decorated with k − , , . . . , k , the value 1being reserved for the original direction , and each new direction can take only two “values” — itcan agree or disagree with the original direction, ... k colored ones, say the original direction 1 has black colour, the newdirection 2 is red so on, because we do not assume that the set [ k ] of all directions on each edge is ordered . Hence we call oftenthe direction on an edge labeled by integer l ∈ [ k ] the l - coloured direction. This new decoration of edges has no impact on thedegree of graphs which is given by the standard formula | Γ | = d ( V (Γ) − 1) + (1 − d ) E (Γ), ∀ Γ ∈ d k fcGC d . ote that the Lie bracket in dfcGC d does not change the number and the “internal structure” ofedges in graphs so that exactly the same formula as in dfcGC d gives us a Lie bracket in d k fcGC d forany k ≥ 1. It is not hard to see that the element γ k := ... k + ... k + ... k + . . . (2 k − terms)where the summation runs over all possible way to decorate the directed edge with new directions2 , , . . . k , is a Maurer-Cartan element and hence makes d k fcGC d into a dg Lie algebra with thedifferential δ Γ := [Γ , γ k ]. Of course, the case k = 1 gives nothing new, d fcGC d ≡ dfcGC d . In fact,the general case k ∈ Z ≥ also does not give us immediately anything really new as well: the naturalmonomorphisms which add to each edge of a graph one extra direction in all possible ways (cf.(41)), fcGC d −→ d fcGC d −→ d fcGC d −→ d fcGC d −→ . . . are quasi-isomorphisms. However we can consider now lots of interesting subcomplexes in themultidirected graph complex ( d k fcGC d , δ ). Given any two non-negative integers p and q with p + q ≤ k , and any two disjoint subsets I p and I q of the set [ k ] of cardinalities p and q respectively(the particular choice of such subsets plays no role), one can define a subcomplex s p o q d k fcGC d ⊂ d k fcGC d of k -directed p -sourced and q -oriented graphs as the span of those multioriented graphs Γ whichsatisfy the conditions(i) Γ is sourced with respect to every direction in the subset I p ⊆ [ k ], i.e. for each direction c ∈ I p there is at least one vertex v ∈ V (Γ) which has valency ≥ c ;(ii) Γ is oriented with respect to every direction in the subset I q ⊂ [ k ], i.e. for each direction c ∈ I p the graph Γ has no closed directed paths of edges ( wheels ) with respect to c .By analogy to the previous examples, there is no loss of generality when working with a subcomplex s p o q d k GC d ⊂ s p o q d k fcGC d generated by graphs whose vertices are at least bivalent as this inclusion is a quasi-isomorphism.The major result of [Z2] is the following For any integer k ≥ and any non-negative integers p and q with p + q ≤ k there is an isomorphism of cohomology groups H • ( GC ≥ d ) = H • ( s p o q d k GC d + p + q ) . Therefore extra directions which are neither oriented nor sourced can be omitted as irrelevant, i.e.one can set k = p + q without loss of important information and work only with dg Lie algebras s p o q GC d + p + q := s p o q d p + q GC d + p + q . M. ˇZivkovi´c Theorem gives us infinitely many graph incarnations of the Grothendieck-Teichm¨ullerLie algebra, grt = H ( GC ) = H ( s p o q GC p + q ) ∀ p, q ∈ Z ≥ , and hence a clue (cf. [Me4]) to what could be a non-trivial deformation quantization theory indimension d ≥ d = 2 and d = 3 are well understood by now, see below). .5.2. Remark on hairy graphs. There is an important version of the graph complexes GC d which is based on graphs with hairs [KWZ2, W6, W4]. These graph complexes control the rationalhomotopy groups of of long embeddings (modulo immersion) of R m in R n . Very recently, theirmarked version has been used in the study of cohomology groups of moduli spaces M g,n of genus g algebraic curves with n punctures [CGP2]. . Some applications of the theory of Drinfeld’s associators, GRT and graph complexes8.1. Universal quantizations of Lie bialgebras. A Lie bialgebra is a graded vector space V equipped with two linear maps,[ , ] : ∧ V → V , △ : V → V ∧ V, such that the first map [ , ] makes V into a Lie algebra, the second map △ makes V into a Liecoalgebra, and the compatibility condition △ [ a, b ] = X a ⊗ [ a , b ] + [ a, b ] ⊗ b − ( − | a || b | ([ b, a ] ⊗ a + b ⊗ [ b , a ]) , holds for any a, b ∈ V with △ a =: P a ⊗ a , △ b =: P b ⊗ b . This algebraic structure wasintroduced by V. Drinfeld [Dr1] in the context of studying universal deformations of the standardHopf algebra structure on universal enveloping algebras. More precisely, consider the symmetrictensor algebra ⊙ • V equipped with the standard graded commutative and graded cocommutativebialgebra structure ( · , ∆ ), where · is the canonical multiplication on ⊙ • V and ∆ is the coproducton ⊙ • V uniquely determined by the condition that the elements of V ⊂ ⊙ • V are primitive. Assumethe topological vector space ⊙ • ( V )[[ ~ ]], ~ being a formal parameter of degree zero, has a continuousbialgebra structure ( ⋆ ~ , ∆ ~ ) of the form, ⋆ ~ : ⊙ • V N ⊙ • V −→ ⊙ • V [[ ~ ]]( f ( x ) , g ( x )) −→ f ∗ ~ g = f · g + P ∞ k ≥ ~ k P k ( f, g )∆ ~ : ⊙ • V −→ ⊙ • V N ⊙ • V [[ ~ ]] f ( x ) −→ ∆ ~ f = ∆ f + P ∞ k ≥ ~ k Q k ( f )where all operators P k are bidifferential, and Q k are co-bidifferential (that is, dual to bi-differentialoperators on polynomial functions, see, e.g., § O ( ~ ) imply that the firstorder deformations P and Q are uniquely determined by some Lie bialgebra structure ( △ , [ , ])on V . In this case the data ( ⋆ ~ , ∆ ~ ) on ⊙ • ( V )[[ ~ ]] is called a deformation quantization of that Liebialgebra structure ( △ , [ , ]) on V . Thus Lie bialgebra structures control infinitesimal deformationsof the standard Hopf algebra structure on ⊙ • V . The Drinfeld quantization problem : given any Lie bialgebra structure on a vector space V , doesthere always exist its deformation quantization ( ⋆ ~ , ∆ ~ )? Put another way, given infinitesimaldeformations P and Q of the standard bialgebra on ⊙ • V , can we always find suitable operators P , P , . . . , Q , Q , . . . which make ⊙ • ( V )[[ ~ ]] into a (not necessarily commutative cocommutative)bialgebra?Surprisingly enough, the problem can not be solve by a trivial inductive procedure . It was provenby P. Etingof and D. Kazhdan in [EK] that, given a choice of a Drinfeld associator, there does exista corresponding universal (in the sense, for any vector space V and any Lie bialgebra structure on V ) solution to the Drinfeld quantization problem. It was proven in [Me3] that any solution of the H ( GC ). A “folklore” conjecture says that H ( GC ) =0, but one hast to choose an associator to make every degree 1 cycle in GC into a coboundary; thus an iterative procedure canexist but it can not be trivial. rinfeld quantization problem extends to a quasi-isomorphism — a formality map — of completed L ie ∞ algebras,(43) F V : [ ⊙ •≥ ( V [ − ⊕ V ∗ [ − −→ Y m,n ≥ Hom m O ( ⊙ • V ) , n O ( ⊙ • V ) ! [2 − m − n ]where the l.h.s. is equipped with the Poisson type Lie bracket induced by the standard paring V ⊗ V ∗ → K , and the r.h.s. is the Gerstenhaber-Schack complex of the graded commutativecocommutative Hopf algebra ⊙ • V equipped with the L ie ∞ structure uniquely determined by achoice of a minimal resolution A ss B ∞ of the properad A ss B controlling (associative) bialgebras. Inthe other direction: any quasi-isomorphism as above gives a solution of the the Drinfeld quantizationproblem.Since we are interested in universal solutions, it makes sense to search for a reformulation of thisproblem which does not refer to a particular vector space at all. The theory of operads (see § )is very effective in the study of the homotopy theory of algebraic operations on vector spaceswith many inputs but only one output. It can not be applied to such algebraic structures as Liebialgebras, but there is a very nice extension of that theory called the theory of props, properads,and their wheeled versions . We refer to [V] for an excellent introduction into the theory of props andproperads. Roughly speaking, the theory prop(erad)s is based on (connected) decorated orientedgraphs with legs. The deformation theory of morphisms of properads and props was developedin [MV]. There is a properad L ieb (see, e.g., [MW3, MW5, V] and references cite therein) whoserepresentations in a vector space V are precisely Lie bialgebra structures on V and the homotopytheory of such structures is controlled by the minimal resolution H olieb of L ieb . There is also aproperad A ss B whose representations in a vector space W are precisely bialgebra structures on W and the homotopy theory of such structures is controlled by a minimal resolution A ss B ∞ of A ss B (which exists but is not unique [Ma]). V. Drinfeld deformation quantization problem (in theextended formality version as explained above) can be reformulated as existence of a morphism ofproperads [MW3](44) F : A ss B ∞ −→ D \ H olieb satisfying certain non-triviality conditions. Here \ H olieb is the genus completion of H olieb and D : category of dg props −→ category of dg propsa polydifferential (exact) endofunctor which creates out of any prop P another prop DP withthe property that there is a one-to-one correspondence between representations of P in a vectorspace V and representations of DP in ⊙ • V given in terms of polydifferential operators. TheEtingof-Kazhdan theorem can be used to show that for any Drinfeld associator there exists amorphism F of dg props as above. To classify all solutions to the Drinfeld problem, i.e. to classifyall possible homotopy non-trivial formality maps as in (44), one has to compute the cohomology ofthe deformation complex of any map F as above, Def (cid:16) A ss B ∞ F → D \ H olieb (cid:17) This was done in [MW3] where it was proven H • ( fGC ) ⊕ K = H • +1 (cid:16) Def (cid:16) A ss B ∞ F → D \ H olieb (cid:17)(cid:17) This result implies in particular that H (cid:16) Def (cid:16) A ss B ∞ F → D \ H olieb (cid:17)(cid:17) = grt ⊕ K , .e. the Grothendieck-Teichm¨uller group GRT = GRT ⋊ K ∗ acts faithfully and transitively on theset of solutions of the Drinfeld quantization problem which in turn implies that this set can beidentified with the set of Drinfeld associators.An explicit transcendental formula for a universal quantization of Lie bialgebras has been con-structed in ([MW5]). Let C ∞ ( R n ) be the commutative algebraof smooth functions in R n . A star product on C ∞ ( R n ) is a continuous associative product on C ∞ ( R n )[[ ~ ]], ~ being a formal parameter, ∗ ~ : C ∞ ( R n ) × C ∞ ( R n ) −→ C ∞ ( R n )[[ ~ ]]( f ( x ) , g ( x )) −→ f ∗ ~ g = f g + P ∞ k ≥ ~ k P k ( f, g )where all operators P k are bidifferential. It is not hard to check that the associativity conditionon ∗ ~ implies that π ( f, g ) := B ( f, g ) − B ( g, f ) is a Poisson structure in R n ; then ∗ ~ is called a deformation quantization of π ∈ T poly ( R n ).The deformation quantization problem addresses the question: given a Poisson structure π on R n , does there exist a star product ∗ ~ on C ∞ ( R n ) which is a deformation quantization of π ? Aspectacular solution of this problem was given by Maxim Kontsevich [K2] in the form of a explicitmap between the two sets (cid:26) Poissonstructures in R n (cid:27) depends onassociators / / (cid:26) Star products ∗ ~ in C ∞ ( R n )[[ ~ ]] (cid:27) given by transcendental formulae. In fact a stronger statement was proven — the formality theorem which says that for any n there is an explicit L ie ∞ quasi-isomorphism of dg Lie algebras, F K : T poly ( R n ) −→ C • ( C ∞ ( R n ) , C ∞ ( R n ))where T poly ( R n ) is the Lie algebra of polyvector fields on R n equipped with the Schouten-Nijenhuisbracket, and C • ( C ∞ ( R n ) , C ∞ ( R n )) is the Hochschild complex of the algebra C ∞ ( R n ), that is, the dgLie algebra controlling deformations of C ∞ ( R n ) as an associative (but not necessarily commutative) R -algebra. Moreover, the formality theorem holds true for any manifold M , not necessarily for R n [K2]. Later D. Tamarkin has proven the existence theorem for deformation quantizations whichexhibited a key role of Drinfeld’s associators [T2], and V. Dolgushev [Do] has proven that the set ofhomotopy classes of universal formality maps can be identified with the set of Drinfeld’s associatorsso that GRT acts faithfully and transitively of homotopy classes of formality maps. GRT on Lie bialgebras. One can consider a degree shifted version of thenotion of Lie bialgebra. For any integers c, d ∈ Z one defines [MW4] a Lie ( c, d ) -bialgebra as agraded vector space equipped with linear maps[ , ] : ⊙ ( V [ d ]) → V [ d + 1] , △ : V [ c ] → ⊙ ( V [ c ])[1 − c ] , making V into a degree shifted Lie algebra and degree shifted Lie coalgebra satisfying an analogueof the Drinfeld compatibility condition. There exist a prop(erad) L ieb c,d which governs thesestructures and its minimal resolution H olieb c,d which governs their homotopy theory [MW4]. Thecase c = 1, d = 1 gives us ordinary Lie bialgebras discussed above, while the case c = 1, d = 0 isimportant in the theory of Poisson structures on manifolds as there is a one-to-one correspondence[Me1] between representation of the pro(erad) H olieb , on a graded vector space V and formalgraded Poisson structures on V viewed as a formal manifold which vanish at the distinguished point0 ∈ V .The deformation theory [MV] of the identity map Id : \ H olieb c,d → \ H olieb c,d gives us a clue to thesymmetry group of the genus completion \ H olieb c,d of the dg prop H olieb c,d (and hence of \ L ieb c,d ). his problem was settled in [MW5] where it was proven that there is a quasi-isomorphism (up toone rescaling class) of complexes GC or c + d +1 −→ Def (cid:16) \ H olieb c,d Id → \ H olieb c,d (cid:17) [1]implying H • ( GC ≥ c + d ) ⊕ K = H • ( GC or c + d +1 ) ⊕ K = H • +1 (cid:16) \ H olieb c,d Id → \ H olieb c,d (cid:17) so that grt ⊕ K = H ( \ H olieb Id → \ H olieb )Hence the Grothendieck-Teichm¨uller group GRT = GRT ⋊ K ∗ is essentially the automorphismgroup of the completed prop(erad) d L ieb .There is a further generalization of the notion of Lie ( c, d )-bialgebra to the multioriented case, and GRT acts on such structures for any c, d ∈ N with c + d ≥ Let c ass and b lie be the completedfree associative and, respectively, Lie algebra generated by formal variables x and y (see § ).Consider the quotient space d Cyc := c ass / h AB − BA | ∀ A, B ∈ c ass i ≡ c ass / [ c ass , c ass ]the (completed) vector space spanned by cyclic words in two letters. There is a canonical projection tr : c ass → d Cyc . Note that every element A ∈ c ass has a unique decomposition, A = A + ∂ x ( A ) x + ∂ y ( A ) y for some A ∈ K , ∂ x ( A ) , ∂ y ( A ) ∈ c ass .Recall the Bernoulli power series series x e x = 1 + x X n ≥ b n n ! x n =: 1 + x b ( x )and the Baker-Cambell-Hausdorf power series bch ( x, y ) defined in (2). A triple ( A, B, g ) consistingof two Lie series A, B ∈ b lie and a formal power series g ( x ) = P n ≥ g n x n is called a solution of the(generalized) Kashiwara-Vergne problem if they satisfy the equations x + y − bch ( x, y ) = (1 − e − ad x ) A + ( e ad y − B in b lie and tr ( ∂ x ( A ) x + ∂ y ( B ) y ) = 12 tr ( g ( x ) − g ( bch ( x, y )) + g ( y )) . Given a Lie group G with the Lie algebra g , if a solution of the KV problem exists, then the G -invariant harmonic analysis on g is related with G -invariant harmonic analysis on the group G itself by the standard exponential map. Moreover, if a solution exists, then g even := P n ≥ g n x n must be equal to the Bernoulli series b ( x ).Existence of solution of the KV problem was established in [AM]. An alternative solution of the KV problem which classified all such solutions in terms of so called KV associators was given in[AT2]. Any Drinfeld associator is a KV associator, and there is a conjecture which says that thisassociation is one-to-one. .4. Formality theorem in the Goldman-Turaev theory. Let Σ be a Riemann surface ofgenus g with N + 1 boundary components, N ∈ N . The group algebra K h π (Σ) i of the fundamentalgroup π (Σ) is canonically filtered by powers of the augmentation ideal (see § ) and hence admits acanonical completion \ π (Σ). The associated (completed) vector space spanned by conjugacy classesin π (Σ), d g [Σ] := \K h π (Σ) i [ \K h π (Σ) , \K h π (Σ) i ]that is, by free homotopy classes of loops in Σ, has a canonical Goldman-Turaev Lie bialgebrastructure . It is filtered, and the associated graded vector spacegr d g [Σ] := c ⊗ • H (Σ)[ c ⊗ • H (Σ) , c ⊗ • H (Σ)] , where H (Σ) is the first homology group of Σ over K , has an induced Lie bialgebra structure whichadmits a rather simple combinatorial description. The formality theorem [AKKN1, AKKN2] (seealso references cited therein) says that for any solution of the KV problem, in particular, for anyDrinfeld associator there is an associated isomorphismΘ : d g [Σ] −→ gr d g [Σ]of Lie bialgebras. In the case g = 0 and K = C a nice explicit formula for such an isomorphismwas constructed in [AN] with the help of the Knizhnik-Zamolodchikov connection. In particular,the group GRT acts as automorphisms on gr d g [Σ] for any Riemann surface Σ. grt . Let M g be the modulispace of Riemann surfaces of genus g . 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(2015), no. 3, 1649-1666.[W4] T. Willwacher, Deformation quantization and the Gerstenhaber structure on the homology of knot spaces , preprintarXiv:1506.07078 (2015)[W5] T. Willwacher, The Homotopy Braces Formality Morphism , Duke Math. J. (2016), no. 10, 1815-1964.[W6] T. Willwacher, Pre-Lie pairs and triviality of the Lie bracket on the twisted hairy graph complexes , preprintarXiv:1702.04504 (2017)[Z1] M. ˇZivkovi´c, Multi-oriented graph complexes and quasi-isomorphisms between them I: oriented graphs , preprintarXiv:1703.09605 (2017)[Z2] M. ˇZivkovi´c, Multi-oriented graph complexes and quasi-isomorphisms between them II: sourced graphs , preprintarXiv:1712.01203 (2017) Sergei Merkulov: Mathematics Research Unit, University of Luxembourg, Grand Duchy of Luxembourg E-mail address : [email protected]@uni.lu