Associahedra, cellular W-construction and products of $A_\infty$-algebras

Our aim is to construct a functorial tensor product of $A_\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of $A_\infty$-algebras.


Introduction.
In this paper we study tensor products of A ∞ -algebras.More precisely, given two A ∞algebras A = (V, ∂ V , µ V 2 , µ V 3 , . ..) and B = (W, ∂ W , µ W 2 , µ W 3 , . ..), we will be looking for a functorial definition of an A ∞ -structure A ⊙ B that would extend the standard (non-associative) dg-algebra structure on the tensor product A ⊗ B. This means that the A ∞ -algebra A ⊙ B will be of the form (V ⊗ W, ∂, µ 2 , µ 3 , . ..),where ∂ is the usual differential on the tensor product, and the bilinear product µ 2 is given by another standard formula where v, v ′ , v ′′ ∈ V and w, w ′ , w ′′ ∈ W .
A "coordinate-free" formulation of the problem is the following.Let A be the non-Σ operad describing A ∞ -algebras (see [8, page 45]), that is, the minimal model of the non-Σ operad Ass for associative algebras.The above product is equivalent to a morphism of dg-operads (a diagonal ) ∆ : A → A ⊗ A such that ∆ induces the usual diagonal ∆ Ass on the non-Σ operad Ass = H * (A).
The existence of such a diagonal is not surprising and follows from properties of minimal models for operads, see [8,Proposition 3.136].On the other hand, there is no way to control the co-associativity of diagonals constructed using this general argument and we will see below, in Theorem 13, that there, surprisingly enough, does not exist a co-associative diagonal.
For practical purposes, such as applications in open string theory [2], one needs a tensor product (and therefore also a diagonal) given by an explicit formula.Such an explicit diagonal was constructed by Umble and Saneblidze in [9].Our work was in fact motivated by our unsuccessful attempts to understand their paper.We will denote this diagonal by ∆ su and call it the SU-diagonal .In this article we recall the definition of this diagonal and give a conceptual explanation why it is well-defined.The operad A can be identified with the operad of the cellular chain complexes of the non-Σ operad of associahedra, A ∼ = C * (K) (see [8, page 45]), therefore, the required diagonal is given by a family of chain maps commuting with the induced operad structures and such that H * (∆ Kn ) = ∆ Ass .
The cells of the associahedra are not conducive to the definition of a diagonal.There is, however, a cubical decomposition of the associahedra provided by the W-construction of Boardman and Vogt [1], which is a homotopically equivalent non-Σ operad W = {W n } n≥1 , for which there is a canonical diagonal induced by the cubical structure (see ( 14)).A suitable diagonal on the associahedra can be then obtained by transfering ∆ Wn from W to K.More precisely, let be arbitrary operadic maps such that H * (p n ) and H(q n ) are identity endomorphisms of Ass(n), via the canonical identifications Then the formula ∆ Kn := (p n ⊗ p n ) • ∆ Wn • q n (4) clearly defines a diagonal.In fact, it can be proved that the operadic maps p = {p n } n≥1 and q = {q n } n≥1 with the above properties are homotopy inverses, but we will not need this statement.
It remains to find maps in (3).While there is an obvious and simple definition of q n , finding a suitable formula for p n is much less obvious.We give an explicit and very natural definition inspired by a formula in [9].
We will see that the operad of cellular chains C * (W ) can be described in terms of metric trees.Similar cellular W -constructions on a given dg-operad were considered by Kontsevich and Soibelman in [4].In this terminology, the chain maps p and q are explicit homotopy equivalences, defined over the integers, between the chain W -construction on the operad Ass and the minimal model A of Ass, which give rise to explicit equivalences of the categories of algebras over these dg-operads.

Categorial properties of diagonals and tensor products.
Recall [5] that there are two notions of morphisms of A ∞ -algebras.A strict morphism of A ∞ -algebras (X, ∂, µ 2 , µ 3 , . ..) and (Y, ∂, ν 2 , ν 3 , . ..) is a linear map f : X → Y that commutes with all structure operations.A weaker notion is that of a strongly homotopy (sh) morphism, given by a sequence of maps f n : X ⊗n → Y , n ≥ 1, satisfying rather complicated set of axioms (see, for example, [5,7]).Such a map is invertible if and only if f 1 : X → Y is an isomorphism.We will denote by strA ∞ the category of A ∞ -algebras and their strict morphisms, and shA ∞ the category of A ∞ -algebras and their sh morphisms.
As proved in [8, Proposition 3.136], any two diagonals ∆ ′ , ∆ ′′ : A → A ⊗ A are homotopic as maps of operads.Let ⊙ ′ (resp.⊙ ′′ ) denotes the tensor product induced by ⊙ ′ (resp.⊙ ′′ ).Although A ⊙ ′ B and A ⊙ ′′ B are, in general, not strictly isomorphic, the homotopy between ∆ ′ and ∆ ′′ can be shown to induce a strongly homotopy isomorphism between A ⊙ ′ B and A ⊙ ′′ B. Therefore we obtain the following uniqueness: Proposition 1.For any two A ∞ -algebras A, B, the A ∞ -algebras A ⊙ ′ B and A ⊙ ′′ B are isomorphic in shA ∞ .
We will prove in Theorem 13 that there are no co-associative diagonals.This means that in general in the 'strict' category strA ∞ .On the other hand, as argued in [8,Proposition 3.136], each diagonal ∆ is homotopy associative in the sense that the maps (∆ ⊗ 1 1)∆ and (1 1 ⊗ ∆)∆ are homotopic maps of operads, from which we infer: By the same argument, one can also prove Proposition 3.For any two A ∞ -algebras A and B, This naturally rises the question whether shA ∞ with a product ⊙ based on an appropriate diagonal is a (possibly symmetric) monoidal category.Even to formulate this question precisely, one more step should be completed.
While it is clear that ⊙ is a functor strA ∞ × strA ∞ → strA ∞ , to make it a functor shA ∞ ×shA ∞ → shA ∞ , one should define, for two sh morphisms f : A ′ → A ′′ and g : The above objects exist by general nonsense, but it is not clear whether they fulfill the axioms of a (symmetric) monoidal category (the pentagon and the hexagons), although it is quite possible that for some special choices of the above data these axioms are satisfied.On a more abstract level, the 'full' functorial monoidal product A, B → A ⊙ B, f, g, → f ⊙ g in shA ∞ means to construct a 'diagonal' in the minimal model of the two-colored operad Ass •→• describing homomorphisms of associative algebras, satisfying some additional properties which do not follow from a general nonsense.
3. Calculus of oriented cell complexes of K n and W n .
All operads P considered in this paper are such that P(0) is trivial and that P(1) is isomorphic to the ground field.The category of operads with this property is equivalent to the category of pseudo-operads P such that P(0) = P(1) = 0, the equivalence being given by forgetting the n = 1 piece.This, roughly speaking, means that we may ignore operadic units, see [6, Observation 1.2] for details.Therefore, for the rest of this paper, an operad means a pseudo-operad with P(0) = P(1) = 0.
First, we establish some notation.Let K = {K n } n≥2 be the non-Σ operad of associahedra.The topological cell complex K n can be realized as a convex polytope in R n−2 , with kcells labeled by the planar rooted trees with n leaves and n − k − 2 internal edges, or equivalently by (n − k − 2)-fold bracketings of n elements, see [8,II.1.6].For example, 0-cells correspond to binary trees with n leaves, or equivalently, full bracketings of n elements.All our constructions will be expressed in terms of rooted planar trees although there is clearly an underlying geometric meaning based on the polytope realization of K n .Boardman and Vogt have defined in [1] a cubical subdivision of the cells of K n , for n ≥ 2, giving rise to a cubical cell complex known as the W -construction, W n .See Figure 6 of [8, Section II.2.8] for W 4 represented as a cubical subdivision of K 4 .
The cells of W n are in one-to-one correspondence with "metric n-trees," that is, planar rooted trees with n leaves and with internal edges labeled either "metric" or "non-metric."The metric n-trees with k metric edges label the topological k-cells of W n .A cubical cell is called an interior cell if the labeling tree has only metric edges.In the geometric realization the interior cells are in the interior of the convex polytope.
In order to define the boundary operators on the complexes C * (K) := {C * (K n )} n≥2 and C * (W) := {C * (W n )} n≥2 (non-Σ operads in the category of chain complexes), we have to introduce an orientation on the cells.Let T be a planar rooted tree with internal edges labeled e 1 , . . ., e m .Two orderings e i 1 , . . ., e im and e j 1 , . . ., e jm will be called equivalent if they are related by an even permutation.The equivalence class corresponding to an ordering e i 1 , . . ., e im will be called an orientation and denoted e i 1 ∧ • • • ∧ e im .Definition 4.An oriented k-cell in K n is a pair (T, ω) where T is a planar rooted tree with n leaves and n − k − 2 internal edges and ω is an orientation.Let C k (K n ) be the vector space spanned by the oriented k-cells in K n modulo the relation (T, ω) = −(T, ω ′ ) where ω and ω ′ are the two distinct orientations.
An oriented metric k-cell in W n is a pair (T, ω) where T is a metric tree with n leaves and k metric edges and ω is an orientation of the metric edges.Let C k (W n ) be the vector space spanned by the oriented k-cells in W n modulo the relation (T, ω) = −(T, ω ′ ) where ω and ω ′ are the two distinct orientations.

The operad composition law
is defined on the basis elements by where • i is defined on planar rooted trees in the standard way, grafting the second tree onto the i-th leaf of the first, and ω ∧ ω ′ ∧ e is the concatenation of the two orientations, with the new edge created by grafting labeled e.The operad composition law is defined on the basis elements by A heuristic explanation of why we don't need any signs in the above display is that the orientation of the cells of W defined in terms of metric edges is geometric in the sense that the number of metric edges is the same as the dimension of the cell.In the case of C * (W) the new edge created by the grafting is non-metric and so does not appear in the ordering of metric vertices.
The boundary operator on C * (K n ) is defined by where the sum is over all trees T ′ with an edge e ′ , such that when e ′ is collapsed T ′ reduces to T .The condition ∂ 2 K = 0 follows immediately from the identities Next we define the boundary operator on the complex C * (W n ).Let T be a metric tree and e 1 ∧ • • • ∧ e k an orientation: The partial order on the set of binary trees.The first on the right arrow moves the vertex α and the second arrow moves the vertex β.
. .where T i is the same (unlabeled) tree as T but with the metric edge e i changed to a nonmetric edge.As above, the condition ∂ 2 W = 0 follows from the relations for the orientation elements.
In the rest of this section we introduce 'standard orientations' for top dimensional cells of W n and 0-dimensional cells of K n .There is a partial order relation on rooted planar binary trees given by the associator which moves a vertex to the right and changes the outgoing edge from a right leaning position to a left-leaning position, as shown in Figure 1.The standard orientation ω b(n) of the maximal fully metric binary tree b(n) (all internal edges leaning to the left) is given by enumerating the internal edges in sequence, starting with e 1 , the edge adjacent to the root, and continuing e 2 , . . ., e n−2 in sequence going away from the root, see Figure 2.
The standard orientation ω T of a non-maximal fully metric binary tree T is determined by a sequence of sign changes and relabelings along a path from b(n) to T in the associahedron.See Figure 3 for the standard orientations of binary trees with four leaves.To check that this rule gives and unambiguous definition of the orientation, it is sufficient (thanks to Mac Lane's Coherence Theorem) to verify that the definition is independent of path in the pentagon (expressing coherence of the associator) and in the square (expressing naturality).The verification for the pentagon is given in Figure 3.The verification for the square is straightforward and follows from the functoriality.Therefore each fully metric binary n-tree T together with its standard orientation ω T determines an element (T, We also define the standard orientation ξ T of a binary n-tree T representing a 0-cell of C 0 (K n ) inductively as follows.The only binary 2-tree representing a 0-cell of C 0 (K 2 ) has no internal edges, and its canonical orientation is given by assigning the +1-sign to this tree.The canonical orientation of any binary tree would be then determined by the formula The rule for defining the standard orientation of fully metric trees is illustrated for the pentagon in the figure above.This example also verifies that the definition is independent of the path.once we checked that there was no ambiguity.This can be done exactly as in the previous paragraph for ω T .For example, we immediately get the following standard orientations: We recommend as an exercise to verify that the standard orientation ξ b(n) of the maximal binary tree in Figure 3, this time considered as a 0 cell of and that the standard orientation of the minimal binary n-tree b(n) with the interior edges (all are right-leaning) enumerated in sequence going away from the root, is given as

The chain maps p and q.
The goal of this section is to construct maps p : C * (K) → C * (W) (Definition 7) and q : C * (W) → C * (K) (Definition 5) with the properties discussed in Section 1.The proofs that that these maps are indeed chain maps (Proposition 10 and Proposition 6) are postponed to Section 7.
As an operad in the category of vector spaces, C * (K) is a free operad generated by the collection with arity n component, a one-dimensional subspace concentrated in degree n − 2 spanned by corolla with n leaves, and C * (W) is a free operad generated by the collection with arity n component, the vector space with basis the set of purely metric planar rooted [March 29, 2022] trees with n leaves.Since a operadic map of a free operad is determined by its value on generators, the operadic chain map C * (K) q −→ C * (W) is determined by its value on corollae, and the operadic chain map C * (W) p −→ C * (K) is determined by its value on purely metric trees.
Let c(n) be the corolla with n leaves; since there are no internal edges, we denote the orientation by the symbol 1, and adopt the convention that Definition 5. Let mBin(n) be the set of n − 2 cells of W n corresponding to the fully metric planar rooted binary trees with n leaves and standard orientation.Then q(c(n), 1) is defined as a sum over mBin(n): The operadic extension of q to the free operad C(K) , which map will also be denoted q, defines a morphism of operads in the category of graded vector spaces.Proposition 6.The morphism q described in Definition 5 commutes with the boundary operators, and therefore is a morphism of operads in the category of chain complexes.
The proof of Proposition 6 is postponed to Section 7. The operad chain map p : C * (W) → C * (K) is determined by its value on fully metric trees.Before giving the precise definition, we will give a conceptual description.As a topological cell complex, the associahedron can be realized as a convex polytope K n ⊂ R n−2 .The cubical cell complex W n is a decomposition of the associahedral k-cells into k-cubes.The interior k-cell of W n labeled by a purely metric tree T with k edges is transverse to the n − 2 − k cell of K n labeled by the same tree.Let T min be the binary tree labeling the minimal vertex of this transverse cell in K n .
The image p(T ) is defined as the sum with appropriate signs of all the k-cells in C k (K n ) all of whose vertices are labeled by binary trees less than or equal to T min relative to the partial order on binary trees.
The tree T min is created by "filling-in" the non-binary vertices of T .A vertex in T with r input edges, r > 2, is replaced in T min by the minimal binary tree with r leaves, which introduces r − 2 new right-leaning edges.When this procedure is carried out at all the non-binary vertices of T , it adds n − 2 − k new edges, all of which are right-leaning.See Figure 4 for an example of this procedure.In exactly the same way, one defines T max as the binary tree obtained from T by filling-in the non-binary vertices by left-leaning edges.
In order for a binary tree S to be the maximal vertex of a k-cell in K n , it must contain at least k left-leaning edges, since an associativity move applied to a binary tree replaces a right-leaning edge with a left-leaning edge (see Figure 1) and the tree labeling the maximal vertex of k-cell is the output of at least k distinct associativity moves, corresponding to the k one-cells of the associahedron which meet at the given vertex.If T is an interior k-cell in W n , then T min cannot have more than k left-leaning edges, since the new edges in T min are all right-leaning.Since the number of left leaning edges in a binary tree is a non-decreasing function relative to the partial order, if T min has less than k left-leaning edges, there are no k-cells less than T min and we put p(T ) = 0.
Given (T, , such that T min has k left-leaning edges, then each edge e i corresponds to an edge in T min which we also denote e i .Choose any labeling of the new edges, and let ξ T min be the standard orientation of T min considered as the label for a 0-cell of K n , In the above display, is the contraction relative to the pairing e i , e j := δ i j .For any binary tree S < T min with k left-leaning edges, we will describe a method (analogous to the definition of the standard orientation) of assigning in a unique way a labeling of the left-leaning edges by the labels e 1 , . . ., e k .First, we describe a rule which determines the labeling of the left-leaning edges in a tree given the labeling of the left-leaning edges in an adjacent tree (related by one associativity).Consider a binary tree with labels only on the left-leaning edges, adjacent trees are related by replacing configuration of two edges by the configuration (going from the greater tree to the lesser tree).If both edges are internal, the rule is simply to use the same label for the left-leaning edge in both configurations.The ambiguity of the path connecting two trees resolves into a sequence of pentagons and squares and the validity of the definition is checked by considering these two figures.On the other hand, if the lower edge is a leaf, the new configuration has a right-leaning edge in place of a left-leaning edge and the resulting binary tree has less than k left-leaning edges so that there are no k-cells less than it and the contribution to p(T ) is zero.For example, in Figure 1, one of the associativity moves preserves the number of left-leaning edges and the other changes the number by one.
We can now give the full definition of p: where the sum is over binary trees S less than or equal to T min with k left-leaning edges labeled e 1 , . . ., e k according to the procedure described above, ξ is the standard orientation of the binary tree S and is the same contraction as in (11).
The function p has a unique extension to a morphism (denoted also by the same symbol) p : C * (W * ) → C * (K * ) of operads in the category of graded vector spaces.Let us close this section by the following proposition whose proof is postponed to Section 7.
Proposition 10.Let (T, ω T ) be an oriented fully metric tree, then Since p is a operad morphism, this implies that p commutes with the differential on C * (W * ) and therefore is a morphism of operads in the category of chain complexes.

The Saneblidze-Umble diagonal.
In this section we define the SU-diagonal [9].Let us start with a definition of the cubical diagonal ∆ W adapted from [10, Section 2]: where the summation runs over all disjoint decompositions L ⊔ R = {i 1 , . . ., i l } ⊔ {j 1 , . . ., j r } of {1, . . ., k} into ordered subsets, T /e L is the tree obtained from T by contracting edges {e i ; i ∈ L}, T R is the tree obtained by changing the metric edges {e j ; j ∈ R} to non-metric ones, and ρ L,R is the number of couples i ∈ L, j ∈ R such that i < j.We leave as an exercise to prove: Proposition 11.The diagonal ( 14) is co-associative and commutes with the • i -operations introduced in (6), therefore the W -construction (W, ∆ W ) is a Hopf non-Σ operad.
The SU-diagonal is then defined by formula (4), that is Exercise 12. Derive from definition that, in the shorthand of Example 9, Prove also that ∆ su (c(n), 1) always contains the terms Let us analyze formula (15) applied to (c(n), 1).The map q n applied to the oriented corolla (c(n), 1) ∈ C n−2 (K n ) is, by definition, the sum of all fully metric binary trees with standard orientations.The diagonal ∆ Wn acts on such a tree (S, ω S ) as follows.Divide interior edges of S into two disjoint groups, {f 1 , . . ., f s }, {e 1 , . . ., e t }, t + s = n − 2, and let with some η ∈ {−1, 1}, be the standard orientation.
Then ∆ Wn (S) contains the term (S L , e 1 ∧ . . .∧ e t ) ⊗ (S R , f 1 ∧ . . .∧ f s ), where S L = S/{f 1 , . . ., f s } and S R is obtained by replacing edges {e 1 , . . ., e t } of S by non-metric ones.We must then evaluate One can also describe the pair S L , S R as follows: S R is the same binary tree as S, but with only a subset of the edges retaining the metric label, S L is the fully-metric tree formed from S by collapsing the same subset of edges.
Let us pause a little and observe that the expression in ( 17) is nonzero only for trees S of a very special form.Since the value p(U, ω) is, for a binary fully metric tree U, nonzero only when U is maximal, p n (S R , f 1 ∧ . . .∧ f s ) is nontrivial only when S R is build from maximal binary fully metric trees, using the •-operation t-times.Similarly, as we saw in Section 4, p n (S L , e 1 ∧ . . .∧ e t ) is nonzero if and only if (S L ) min has exactly t left leaning edges.
A moment's reflection convinces us that the above two conditions are satisfied if and only if S R is build up from t + 1 fully metric maximal binary trees, using t times n denote the set of such n-trees and We recommend to prove as an exercise that M n is the set of all n trees T whose number of interior edges is the same as the number of left leaning edges of T min .
Let us reverse the process and start with an oriented n-tree (T, ξ) ∈ C s (K n ) such that T ∈ M t n and ξ = e 1 ∧ . . .∧ e t .Let T be the tree obtained from T by filling all nonbinary vertices by left-leaning metric edges.Let us denote these newly created metric edges f 1 , . . ., f s .Observe that to be the standard orientation of T .It is not hard to prove that η T indeed depends only on T and not on the choices of the labels e 1 , . . ., e t , f 1 , . . ., f s , as suggested by the notation.For example, η T = 1 for all trees from T ∈ M n with n ≤ 1 except T = = b(4) for which Observe finally that T L = T .Equation (15) can then be rewritten as Let us notice that the above display contains the symbol (T, e 1 ∧ . . .∧ e t ) twice.The first occurrence of this symbol denotes a cell of C t (W n ), the second occurrence a cell of C s (K n ).The sign η T then accounts for the difference between these two interpretations of the same symbol.
We already observed in Example 8 that, modulo orientations, p n (T, e 1 ∧ . . .∧ e t ) in ( 18) is the sum of all n-trees U with s interior edges such that U max ≤ T min .This leads to the following formula for the SU-diagonal whose spirit is closer to [9]: where, as usual, c(n) is the n-corolla representing the top dimensional cell of K n , the summation is taken over all (U, ω U ), (T, ω T ) with U max ≤ T min and dim(S, ω S ) + dim(T, ω T ) = n, and ϑ is a sign which can be picked up by comparing this formula to (18).

Non-existence of a co-associative diagonal.
As we already indicated, the SU-diagonal is not co-associative, that is, the co-associativity breaks already for , explicitly: The SU diagonal is also not co-commutative.This means that In the rest of this section we show that the non-coassociativity of ∆ su is not due to bad choices in the definition, but follows from a deeper principle, namely: Theorem 13.The operad A does not admit a co-associative diagonal.Therefore the operad A for A ∞ -algebras is not a Hopf operad in the sense of [3].
Proof.The proof is boring and the reader is warmly encouraged to skip it.The idea is to try to construct inductively a co-associative diagonal ∆ and observe that at a certain stage there is a non-trivial co-associativity constraint.Let us start with the construction.For we are forced to take ∆( ) := ⊗ .
The most general form of ∆( ) is with some a, b, c, d ∈ k.The compatibility with the differential ∂ of A means that must be the same as This is clearly equivalent to It can be equally easily verified that the co-associativity We conclude that the only two co-associative solutions are either ( Let us assume solution (22) which coincides with the SU-diagonal (compare (16)) -solution (23) is just the flip T (∆ su ( )) and this case can be discussed by flipping all the steps below.We will be looking for ∆ of the form ∆ = ∆ su + δ with some perturbation δ : A → A ⊗ A satisfying, of course, δ( ) = δ( ) = 0. Since we know that ∆ su is a chain map, δ must be a chain map as well.
Observe that δ( ) depends on 35 parameters.Therefore the co-associativity of ∆ and the chain condition on δ is expressed by a system of linear equations in 35 variables!We are going to show that this system has no solution.This might be a formidable task, but we will simplify it by making some wise guesses.Let us write where A, B ∈ A 0 (4) and J i (1) ⊗ J i (2) ∈ A 1 (4) ⊗ A 1 (4).Let us also denote The co-associativity of ∆ at of course means that LHS = RHS .An easy calculation shows that the only term of LHS of the form ⊗ something is while the only term of RHS of the same form is Associativity RHS = LHS then evidently means which, since char (k) = 2, clearly implies B = 0. Using the same trick we see also that A = 0, therefore δ( ) must be of the form Since δ is a chain map, trivial on and , ∂δ( ) = 0, which means that .
Looking separately at the components of bidegrees (1, 0) and (0, 1), respectively, and assuming, without loss of generality, that the elements for some scalar α ∈ k.So we managed to cut 35 parameters in (24) to one!Now The only terms of the LHS of the form ⊗ something are while in the RHS , there is only one term of this form, namely The only term of the form ⊗ ⊗ something in the above two displays is coming from the first term of the first display.This implies that α = 0, therefore δ = 0 and ∆ = ∆ su .But this is not possible, because the co-associativity of ∆ su is violated already on , as we saw in (21).
The sign comes from formula (5) setting l = s − 2, since c(s) ∈ C s−2 (K s ).Applying q gives q(∂ K (c(n), 1)) = r+s=n+1 1≤i≤r (−1) (r+i)s+i q(c(r), 1) • i q(c(s), 1).( 25) expression on the right is a sum over all binary rooted planar metric trees with n leaves and one non-metric edge.On the other hand, According to (8), the terms in ∂ W (T, ω T ) are of two types: Type A, in which a metric edge has been changed to a non-metric edge and Type B, in which a metric edge has been collapsed, creating a fully metric tree which is binary except for one tertiary vertex.In the sum of type B terms the same cell appears twice with opposite signs, since there are exactly two binary trees which give rise to the same tree with a unique tertiary vertex.The terms of type A, with one non-metric edge, run over the set of all binary rooted planar metric trees with one non-metric edge, which is the same set as that appearing in the sum on the right of equation (25).It only remains to compare the orientations of the corresponding terms on the two sides of (10).According to Definition 5, (−1) (r+i)s+i q(c(r), 1) where the term shown explicitly on the right is the leading order term relative to the order relation on binary trees, e 1 , . . ., e r−2 label the edges in b(r) and f 1 , . . ., f s−2 label the edges in b(s).Since the definition of the standard orientation on an arbitrary fully metric binary tree involves the same associativities independent of the size of the tree, it is sufficient to compare the orientation of the leading order term in (26) with the orientation of the corresponding term in ∂ W (q(c(n), 1)).If these orientations agree, so will the orientations of all the other terms.Assume i < r.Applying ∂ W to the fully metric binary tree with standard orientation appearing in Figure 5, we get (among others) the term (−1)   with e i changed to a non-metric edge, and the edges labeled e i+1 , . . ., e i+s−2 corresponding to the edges in b(s).Reordering the terms in the orientation element appearing in (26) so that f 1 , . . ., f s−2 appear in sequence between e i−1 and e i introduces a sign factor (−1) (s−2)(r−i−1) .But (−1) (s−2)(r−i−1) (−1) (r+i)s+i = (−1) i+s , since so the signs agree.For i = r, when b(r) • r b(s) = b(r + s − 1), the analysis is much easier and we leave it to the reader.
Proof of Proposition 10.The case n = 2 is trivial.Assuming the proposition is true for fully metric trees (T, ω T ) ∈ C * (W m ) for m < n, we will prove it for C k (W n ), starting with k = n − 2 and descending.In the case C n−2 (W n ), which involves binary fully metric trees, we begin with the maximal binary metric tree.We need to prove the commutativity of Figure 6, which follows from the equations in Figure 7 once we check the signs.Let us start with the second equation in Figure 7.The tree in parentheses on the left ) and therefore its image under p is as required.The orientation element for the trees on the right side of the first equation in Figure 7 with s leaves is Thus p commutes with ∂ on the maximal binary fully metric tree.Next we will show that p commutes with ∂ for all binary fully metric trees.To simplify notation, we will not indicate the orientation element.For a non-maximal fully metric binary tree T , p(T ) = 0, because T min = T has less than n − 2 left leaning edges.The only binary fully metric trees for which p(∂T ) = 0 are trees of the type appearing in Figure 5 with only one right-leaning internal edge.
Let T i,s be the tree in Figure 5, and T i,s j the term in ∂T i,s with edge e j non-metric.Then, for i = j, T i,s j is a •-composition of two fully metric binary trees, one of which is not maximal.Since p is a operad map, the image p(T i,s j ) is also a •-composition, but one of the two components is zero, since p(T ) = 0 when T is fully metric binary but not maximal.
For j = i, i − 1 we also have p(T i,s /e j ) = 0, because the binary tree (T i,s ) min has two right leaning edges.Thus the only terms in ∂T i,s whose image under p is not zero are T i,s /e i−1 , T i,s /e i , and T i,s i .It follows immediately from the definition of p that p(T i,s /e i ) = p(T i,s i ) + p(T i,s /e i−1 ).( 27) The subtree of both T min and (T /e i ) min The only configuration of edges in T which is relevant in the calculation of p(∂T ).
The configuration may occur at any vertex, not necessarily at the root.In fact, the one term appearing in p(T i,s /e i ) and not appearing in p(T i,s /e i−1 ) is p(T i,s i ).Therefore, p(∂T i,s ) = p((−1) (i−1) T i,s /e i−1 + (−1) i−1 T i,s i + (−1) i T i,s /e i ) = 0 = ∂p(T i,s ).This completes the proof of (13) for T ∈ C n−2 (W n ).Now, assuming that (13) is true for all T ∈ C j (W n ) for k < j ≤ n − 2 for all T ∈ C * (W m ) for m < n, we need to prove it for T ∈ C k (W n ).Let T be a fully metric tree with k edges labeled e 1 , . . ., e k and T min the binary tree given by filling in, and label the k edges in T min corresponding to the original edges by the same labels.All the other edges of T min are right leaning.If less than k − 1 of the edges e 1 , . . ., e k in T min are left-leaning, then p(T ) = 0 = p(∂T ) and therefore, ∂p(T ) = p(∂T ).Suppose first that T min has k − 1 leftleaning edges, and e i is right-leaning.Just as for the binary metric trees, the only tree in ∂T for which the image under p is non-zero are T /e i−1 , T /e i and T i , where e i−1 and e i are adjacent edges in T .The configuration is illustrated in Figure 8.The subtree on the right of Figure 8 appears as a subtree in both in T min and (T /e i ) bin , and the tree on the right of Figure 9 appears as a subtree in (T /e i−1 ) min .The following equation analogous to (27) applies in this case p(T /e i ) = p(T i ) + p(T /e i−1 ).( 28) and the remainder of the proof of (13) in this case is the same as before.
Next we consider the case when there are exactly k left-leaning edges in T min .In general, for any k cell T such that T min has k left-leaning edges we can choose a k + 1 cell T + such [March 29, 2022] that ∂T + contains T as a summand and all other summands of the type R i := T + /e i for i = 1, . . ., k have the property that (R i ) min has k − 1 left-leaning edges.The tree T + can be defined as follows: pick any non-binary vertex v in T with r ≥ 3 incoming edges and replace the corolla with vertex v by subtree of type c(2) • 1 − 1) with the new edge labeled e k+1 .Then T + /e k+1 = T and for i = 1, . . ., k, (T + /e i ) min has k − 1 left-leaning edges, since the edge corresponding to e k+1 in (T + /e i ) min is right-leaning.Denote the faces of type 1 T + j in which a metric edge is changed to a non-metric edge by S j , j = 1, . . ., k + 1.By the operad morphism property and the induction assumption we know that (13) is true for each S j .
Lemma 14.The validity of (13) for the faces R i , i = 1, . . ., k, implies its validity for T .
Proof.By definition of R i and S j and the property ∂ 2 = 0, Therefore, −∂T = ∂R i + ∂S j .The summations in the above displays are taken over 1 ≤ i ≤ k and 1 ≤ j ≤ k + 1.This completes the proof of Lemma 14 and the induction in the proof of (13).

Figure 4 :
Figure 4: An example of the filling-in procedure passing from a fully metric tree T to the binary tree T min .

Exercise 8 .Example 9 ., 1 ,
Verify that p(b(n), ω b(n) ) = (c(n), 1) and p(c(n), 1) = (b(n), ξ b ).Note that the first equation involves (n − 2)-cells and the second involves 0-cells.Observe also that, modulo orientations, (12) is the sum of all trees U with n − k interior edges such that U max ≤ T min .Let us describe explicitly the map p : C * (W n ) → C * (K n ) for some small n.For n = 1 and 2, p is given by p , 1 := , 1 and p ( e denotes a metric edge of W 3 .Finally, for n = 4, where e and f are metric edges of W 4 .The above equations can be written in a more condensed form asp( ) = , p( ) = , p( ) = , p( ) = , p( ) = 0, p( ) = , p( ) = + , p( ) = , p( ) = − and p( ) = ,with the convention that binary trees are endowed with their canonical orientations, corollas are oriented with the + sign and trees T with one binary and one ternary vertex are oriented as (T, e), where e denotes the unique interior edge of T .

Figure 5 :
Figure 5: The binary tree in the figure is derived from the maximal binary tree b(n) by moving the s − 1 vertices between edges e i and e i+s−1 ; therefore, its standard orientation is (−1) s−1 e 1 ∧ • • • ∧ e n−2 .

Figure 6 :
Figure 6: The commutative diagrams for the chain map p in degrees n − 2 acting on the maximal binary metric tree.

Figure 7 :
Figure 7: Each of the trees labeling the faces of K n appears precisely once as a term in ∂b(n).The orientation elements (not shown in the figure) are (−1) i−1 e 1 ∧ • • • êi • • • ∧ e n−2 in the upper left, (−1) i e 1 ∧ • • • êi • • • ∧ e n−2 in the lower left, and e for both trees on the right.

Figure 9 :
Figure 9: Collapsing the edge e i−1 from the subtree on the left of Figure 8 creates the subtree S shown here on the left.Filling-in to get a binary subtree S min creates the subtree of (T /e i−1 ) min shown here on the right, with edge e i left-leaning and e i−1 right-leaning i ) = ∂p(R i ) and p(∂S j ) = ∂p(S j ).Thus p(−∂T ) = p(∂R i ) + p(∂S j ) = ∂p(R i ) + ∂p(S j ) = ∂p( R i + S j )= ∂p(∂T + − T ) = ∂p(∂T + ) − ∂p(T )∂∂(T + ) − ∂p(T ) = −∂p(T ).