ARCHIVUM MATHEMATICUM (BRNO) Tomus 53 (2017), Jakub Kopřiva. Dedicated to the memory of Martin Doubek

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1 ARCHIUM MATHEMATICUM BRNO Tomus , ON THE HOMOTOPY TRANSFER OF A STRUCTURES Jakub Kopřiva Dedicated to the memory of Martin Doubek Abstract. The present article is devoted to the study of transfers for A structures, their maps and homotopies, as developed in [7]. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma. 1. Introduction The notion of strongly homotopy associative or A algebras is a generalization of the concept of differential graded algebras. These algebras were introduced by J. Stasheff with the aim of a characterization of delooping and bar construction in the category of topological spaces. Since then they found many applications ranging from algebraic topology and operads to quantum theories in theoretical physics. We consider the following situation: let, and W, W be two chain complexes of modules, and f :, W, W and g : W, W, two mappings of chain complexes such that gf is homotopic to the identity map on and, is equipped with A algebra structure. Then a natural question arises can A structure be transferred to W, W and secondly, what is its explicit form in terms of A algebra structure on, and and in which sense is it unique? While the existence of a transfer follows from general model structure considerations, an unconditional and elaborate answer producing explicit formulas for the transferred objects was formulated in [7]. The present article contributes to the problem of transfer of A structures. Its modest aim is to supply detailed proofs of many claims omitted in the original article [7], thereby facilitating complete subtle proofs to a reader interested in this topic. This exposition also extends the results of the aforementioned article in several ways, and sheds a light on its relationship with the homological perturbation lemma Mathematics Subject Classification: primary 18D10; secondary 55S99. Key words and phrases: A structures, transfer, homological perturbation lemma. Received March 11, 2017, revised September Editor M. Čadek. DOI: /AM

2 268 J. KOPŘIA The content of our article goes as follows. In the Section 2 we recall a well-known correspondence between A algebras and codifferentials on reduced tensor coalgebras. This allows us to simplify the proofs in Section 3 considerably. The Section 3 is devoted to the problem of homotopy transfer of A algebras. We first derive the formulas introduced in [7], and then give their self-contained proofs. Here we achieve a substantial simplification of all proofs due to the reduction of sign factors. We also comment on another remark in [7], namely, the relationship between the homological perturbation lemma and homotopy transfer of A algebras. We prove that on certain assumptions the explicit formulas in [7] do coincide with those coming from the homological perturbation lemma. We shall work in the category of Z-graded modules over an arbitrary commutative unital ring R, and their graded R-homomorphisms. We first briefly recall the concepts of A algebra, A morphism of A algebras and A homotopy of A morphisms, cf. [7], [4]. Definition 1.1. Let, be a chain complex of modules indexed by Z, i.e., is a Z-graded modules = i= i with i i 1 and = 0. Let µ n : n be a collection of linear mappings of degree n 2 n 2, satisfying µ n 1 n µ n 1 i 1 1 n i 1 = An 1 il1n µ k 1 i 1 µ l for all n 2 and An = {k, l N k l = n 1, k,l 2, 1 i k}. The structure,, µ 2, µ 3,... is called A algebra. Throughout the article, we use the Koszul sign convention. This means that for U, a W graded modules and f : U, g : U, h: W and i: W linear maps of degrees f, g, h and i, respectively, holds h if g = 1 f i hf ig. Similarly for u 1, u 2 U of degree u 1 and u 2, respectively, holds f gu 1 u 2 = 1 u1 g fu 1 gu 2. Definition 1.2. Let,, µ 2,... and W, W, ν 2,... be A algebras. Then the set {f n : n W, f n = n 1} n 1 is called A morphism if W f n Bn 1 ϑr1,...,r k ν k f r1 f rk 2 = f 1 µ n An 1 n f n 1 i 1 1 il1n f k 1 i 1 1 n i µ l

3 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 269 holds for all n 1 with Bn = {k, r 1,..., r k N k 2, r 1,..., r k 1, r 1 r k = n} and ϑr 1,..., r k = 1 i<j k r ir j 1. Morphisms of A algebras can be composed: for U, U, ϱ 2,...,,, µ 2,... and W, W, ν 2,... A algebras, {f n : U n } n 1 and {g n : n W } n 1 A morphisms, their composition {gf n : U n W } n 1 is defined as 3 gf n = g 1 f n Bn 1 ϑr1,...,r k g k f r1 f rk. Definition 1.3. Let {f n : n W } n 1 and {g n : n W } n 1 be morphisms between A algebras,, µ 2,... and W, W, ν 2,.... The set of linear mappings {h n : n W, h n = n} n 1 is an A homotopy between A morphisms {f n : n W } n 1 and {g n : n W } n 1 provided 4 f n g n = h 1 µ n An Bn 1 n h n 1 i 1 1 il1n h k 1 i 1 1 i k 1 ϑr1,...,r k ν k 1 n i µ l δw h n f r1 f ri 1 h ri g ri1 g rk, is true for all n 1 with Bn = {k, r 1,..., r k N k 2, r 1,..., r k 1, r 1 r k = n}. 2. Reduced tensor coalgebras In the present section we introduce a bijective correspondence between A algebras and codifferentials on reduced tensor coalgebras, cf. [4]. We retain the notation = i= i for Z-graded modules as well as A An = {k, l N k l = n 1, k,l 2, 1 i k}, B Bn = {k, r 1,..., r k N k 2, r 1,..., r k 1, r 1 r k = n} for n N, and A1 = A2 = B1 =. We use a few natural variations on this notation, e.g. A n = {k, l N k l = n 1, k,l 2, 1 i k } Codiferentials on tensor coalgebras. Definition 2.1. Let T = n=1 n, where the elements in i have degree or homogeneity i, and let the mapping C : T T T be defined in such a way that C : v 0 for v 1 = and 5 C : v 1 v n v 1 v i v i1 v n, for n 2 and v 1,..., v n. The pair T, C is called the reduced tensor coalgebra.

4 270 J. KOPŘIA Definition 2.2. A linear mapping δ : T T of degree 1 is called coderivation if C δ = δ 11 δ C. Moreover, if δ satisfies δ δ = 0, it is called codifferential. Remark 2.3. We notice that C is coassociative, 1 C C = C 1 C. For all v T holds Cv = 0 if and only if v is of homogeneity 1. For all maps ϕ: n T W, n 1, holds C T W ϕ = 0 if and only if ϕ n W. For all v = v 1... v n T and w = w 1... w m T, we have m 1 Cv w = v 1,i v i1,n w v w v w 1,i w i1,m, with v i,j = v i... v j, i j, i,j {1,..., n}, and analogously for w i,j. This little calculation expresses a fact that T is a bialgebra which is, as a conilpotent coalgebra, cogenerated by. Lemma 2.4. Let E : T T W be a linear mapping for which there exist {e n : n W } n 1 with E n = e n Bn e r 1... e rk, and Bn given in B. Then 6 C T W E n = E i E n i. Proof. Obviously, we can write E n = e n e i E n i. The proof is by induction on n: the claim holds for n = 1 and we assume it is true foll all natural numbers less than n. Then C T W E n = C T W e n e i E n i = C T W e i E n i = e i E n i i j=1 e i E j E n i j l 1 = e i E n i e j E l j E n l l=2 l=2 j=1 l 1 = e 1 E e l e j E l j E n l, and the proof follows by induction hypothesis from E l = e l l 1 e i E l i. Theorem 2.5. Let E : T T W and G: T T W be linear mappings for which there exist linear mappings {e n : n W } n 1, {g n : n W } n 1 such that E n = e n Bn e r 1 e rk and G n = g n Bn g r 1... g rk j=1

5 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 271 with Bn given in B. Given a linear mapping F : T T W, the following conditions are equivalent: 1 C T W F = E F F G C T, 2 there exist linear mappings {f n : n W } n 1 such that F n = f n e r1 e ri 1 f ri g ri1 g rk. Bn 1 i k Proof. 2 1: We have F n = f n E i f n i f i G n i j=1 E j f i G n i j for all n 1. We now verify 1 by expanding n i 1 both sides: E F F G C T n = E F F G = 1 n i [ E n i F i F n i G i ], and by Lemma 2.4 we get C T W C T W C T W = E i f n i = E n i f i f i G n i n i 1 j=1 = f i G n i n i 1 j=1 E j f i G n i j i j=1 i j=1 E n i j f i G j n i 1 j=1 n i 1 j=1 n i 1 j=1 1 i E n i j E j f i, f i G j G n i j, E j f i G n i j j 1 E n i j k E j f i G k k=1 j 1 E j f i G k G n i j k. k=1

6 272 J. KOPŘIA The summation in the variables i j and i j k, respectively, yields C T W = C T W C T W = E i f n i l 1 E n i f i E n l E l j f j, = f i G n i l=2 j=1 l 1 f i G n i f j G l j G n l, l 1 n i 1 j=1 l=2 j=1 E j f i G n i j E n l f j G l j l=2 j=1 l 1 E l j f j G n l l=2 j=1 m 1 l=3 j=1 m 1 l=3 j=1 j 1 E n l E l j f i G j i j 1 E j i f i G l j G n l. Taking all terms of the form E n i and G n i results in [ C T W F n = E n i F i F n i G i ] and the implication is proved. Notice that we also proved, on the assumption F m = f n Bm 1 i k e r 1 e ri 1 f ri g ri1 g rk for n > m 1, that n i 1 C T W E i f n i f i G n i E j f i G n i j = j=1 [ E n i F i F n i G i ]. 1 2: The proof is again by induction. For all v holds C T W F v = 0, which gives F W and so there exists a linear mapping f 1 : W such that

7 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 273 F = f 1. Assume now the claim of the implication is true for all natural numbers less than n, i.e. F m = f m Bm 1 i k e r 1 e ri 1 f ri g ri1 g rk, for n > m 1. The proof of the previous implication claims for F m = f m Bm,r e i>0 r 1 e ri 1 f ri g ri1 g rk with n > m 1, that [ C T W F n = E n i F i F n i G i ] = C T W E i f n i f i G n i E j f i G n i j. n i 1 Because C T W is linear, F n differs from E i f n i f i G n i n i 1 j=1 E j f i G n i j by a linear map f n : n W. This means F n is of the required form and the proof is complete. Theorem 2.6. A linear mapping δ : T T of degree 1 fulfills C δ = δ 1 1 δ C if and only if there exist a set of maps {δ n : n } n 1 of degree 1 such that δ = δ 1 and for n 2 holds δ n = δ n n 1 i 1 δ 1 1 n i An 1 i 1 δ l, where An is given by A. Proof. In Theorem 2.5 we take E = G = 1, where e 1 = g 1 = 1 and e n = g n = 0 for n 2. Lemma 2.7. Let δ : T T be a linear map of degree 1 such that δ = δ 1 and for n 2 holds δ n = δ n n 1 i 1 δ 1 1 n i An 1 i 1 δ l. Then the following conditions are equivalent: 1 δ δ = 0, 2 δ 1 δ 1 = 0 and for all n 2 we have 7 δ 1 δ n δ n 1 i 1 where An is given by A. j=1 δ 1 1 n i δ k 1 i 1 An δ l = 0, Proof. 1 2: The proof goes by induction. By assumption we have for v δδ 1 v = 0, so δ 1 : implies δ 1 δ 1 v = 0. Now assume 7 is true for all natural numbers less than n. Then δ 2 n = δ 1 δ n An δ k δ n 1 i 1 1 i 1 δ 1 1 n i δ l.

8 274 J. KOPŘIA Schematically, this means δ 2 n = δ 1 δ n An 1 a δ k 1 i 1 δ n 1 i 1 δ 1 1 n i δ l 1 a δ bd11 b δ b 1 c δ d 1 e 1 a δ b 1 c δ c 1 d 1 e δ d 1 e, where the last row is a consequence of the Koszul sign convention: 1 a δ b 1 c1e 1 abc δ d 1 e = a 1 δ b 1 c δ d 1 e, 1 a1c δ d 1 e 1 a δ b 1 cde with δ n = 1 for all n N. The term 1 a can be written as 1 a δ bd1 1 b δ c 1 d e 1 = 1 a = 1 δ b δ d 1 a δ bd1 1 b δ bd1 1 e δ b 1 c 1 ab δ c 1 d δ d 1 e 1 e δ c 1 de. We have a b c d e = n, choose arbitrary a, e 0, 1 a e < n and sum over all b, c, d such that 0 b, d n a e and 1 c n a e such that b c d = n e a: b,c,d δ bd1 1 b δ c 1 d = δ1 δ n n An δ n 1 i 1 δ k 1 i 1 δ 1 1 n i δ l, where n = n a e. By induction hypothesis, the last display is equal to 0, and we have 1 a δ bd1 1 b δ c 1 d e 1 = 1 a δ bd1 1 b δ c 1 d a,e 1 e = a,e 1 a 0 1 e = 0. Consequently, 7 is true for n and δ 1 δ n δ n 1 i 1 δ 1 1 n i δ k 1 i 1 An b,c,d δ l = δ 2 n = : The second implication can be easily deduced from the first one Morphisms and homotopies. Definition 2.8. Let δ be a codifferential on T, C and δ W be a codifferential on T W, C. A linear mapping F : T, C, δ T W, C, δ W of degree 0 is called morphism provided C T W F = F F C T and δ W F = F δ.

9 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 275 Lemma 2.9. Let F : T, δ T W, δ W be a linear map of degree 0. Then the following claims are equivalent: 1 C T W F = F F C T, 2 there is a set of linear mappings {f n : n W } n 1 of degree 0 such that F n = f n Bn f r 1... f rk, with Bn given in B. Proof. 2 1: A consequence of Lemma The proof goes by induction. For v we have Cv = 0, which implies 0 = F F C T = C T W F and so F v W. Assuming the claim is true for all natural numbers less than n, F F C n = F F 1 i 1 n i = F i F n i and by induction hypothesis F m = f m Bm f r 1 f rk for all n > m 1. Lemma 2.4 gives F i F n i = C T W f i F n i and because C T W is linear, F n differs from f i F n i by a linear map f n : n W. Then F n is of the required form and the proof is complete. Lemma Let F : T, δ T W, δ W be a linear map of degree 0 such that F n = f n Bn f r 1... f rk, with all {f n : n W } n 1 linear of degree 0. Then the following are equivalent: 1 δ W F = F δ, 2 for all n 1 holds δ1 W f n δ W k f r1 f rk = f 1 δ n Bn 8 f n 1 i 1 δ1 1 n i f k 1 i 1 δl. An Proof. 1 2: The proof goes by induction. The restriction to, δ W F = F δ, corresponds to δ1 W f 1 = f 1 δ1. We now assume 8 applies to all natural numbers less than n. We expand both sides of 8, δ W F n = δ W 1 f n Bn f r1 f ra δb W a,b fra1 f r ab frab1 f r k, F δ n = f 1 δ 1 Bn j,l f r1 f ri 1 f ri 1 j δ l 1 ri j 1 fri1 f rk

10 276 J. KOPŘIA and compare the terms of same homogeneities. We fix j 1 and r 1,..., r j 1, r 1 r j < n and 0 m j, and focus on terms of the form f r1 f ri 1 f ri f rj, where is an expression of the form δ W f f or f 1 δ 1. Terms on the right hand side of the form f r1 f ri 1 δ W f f f ri f rj correspond to f r1 f rm δ1 W f n δk W f r 1 f r f rm1 f rj, k Bn while the terms of the form f r1 f ri 1 f 1 δ 1 fri f rj correspond to n f r1 f rm f 1 δ n f n 1 i 1 δ1 1 n i An f k 1 i 1 δl f rm1 f rj with n = n r 1 r j. Because n < n, they fulfill the equality 8 and hence are equal. Subtracting from both sides all elements of homogeneity greater than 1, we arrive at δ W 1 f n Bn δ W k f r1 f rk = f 1 δ n n However, this equality is true by 8 for n. f n 1 i 1 δ1 1 n i f k 1 i 1 δl. An 2 1: This implication can be again reduced to the previous one. Definition Let δ be a codifferential on T, C and δ W be a codifferential on T W, C. Let F : T, C, δ T W, C, δ W and G: T, C, δ T W, C, δ W be morphisms. F and G are homotopy equivalent provided there exist linear maps H : T T W of degree 1 such that C T W H = F H H G C T and F G = Hδ δ W H. The map H is a homotopy between F a G. Remark Theorem 2.5 implies that H : T T W of degree 1 fulfills C T W H = F H H G C T if and only if there is a set of maps {h n : n W } n 1 of degree 1 such that H n = h n Bn,r f i>0 r 1 f ri 1 h ri g ri1 g rk. Theorem We retain the assumptions of Definition 2.11, and in addition assume the existence of the set of linear maps {e n : n W } n 1, {g n : n W } n 1 of even degree d such that E n = e n Bn e r 1 e rk and G n =

11 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 277 g n Bn g r 1 g rk. Let F : T T W be a linear mapping for which there exists a set of linear maps {f n : n W } n 1 of odd degree d 1 fulfilling F n = f n e r1 e ri 1 f ri g ri1 g rk. Bn,r i>0 Then the following assertions are equivalent: 1 E G = F δ δ W F, 2 e n g n = f 1 δ n n f n1 i 1 δ1 1 n i An f k1 i 1 δl 1 k i δ W 1 f n Bn,r i>0 δw k e r 1 e ri 1 f ri g ri1 g rk for all n 1. Proof. The proof can be done along the same lines as the proofs of Lemma 2.7 and Lemma Codifferentials and A algebras. Definition For graded we define s in such a way that s i = i 1. The graded modules and s are canonically isomorphic: s: s is a linear map of degree 1 called suspension, ω : s is a linear map of degree 1 called desuspension. Remark We have s n ω n = 1 n 2 by the Koszul sign convention. Theorem The following claims are equivalent: 1 {µ n : n ; µ n = n 2} n 1 is A structure on, 2 The linear maps δ n = s µ n ω n are of degree 1, and are the components of a codifferential on T s in the sense of Theorem 2.6. Proof. 2 1: δ n = s µ n ω n are the components of a codifferential, and so we have for all n 1 δ 1 δ n δ n 1 i 1 δ 1 1 n i δ k 1 i 1 δ l = 0. An This can be rewritten, by Koszul sign convention, as δ n 1 i 1 δ 1 δ n = s µ 1 ω s µ n ω n = s µ 1 µ n ω n, = = δ 1 1 n i = n s µ n ω n 1 i 1 1 n i s µ n ω i 1 µ 1 ω ω n i 1 n i 1 i 1 s µ n 1 i 1 s µ 1 ω 1 n i µ 1 1 n i ω n,

12 278 J. KOPŘIA An δ k 1 i 1 δ l = s µ k ω k 1 i 1 An s µ l ω l = 1 k i s µ k ω i 1 µ l ω l ω k i An = 1 k i 1 li 1 s µ k 1 i 1 µ l ω n. An The mappings s and ω are linear, hence s µ 1 µ n 1 µ n 1 i 1 µ 1 1 n i An 1 il1 µ k 1 i 1 µ l ω n = : This can be easily reduced to the proof of the previous implication. Theorem The following claims are equivalent: 1 {ϕ n : n W ; ϕ n = n 1} n 1 is A morphism from, µ to W, ν, 2 the mappings f n = s W ϕ n ω n are of degree 0, and are the components of A morphism from T s, δ to T sw, δ W in the sense of Lemma 2.9. The codifferentials are given by A structures on and W, respectively, via Theorem The following claims are equivalent: 1 {h n : n W ; h n = n} n 1 is A homotopy between A morphisms ϕ with components {ϕ n : n W ; ϕ n = } n 1 and ψ with components {ψ n : n W ; ψ n = n 1} n 1, respectively, from, µ to W, ν, 2 h n = s W h n ω n are of degree 1, and are the components of A homotopy between morphisms F and G from T s, δ to T sw, δ W, where F corresponds to ϕ and G corresponds to ψ in the sense of the first equivalence in the theorem. The codifferentials are given by A structures on and W, respectively, as in Theorem Proof. The proof goes along the same lines as in Theorem Homotopy transfer of A algebras The starting point for the present section are the chain complexes, and W, W, f : W, g : W their morphisms such that gf is homotopy equivalent to 1 by a homotopy h. Let, be equipped with A algebra structure, which means that there is a set of multilinear maps µ = µ 2, µ 3,... satisfying the relations 1. We would like to induce A structure W, W, ν 2, ν 3,... on W, W by transferring,, µ 2, µ 3,..., as well as the morphisms of A

13 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 279 algebras ψ = g, ψ 2, ψ 3,... from W, W, ν to,, µ and ϕ = f, ϕ 2, ϕ 3,... acting in the opposite direction such that their composition ψϕ is A homotopy equivalent with the identity map via H = h, H 2, H 3,.... The strategy to solve this problem, cf. [7], suggests to construct the set of maps {p n : n } n 2 of degree n 2 called p-kernels, and the set of maps {q n : n } n 1 of degree n 1 called q-kernels in such a way that ν n, ϕ n, ψ n and H n defined by 9 ν n := f p n g n, ϕ n := f q n, ψ n := h p n g n, H n = h q n, fulfill the transfer problem of A algebra as discussed in the previous paragraph. We shall first introduce the p-kernels and based on them we introduce the q-kernels later on. Apart from A a B, we shall rely on the notation cf., [7] C Cn ={k,i, r 1,..., r i N 2 k n, 1 i k, r 1,..., r i 1, r 1 r i k i = n}, for n N, and ϑ ϑu 1,..., u k = u i u j 1, for arbitrary u 1,..., u k, k N. 1 i<j k 3.1. p-kernels. Lemma 3.1. The p-kernels together with W constitute an A structure on W, W via 9 if and only if for all n 2 holds f p n 1 n p n 1 1 il1n p k 1 i 1 An 1 n u gf p l g n = 0. Proof. W, W, ν 2,... is an A algebra if we have for all n 1 W ν n 1 n ν n 1 W 1 il1n ν k 1 i 1 W An W 1 n u W ν l W = 0.

14 280 J. KOPŘIA This is true for n = 1, because W, W is the chain complex f = W f and analogously for g. Now expand ν n following 9: W ν n 1 n ν n 1 W W 1 n u W 1 il1n ν k 1 i 1 W ν l An = W f p n g n 1 n f p n g n 1 W W 1 n u W 1 il1n f p k g k 1 i 1 W An f p l g l W. W Because both f and g are linear maps of degree 0, this equals to n f p n g n f 1 n p n g u 1 g W g n u which is f 1 il1n p k g i 1 gf p l g l g k i, An n f p n g n f 1 n p n 1 f 1 il1n p k 1 i 1 An 1 n u g n gf p l g n = 0. Lemma 3.2. Let us assume that p-kernels induce the transfer of A algebra as formulated above, and they fulfill n 2 10 p n 1 n p n 1 1 il1n p k 1 i 1 An 1 n u gf p l = 0. Then p n g n = 1 ϑr1,...,rk µ k h p r1 h p rk g n, Bn where we define h p 1 = 1. Proof. According to 10 these p-kernels induce A structure on W, W by Lemma 3.1. It remains to verify that they give A morphism from,, µ to

15 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 281 W, W, ν, i.e. ψ n Bn 1 ϑr1,...,r k µ k ψ r1 ψ rk = ψ 1 ν n 1 n ψ n 1 W 1 il1n ψ k 1 i 1 W An W 1 n u W ν l W, which by 9 can be formulated as h p n g n 1 ϑr1,...,rk µ k h p r1 h p rk g n Bn n = gf p n g n h 1 n p n 1 h 1 il1n p k 1 i 1 An Due to gf 1 = h h, we have Bn 1 ϑr1,...,rk µ k h p r1 h p rk g n = p n g n h p n 1 n p n 1 h 1 il1n p k 1 i 1 An By assumption 10, we obtain h p n 1 n p n 1 1 n u g n gf p l g n. 1 n u g n gf p l g n. 1 n u g n h 1 il1n p k 1 i 1 An gf p l g n = 0, which reduces to p n Bn 1 ϑr1,...,rk µ k h p r1 h p rk g n = 0.

16 282 J. KOPŘIA Remark 3.3. The assumption of Lemma 3.2 can be weaken to p n g n 1 n p n 1 1 n u g n 11 1 il1n p k 1 i 1 An gf p l g n = 0, where 11 is fulfilled if the p-kernels define A structure on W, W, and f is a monomorphism. In the situation of interest is f, however, assumed to be an epimorphism. Definition 3.4 p-kernels, [7]. We define for each n 2: 12 p n = Bn 1 ϑr1,...,r k µ k h p r1 h p rk, where h p 1 = 1, with Bn given in B and ϑr 1,..., r k given in ϑ. Remark 3.5. For p-kernels there exists a non-inductive explicit expression. Each term in the p-kernel can be represented by a rooted plane tree, and there is a function which associates to a rooted plane tree a sign corresponding to our inductive definition. Theorem 3.6. The p-kernels introduced in [7] satisfy p n 1 n p n 1 1 n u 10 for all n 2. 1 il1n p k 1 i 1 An gf p l = 0, Proof. Let us first simplify our situation by passing to the suspension T s with the induced codifferential δ. Because s and ω are by Definition 2.14 izomorphisms, 10 is true if and only if s p n 1 n p n 1 = s 1 il1n p k 1 i 1 An 1 n u ω n gf p l ω n. Introducing ˆp m = s p m ω m, ĝ = s g ω and ˆf = s f ω ˆp m = 1, ĝ = ˆf = 0, we have δ 1ˆp n ˆp n 1 δ 1 1 n u ˆp k 1 i 1 ĝ ˆf ˆp l = 0. An The proof of the last claim goes by induction. The case n = 2 corresponds to δ 1 δ 2 δ 2 1 δ 1 δ 2 δ 1 1 = 0,

17 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 283 which is certainly true because {δ n : n } n 1 are the components of the codifferential on T s cf., 7 for n = 2 in Lemma 2.7. By induction hypothesis, we assume the claim is true for all natural numbers less than n. The proof is naturally divided into three steps: I. We shall first expand the term δ 1ˆp n : we have ˆp n = s p n ω n, so by Definition 3.4 ˆp n = s 1 ϑr1,...,rk µ k h p r1 h p rk ω n Bn = Bn 1 ϑr1,...,r k 1 σ s µ k ω s h p r1 ω r1 ω s h p rk ω r k with σ = 1 i<j k r ir j 1. However, s h p ri ω ri = 11r i 2 r i = 0, so the last display equals to s µ k ω k s h ω s p r1 ω r1 s h ω s p rk ω r k. Bn Consequently, 13 ˆp n = Bn δ k ĥ ˆp r1 ĥ ˆp rk, ĥ = s h ω ĥ = 1, and so δ 1ˆp n = Bn δ 1 δ k ĥ ˆp r1 ĥ ˆp rk k = Bn Bn Ak δ k 1 i 1 δ k 1 i 1 The last summation can be rewritten as Bn Ak δ k 1 i 1 = Bn = Ak Bn, r i>1 δ 1 ĥ ˆp r1 ĥ ˆp rk δ l 1 k i ĥ ˆp r1 ĥ ˆp rk. δ l 1 k i ĥ ˆp r1 ĥ ˆp rk ĥ δ k ˆp r1 δ l ĥ ˆp ri ĥ ˆp ĥ ˆp ril rk δ k ĥ ˆp r1 ˆp ri ĥ ˆp rk,

18 284 J. KOPŘIA where the last equality comes from the summation over all r 1,...,r il with r i r il fixed. We conclude δ 1ˆp n = δ k ĥ ˆp r1 δ 1 ĥ 1 ˆp ri ĥ ˆp rk Bn,r i>1 Bn,r δ k ĥ ˆp r1 δ 1 ĥ ˆp ri ĥ ˆp rk. II. We shall apply the induction hypothesis to δ 1ˆp n. We remind the formal equality ĥ ˆp 1 = 1 and also gf 1 = hh equivalent to δ 1 ĥ1 = ĝ ˆf ĥδ 1. Then δ 1ˆp n = δ k ĥ ˆp r1 ĥδ 1 ĝ ˆfˆp ri ĥ ˆp rk 14 Bn,r i>1 Bn,r δ k ĥ ˆp r1 δ 1 ĥ ˆp ri ĥ ˆp rk. The second part of the first term on the right hand side 14 equals Bn δ k ĥ ˆp r1 ĝ ˆf ˆp ri ĥ ˆp rk = Bn δ k ĥ ˆp r1 ĥ ˆp 1 ĥ ˆp rk 1 s ĝ ˆf ˆp ri 1 n s ri, where s = j<i r j. The second term in 14 equals δ k ĥ ˆp r1 δ 1 ĥ ˆp ri ĥ ˆp rk Bn,r = Bn,r δ k ĥ ˆp r1... ĥ ˆp ri ĥ ˆp rk 1 s By induction hypothesis, we have for all m < n δ 1ˆp m = m ˆp m 1 δ 1 1 n s ri. δ 1 1 m u ˆp k 1 i 1 ĝ ˆf ˆp l. Am Finally, the first part of the first term 14 equals δ k ĥ ˆp r1 ĥ δ 1ˆp ri ĥ ˆp rk Bn,r i>1 = Bn,r i>1 Bn,r i>1 r i δ k ĥ ˆp r1 ĥ ˆp ri 1 δ 1 1 ri u ĥ ˆp rk δ k ĥ ˆp r1 ĥ ˆp k 1 i 1 ĝ ˆf ˆp l ĥ ˆp rk. Ar i

19 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 285 III. Now we pair up the contributions appearing in the previous step: the right hand side of 14 can be rewritten as P1 P2 P3 Bn,r i>1 r i δ k ĥ ˆp r1 ĥ ˆp ri 1 Bn,r i>1δ kĥ ˆp r1 Ar i Bn δ k ĥ ˆp r1 ĥ ˆp 1 ĥ ˆp rk 1 s δ 1 1 ri u ĥ ˆp rk ĥ ˆp k 1 i 1 ĝ ˆf ˆp l 1 k i ĥ ˆp rk ĝ ˆf ˆp ri 1 n s ri P4 Bn,r δ k ĥ ˆp r1 ĥ ˆp ri ĥ ˆp rk 1 s with s = j<i r j, and we get P1 P4 = P2 P3 = An ˆp n 1 ˆp k 1 i 1 δ 1 1 n u, ĝ ˆf ˆp l. δ 1 1 n s ri, Remark 3.7. Theorem 3.6 implies that the p-kernels in [7] fulfill q-kernels. Lemma 3.8. The q-kernels constitute A morphism ϕ = f, ϕ 2, ϕ 3,..., ϕ n = f q n and ν n = f p n g n, from,, µ 2, µ 3,... to W, W, ν 2,ν 3,... if and only if for all n 2: f q n 1 n q n 1 1 n u Bn 1 ϑr1,...,r k p k gf q r1 gf q rk 1 il1n q k 1 i 1 An µ l 1 n k q 1 µ n = 0. Proof. The proof easily follows from the explicit expansion of A morphism ϕ = f, ϕ 2, ϕ 3,..., which maps,, µ 2, µ 3,... to W, W, ν 2, ν 3,... for ϕ n = f q n and ν n = f p n g n cf. 9.

20 286 J. KOPŘIA Lemma 3.9. Let the q-kernels fulfill q n 1 n q n 1 1 n u Bn 1 ϑr1,...,r k p k gf q r1 gf q rk 15 1 il1n q k 1 i 1 An µ l 1 n k q 1 µ n = 0. for all n 2. Then we have q n = Cn 1 nriϑr1,...,ri µ k ψϕr1 ψϕ ri 1 h q ri, for all n 2, where A morphisms ϕ and ψ are given by p-kernels and q-kernels, 9. We also used the notation Cn as in C and ϑr 1,..., r k as in ϑ. Proof. Assuming 15, the set of q-kernels constitutes by Lemma 3.8 A morphism ϕ = f, ϕ 2, ϕ 3,... from,, µ 2, µ 3,... to W, W, ν 2, ν 3,.... We also demand the set of maps H n = h q n gives A homotopy H = h, H 2, H 3,... between ψϕ and 1. This is equivalent by Definition 1.3 to H n 1 n H n 1 1 n u 1 nriϑr1,...,ri µ k ψϕr1 ψϕ ri 1 H ri 1 k i H1 µ n Cn = 1 il1n H k 1 i 1 An µ l 1 n k ψϕ n 1 n for all n 2. According to 3, we have ψϕ m = ψ 1 ϕ m Bm 1 ϑr1,...,r k ψ k ϕ r1 ϕ rk, and so we can write the composition of A morphisms in terms of p-kernels and q-kernels: 16 ψϕ m = gf q m Bm 1 ϑr1,...,r k h p k gf q r1 gf q rk.

21 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 287 By Definition 1.3, the A homotopy H = h, H 2, H 3,... can be rewritten in terms of p-kernels and q-kernels we use again h = gf 1 h and 1 n = 0: gf q n q n h q n 1 n h q n 1 1 n u 1 nriϑr1,...,ri µ k ψϕr1 ψϕ ri 1 h q ri Cn h q 1 µ n = 1 il1n h q k 1 i 1 An µ l 1 n k gf q n Bn 1 ϑr1,...,r k h p k gf q r1 gf q rk. We subtract from both sides of the last display gf q n, and by 15 conclude h q n 1 n h q n 1 1 n u h q 1 µ n = 1 il1n h q k 1 i 1 An µ l 1 n k Bn 1 ϑr1,...,r k h p k gf q r1 gf q rk, which finally results in q n = 1 nriϑr1,...,ri µ k ψϕ r1 ψϕ ri 1 h q ri. Cn Remark The assumption 15 is fulfilled as soon as the q-kernels give a A morphism ϕ = f, ϕ 2, ϕ 3,... from,, µ 2, µ 3,... to W, W, ν 2, ν 3,... and f is a monomorphism. Definition 3.11 q-kernels, [7]. Let n 2 and define q 1 := 1. We define q-kernels inductively by q n = 1 nriϑr1,...,ri µ k ψϕ r1 ψϕ ri 1 h q ri, Cn where ψϕ m = gf q m Bm 1ϑr1,...,r k h p k gf q r1 gf q rk cf., 16, p-kernels were introduced in 3.4, with Cn given in C and ϑu 1,..., u k in ϑ. Remark There is an explicit description of the q-kernels in terms of rooted plane trees, but it is much more complicated when compared to the analogous description for the p-kernels.

22 288 J. KOPŘIA We shall now prove that the q-kernels introduced in Definition 3.11 satisfy 15. Let us consider again the suspension T s with the induced codifferential δ such that δ 1 = s ω and δ n = s µ n ω n, n 2. Then ˆq m = s q m ω m, ˆψ m = s ψ m ω m and ˆϕ m = s ϕ m ω m for m 2 ˆq m = ˆϕ m = ˆψ m = 0, and 15 is equivalent to δ 1ˆq n Bn ˆp k ĝ ˆf ˆq r1 = ˆq n 1 δ 1 1 n u ˆq k 1 i 1 δ l 1 n k ˆq 1 δ n. An In the following two lemmas we prove that ˆψˆϕ is an A morphism. Lemma Let us assume 15 is true for all n m. Then the p-kernels in Definition 3.4 and the q-kernels in Definition 3.11 fulfill m δ 1 ˆψˆϕ m = ˆψˆϕ m 1 δ 1 1 m u Am ˆψˆϕ k 1 i 1 δ l 1 n k ˆψˆϕ 1 δ m Bm ˆp k ĝ ˆf ˆq r1... ĝ ˆf ˆq rk for all m 2. Proof. We shall first expand the composition of morphisms in the suspended form as in 16, and also use the homotopy ĥ between ĝ ˆf and 1 : δ 1 ˆψˆϕ m = ĝ ˆf δ 1 q m Bm δ 1 ĥ ˆp k ĝ ˆf ˆq r1 17 = ĝ ˆf δ 1 q m Bmĝ ˆf 1 ĥδ 1 ˆp k ĝ ˆf ˆq r1. By Theorem 3.6 ĥ δ 1ˆp k ĝ ˆf ˆq r1 Bm k 18 = Bm 19 Bm ĥ ˆp k 1 A k and as ĝ, ˆf and ˆq m are of degree 0, we have 18 = Bm δ 1 1 k u ĝ ˆf ˆq r1 ĝ ˆf ˆq rk ĥ ˆp k 1 i 1 ĝ ˆf ˆp l 1 k k ĝ ˆf ˆq r1 k ĥ ˆp k ĝ ˆf ˆq r1 ĝ ˆf δ 1ˆq ru.

23 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 289 By 15 for n m, we expand the terms of the form δ 1ˆq as Bm Bm Bm Bm k ĝ ˆp k ˆf ˆq r1 r u v=1 A r u ĝ ˆf ˆq ru 1 v 1 δ 1 1 ru v ĝ ˆf ˆq rk k ĝ ˆp k ˆf ˆq r1 ĝ ˆf ˆq k 1 i 1 δ l 1 ru k ĝ ˆf ˆq rk k ˆp k ĝ ˆf ˆq r1 ĝ ˆf ˆq 1 δ ru k ĝ ˆp k ˆf ˆq r1 ĝ ˆf ˆp k ĝ ˆf ˆq r 1 ĝ ˆf ˆq ĝ ˆf r ˆq k rk. B r u In the second contribution 19, which equals to Bm ˆp k ĝ ˆf ˆq r1 ĝ ˆf ˆp l ĝ ˆf ˆq ri ĝ ˆf ˆq ril 1 A k, we sum over all inner positions of ˆp k ĝ ˆf ˆq r ĝ ˆf ˆq rk and get 19 = Bm k ˆp k ĝ ˆf ˆq r1 B r u. ĝ ˆf ˆp k ĝ ˆf ˆq r 1 ĝ ˆf ˆq r k Up to a sign, this is the same expression as the expression on the fourth line of the expansion 18. We substitute into 17 for Bm ĥ δ 1ˆp k ĝ ˆf ˆq r1... ĝ ˆf ˆq rk the combination and also substitute for δ 1ˆq m according to 15: δ 1 ˆψˆϕ m = Bm ĝ ˆf ˆp k ĝ ˆf ˆq r1 m Am ĝ ˆf ˆq m 1 ĝ ˆf ˆq k 1 i 1 δ 1 1 m u δ l 1 n k ĝ ˆf ˆq 1 δ m Bmĝ ˆf 1 ˆp k ĝ ˆf ˆq r1

24 290 J. KOPŘIA Bm Bm Bm k ˆp k ĝ ˆf ˆq r1 k r u v=1 ˆp k ĝ ˆf ˆq r1 A r u ĝ ˆf ˆq ru 1 v 1 ĝ ˆf ˆq k 1 i 1 δ 1 1 ru v k ˆp k ĝ ˆf ˆq r1 ĝ ˆf ˆq 1 δ ru. This completes the proof. δ l 1 ru k Lemma The p-kernels in Definition 3.4 and the q-kernels in Definition 3.11 fulfill δ k ˆψˆϕ r1 ˆψˆϕ rk = ˆp k ĝ ˆf ˆq r1 Bm for all m 2. Proof. By 13, we have Bm Bm ˆp k ĝ ˆf ˆq r1 = Bm B k δ k h ˆp r 1 h ˆp r k ĝ ˆf ˆq r1. Taking into account that ĝ, ˆf and ˆq m are of degree 0, the last display equals to δ k h ˆp ĝ ˆf r ˆq 1 r1 ĝ ˆf ˆq rr 1 Bm B k h ˆp r k ĝ ˆf ˆq rrk 1 1 and the summation over the terms δ k r1 rk in all possible indices j denoting a map j gives [ δ k ĝ ˆf ˆq r1 ˆp k ĝ ˆf ˆq r 1 ĝ ˆf ˆq rk Bm B r 1 ĝ ˆf ˆq rk B r k ˆp k ĝ ˆf ˆq r 1 ]. However this is already 16 composed with the suspension, and the proof is complete. Because the formula for the q-kernels in Lemma 3.9 was based on the assumption 15, we have to prove that it is fulfilled by the q- kernels in Definition 3.11.

25 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 291 Theorem The p-kernels in Definition 3.4 and the q-kernels in Definition 3.11 fulfill 15, i.e. q n 1 n q n 1 1 n u Bn 1 ϑr1,...,r k p k gf q r1 gf q rk 1 il1n q k 1 i 1 An µ l 1 n k q 1 µ n = 0. This means that the objects introduced in 9 solve the problem of the transfer of A structure. Proof. We shall prove an equivalent assertion: δ 1ˆq n Bn ˆp k ĝ ˆf ˆq r1 = ˆq n 1 δ 1 1 n u ˆq k 1 i 1 δ l 1 n k ˆq 1 δ n, An with suspended q-kernels given by Definition 3.11: 20 ˆq n = Cn δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆq ri. The proof goes by induction on n: for n = 2, we have by 7 for n = 2 and 20: δ 1ˆq 2 = δ 1 δ2 ĝ ˆf ĥ δ 2ĥ 1 = δ 2 δ 1 1 ĝ ˆf ĥ δ 21 δ 1 ĝ ˆf ĥ δ 2 δ 1 1 ĥ 1 δ 2 1 δ 1 ĥ 1. By the Koszul sign convention δ 2 δ 1 1 ĝ ˆf ĥ = 1 δ1 ĥ δ 2 ĝ ˆf ĥδ 1 1, δ 2 1 δ 1 ĝ ˆf ĥ = δ 2ĝ ˆf ĝ ˆf 1 ĥδ 1 = δ 2 ĝ ˆf ĝ ˆf δ 2 ĝ ˆf 1 δ 2 ĝ ˆf ĥ1 δ 1, δ 2 δ 1 1 ĥ 1 = δ 2 ĝ ˆf 1 ĥδ 1 1 = δ 2 ĝ ˆf 1 δ δ 2 ĥ 1 δ 1 1, δ 2 1 δ 1 ĥ 1 = 1 δ1 ĥ δ 2 ĥ 1 1 δ 1,

26 292 J. KOPŘIA where 1 δ1 ĥ = 1 is a consequence of ĥ = δ 1 = 1, and so δ 1ˆq 2 = δ 2 ĝ ˆf ĥδ 1 1 δ 2 ĥ 1 δ 1 1 δ 2 ĝ ˆf ĥ1 δ 1 δ 2 ĥ 1 1 δ 1 δ δ 2 ĝ ˆf ĝ ˆf = ˆq 2 δ 1 1 ˆq 2 1 δ 1 ˆq 1 δ 2 ˆp 2 ĝ ˆf ĝ ˆf. The induction step is divided into three steps: I. We first expand the term δ 1ˆq n : by 20 δ 1ˆq n = Cn δ 1 δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆq ri = k δ k 1 δ 1 1 k u Cn ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆq ri δ k 1 i 1 δ l 1 k i Cn A k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆq ri. The first summation can be rewritten as Cn k δ k 1 δ 1 1 k u ˆψˆϕ r1... ˆψˆϕ ri 1 ĥ ˆq ri Q1.1 = i 1 δ k ˆψˆϕ r1... δ 1 ˆψˆϕ ru... ˆψˆϕ ri 1 ĥ ˆq ri Cn Q1.2 Q1.3 Cn δ k ˆψˆϕ r1... ˆψˆϕ ri 1 δ 1 ĥ ˆq ri 1 δ1 ĥ Cn while the second as Cn A k k u=i1 δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆq ri 1 δ k 1 i 1 δ l 1 k i δ 1 1 k u i, ˆψˆϕ r1 ˆψˆϕ ĥ ˆq ri 1 ri 1 k i

27 Q2.1 Q2.2 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 293 = δ k ˆψˆϕ r1 δ ˆψˆϕ ˆψˆϕ Cn ˆψˆϕ ri 1 ĥ ˆq ri Cn δ k ˆψˆϕ r1 δ ˆψˆϕ ˆψˆϕ ri 1 ĥ ˆq ri Q2.3 1 k 1 1 δ l ĥ Cn A k i ĥ ˆq ri 1 i 1 δ l 1 k i. δ k iˆψˆϕ r1 ˆψˆϕ ri 1 The summation over all indices in Q2.1 terms of the form δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆq ri leads to Q2.1 = Cn i 1 δ k ˆψˆϕ r1 B r u ˆψˆϕ ri 1 ĥ ˆq ri δ k ˆψˆϕ r ˆψˆϕ rk 1 Analogously, the summation over all indices in Q2.2 terms of the form δ k ˆψˆϕ r k j gives Q2.2 = δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ˆq ri. Cn r i>1 II. By Lemma 3.14: Q2.1 = Cn i 1 δ k ˆψˆϕ r1 ˆp k ĝ ˆf ˆq r 1 ĝ ˆf ˆq rk B r u ˆψˆϕ ri 1 ĥ ˆq ri and Lemma 3.13 for ˆψˆϕ m Definition 3.11 and definition of Cn in C imply that m is strictly less than n, so that assumptions of Lemma 3.13 are fulfilled by our induction hypothesis gives., Q1.1 Q2.1 = Q a i 1 δ k ˆψˆϕ r1 1 ĥ δ1 ˆψˆϕ ru δ 1 ˆψˆϕ ri 1 ĥ ˆq ri 1 k i Cn r Cn i 1 r i δ k ˆψˆϕ r ĥ δ1 ˆψˆϕ ri 1 r u>1 δ 1 1 ri u Q b ˆψˆϕ ri 1 ĥ ˆq ri

28 294 J. KOPŘIA Cn i 1 δ k ˆψˆϕ r1 1 ĥ δ l ˆψˆϕ k 1 i 1 r u>1 A r u δ l 1 ru k Q c ˆψˆϕ ri 1 ĥ ˆq ri Q d i 1 δ k ˆψˆϕ r1 1 ĥ δru ˆψˆϕ 1 δ ru ˆψˆϕ ri 1 ĥ ˆq ri 1 k i Cn r u>1 Cn i 1 δ k ˆψˆϕ r1 1 ĥ δru ˆψˆϕ 1 δ ru ˆψˆϕ ri 1 ĥ ˆq ri 1 k i r u>1 Q2.1 i 1 Cn r u>1 δ k ˆψˆϕ r ĥ δru ˆψˆϕ 1 δ ru... ˆψˆϕ ri 1 ĥ ˆq ri 1 k i, where the first five terms come from Q1.2 by application of Lemma 3.13, and the fifth one cancels out when combined with Q2.1. Recall that we have δ l = 1 for all l, and so 1 ĥ δl = 1 as well as Q1.2 Q2.2 = δ k ˆψˆϕ r1 ˆψˆϕ ri 1 δ 1 ĥ 1 ˆq ri Cn,r i>1 = Cn,r Cn,r i>1 Cn,r δ k ˆψˆϕ r1 ˆψˆϕ ri 1 δ 1 ĥ ˆq ri δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥδ 1 ĝ ˆfˆq ri δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥδ 1 ĝ ˆf 1 ˆq ri. Thanks to the induction hypothesis we substitute for δ 1ˆq and the last display turns into δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆp k ĝ ˆf ˆq r 1 ĝ ˆf ˆq r Cn,r i>1 Cn,r i>1 B r i δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĝ ˆf ˆq ri Q a r i δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ˆq ri 1 Cn,r i>1 δ 1 1 ri u 1 k i k

29 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 295 Q b δ k ˆψˆϕ r1 ˆψˆϕ ri 1 Cn,r i>1 Q c Cn,r Cn,r A r i ˆq k 1 i 1 δ l 1 ri k δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆq ri δ 1 δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĝ ˆf 1 ˆq ri. ˆq 1 δ ri 1 k i The non-numbered terms first, second and sixth can be further simplified. We notice δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĥ ˆp k ĝ ˆf ˆq r 1 ĝ ˆf ˆq r 1 k i k Cn,r i>1 Cn,r i>1 Cn,r = Cn Cn = Bn B r i δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĝ ˆf ˆq ri δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ĝ ˆf 1 ˆq ri δ k ˆψˆϕ r1 ˆψˆϕ ri 1 ˆψˆϕ ri δ k ˆψˆϕ r1 ˆψˆϕ ri 1 1 δ k ˆψˆϕ r1 ˆψˆϕ rk δn 1 n. By Lemma 3.14, this expression equals to Q d Bn ˆp k ĝ ˆf ˆq r1 ˆq 1 δ n. III. In the last step we pair various contributions together: the first step can be written as δ 1ˆq n =Q1.1 Q1.2 Q1.3 Q2.1 Q2.2 Q2.3, while the second step as Q1.1 Q2.1=Q a Q b Q c Q d and Q1.2 Q2.2=Q a Q b Q c Q d.

30 296 J. KOPŘIA Taken altogether, Q1.3 Q a Q b Q a Q c = ˆq n 1 δ 1 1 n u, Q2.3 Q c Q d Q b = An ˆq k 1 i 1 δ l 1 n k, Q d = Bn ˆp k ĝ ˆf ˆq r1 ˆq 1 δ n. The proof is complete. 4. Homotopy transfer and the homological perturbation lemma In the present section we discuss a motivation to find explicit formulas for the transfer of A algebra structure presented in an apparently arbitrary form in 9. In the following, we recall the homological perturbation lemma and show that it gives a recipe to search for the transfer problem exactly in the form 9. This is the approach with which we develop and formalize [7, Remark 4]. Lemma 4.1 Homological perturbation lemma, [1]. Let, and W, W be chain complexes together with quasi-isomorphisms f : W and g : W such that gf 1 = h h for a linear map h :. Let µ : be a linear map of the same degree as such that µ 2 = 0 and the linear map 1 µh is invertible µ is called in this context perturbation. We define 21 ν = W fag, ψ = g hag, ϕ = f fah, H = h hah, where A = 1 µh 1 µ. Then, µ and W, ν are chain complexes and ϕ:, ψ : W W their quasi-isomorphisms with ψϕ 1 = µh H µ. In our case, on, we have an additional A structure given by a collection of multilinear maps µ = µ 2, µ 3,... fulfilling certain axioms. In order to regard µ as a perturbation, we have to pass to the suspended tensor algebra generated by. Let us consider T s with a coderivation δ and T sw with a coderivation δ W, ˆF and Ĝ morphisms and Ĥ a homotopy between Ĝ ˆF and the identity on T s. Here δ is given by components {s ω : s s } {0: s n s } n 2 in the sense of Theorem 2.6, and it is codifferential by Lemma 2.7 because is a differential on. Analogous conclusions do apply to δ W. The map ˆF : T s, δ T sw, δ W is given by components {s f ω : s sw } {0: s n sw } n 2 Lemma 2.9. By Lemma 2.10, ˆF is a morphism f is a map of chain complexes, i.e. ˆF s n = ˆf n for ˆf = s f ω. Analogous conclusions apply to Ĝ as well. Homotopy Ĥ : T s T s is a map given by {ĝ ˆf : s s } {0 : s n s } n 2 on the left, {s h ω : s s } {0: s n s } n 2 in the middle and

31 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 297 {1 : s s } {0: s n s } n 2 on the right in the sense of Theorem 2.5. Because h is a homotopy between gf and 1, Ĥ is a homotopy between Ĝ ˆF and the identity on T s according to Theorem 2.13; the notation is ĥ = s h ω. Let δ µ be a coderivation on T s corresponding to µ, whose components are given by {0: s s } {s µ n ω n : s n s } n 2 in the sense of Theorem Because and µ form an A structure on, δ δ µ 2 = 0 by Theorem 2.16; we use the notation δ n = s µ n ω n. The remaining assumption in Lemma 4.1 is the invertibility of the map 1 δ µĥ. We know Ĥ s n = ĝ ˆf i ĥ 1 j ij=, i,j 0 so that Ĥs n s n for all n 1, and also δ µ s = 0 implies δ µ s n = δ n An 1 i 1 δ l 1 n k for all n 2 with An as in A. Consequently, for all n 2 holds δ µ s n s s, and its iteration results in δ µĥ s n s, δ µĥn s n = 0. By previous discussion and in accordance with Remark 2.3, [1], 1 δ µĥ 1 s s n = 1 δ µĥn, which means that 1 δ µĥ is invertible. Now all assumptions of Lemma 4.1 are fulfilled and we can write δ W δ ν = δ W ˆF δ µ Ĥδ µ n Ĝ, ˆψ = Ĝ Ĥ δ µ Ĥδ µ n Ĝ, n 0 n 0 ˆϕ = ˆF ˆF δ µĥn, ĤH = Ĥ Ĥ δ µĥn. n 1 Here we see immediately the motivation for 9: δ µ n 0 Ĥδ µ n corresponds to the ˆp-kernels and n 1 δ µĥn corresponds to the ˆq-kernels. For our purposes it is more convenient to write δ W δ ν = δ W ˆF δ µĝ ˆF δ µĥn δ µĝ, 22 n 1 n 1 ˆψ = Ĝ Ĥδ µĝ Ĥ δ µĥn δ µĝ, n 1 ˆϕ = ˆF ˆF δ µĥ ˆF δ µĥn δ µĥ, n 1 ĤH = Ĥ Ĥδ µĥ Ĥ δ µĥn δ µĥ. n 1

32 298 J. KOPŘIA There is a drawback related to these formulas, however: by a direct inspection we see that δ W δ ν is not a coderivation in the sense of Theorem 2.6, ˆϕ and ˆψ do not define a morphism in the sense of Lemma 2.9, and ĤH does not fulfill the first part of morphism definition in the sense of Theorem 2.5. In what follows we prove that on the additional assumptions see [7, Remark 4]: 23 ˆfĝ = 1, ˆfĥ = 0, ĥĝ = 0, ĥĥ = 0, the homological perturbation lemma gives the results compatible with Section 3. Lemma 4.2. Let us assume the formulas in 23 are satisfied. Then 1 ˆq n ĝ n = 0 for n 2, 2 ˆq i1j ĝ ˆf i ĥ 1 j = 0 for all i, j 0, i j 1. Proof. 1: The proof goes by induction. By definition ˆq 2 = δ 2 ĝ ˆf ĥδ2ĥ 1 for n = 2, so that ˆq 2 ĝ 2 = δ 2 ĝ ˆfĝ ĥĝ δ2ĥĝ ĝ and the claim follows from 23 ĥĝ = 0. We assume the assertion is true for all natural numbers less than n N n 2. By definition ˆq n ĝ n = Cn δ k [[ˆψˆϕ] r1 ĝ r1 [[ˆψˆϕ] ri 1 ĝ ri 1 ĥ ˆq ri ĝ ri ĝ k i, where 24 [[ˆψˆϕ] m = ĝ ˆf ˆq m Bm ĥ ˆp k ĝ ˆf ˆq r1 with [[ˆψˆϕ] 1 = ĝ ˆf. In the case r i > 1, the composition ĥ ˆq ri ĝ ri is trivial by the induction hypothesis. If r i = 1, ĥ ˆq 1 ĝ = ĥĝ is trivial by 23. 2: The proof is by induction on n = i 1 j. For n = 2 we prove ˆq 2 ĥ 1 = 0, ˆq 2 ĝ ˆf ĥ = 0. As we know ˆq 2 ĥ 1 = 1 ĥ ĥ δ 2 ĝ ˆfĥ ĥ δ 2ĥĥ 1 and ˆq 2 ĝ ˆf ĥ = δ 2 ĝ ˆfĝ ˆf ĥĥ δ2ĥĝ ˆf ĥ, the claim follows thanks to 23. Let the claim hold for m 2 and all natural numbers less than n, we prove it is true for n. First of all, for n > i j 1 2 we have [[ˆψˆϕ] i 1j ĝ ˆf i ĥ 1 j = 0 and also ĝ ˆf ˆq 1 ĥ = ĝ ˆf ĥ = 0. By definition [[ˆψˆϕ] i 1j ĝ ˆf i ĥ 1 j Bi 1j = ĝ ˆf ˆq i 1j ĝ ˆf i ĥ 1 j ĥ ˆp k ĝ ˆf ˆq r1 ĝ ˆf i ĥ 1 j,

33 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 299 and by induction hypothesis ˆq i i 1j ĝ ˆf ĥ 1 j = 0. The last summation can be conveniently rewritten as ĥ ˆp k ĝ ˆf ˆq r1 ĝ ˆf i ĥ 1 j Bi 1j = Bi 1j ĥ ˆp k ĝ ˆf ˆq r1 ĝ ˆf r1 ĝ ˆf ˆq ru ĝ ˆf ĥ 1, and the induction implies ˆq ru ĝ ˆf ĥ 1 = 0 for r u > 1. We already showed ĝ ˆf ˆq ru ĝ ˆf ĥ 1 = ĝ ˆf ˆq 1 ĥ = 0 for r u = 1. We now return back to the main thread of the proof and show ˆq n ĝ ˆf i ĥ 1 j = 0. We consider k, i, r 1,..., r i 1, r i in Cn given by C, and compute δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri 1 k i ĝ ˆf i ĥ 1 j. After substitution for [[ˆψˆϕ], there are the following three possibilities for indices i a i : i < r 1 r i 1: Then there exist 1 u < i such that [[ˆψˆϕ] ru ĝ ˆf ĥ 1. For r u 2 we already proved [[ˆψˆϕ]] ru ĝ ˆf ĥ 1 = 0, for r u = 1 we have [[ˆψˆϕ] ru ĝ ˆf ĥ 1 = ĝ ˆf ĥ = 0. r 1 r i 1 i < r 1 r i : In the tensor product there is a term of Cn the form ĥ ˆq ĝ ˆf ri ĥ 1, which is by the induction hypothesis 0 for r i > 1. If r i = 1, then ĥ ˆq ĝ ˆf ri ĥ 1 = ĥĥ equals to 0 by 23. r 1 r i i: In this case we get in the tensor product the term of the form ĥ ˆq ĝ ˆf r i ri = ĥ ˆq ri ĝ r i ˆf r i, which is trivial for r i 2 by 1 of the lemma. If r i = 1, then ĥ ˆq ĝ ˆf r i ri = ĥ ĝ ˆf equals to zero again by 23. Because k, i, r 1,..., r i 1, r i in Cn was chosen arbitrarily, we get δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri 1 k i ĝ ˆf i ĥ 1 j = 0, and so finally ˆq n ĝ ˆf i ĥ 1 j = 0. Remark 4.3. We easily observe: 1 For all n 2 and for linear mappings {a n : s n s } n 1, Bn a r1 a rk = Bn, r k >1 u=2 n 3 a r1 a rk a u a n u 1 a n 1, Bn u, r k >1 a r1 a rk a u 1

34 300 J. KOPŘIA where Bn given as in B, 2 For all n 2, we have and if ĥ ˆp 1 = 1 3 For all n 1 and 0 u n 1: [[ˆψˆϕ] n ĝ n = ĥ ˆp n ĝ n, Definition 3.4 the formula is true for n = 1 as well. [[ˆψˆϕ] n ˆfĝ u ĥ 1 u = 0. Lemma 4.4. Let us assume 23 is true for n 2. Then 25 ˆp n ĝ n = δ n ĝ n ˆq n i1 i=2 n i ĝ n. Proof. The proof goes by induction on n. As for n = 2 we have ˆp 2 = δ 2, hence the claim follows. We now assume the assertion holds for all natural numbers greater than 1 and less than n. Let us consider 2 m < n and k,i,r 1,..., r i 1, r i as given in Cm, so that δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri ĝ u δ l ĝ m 1 u = 0 whenever u < r 1 r i 1 or r 1 r i u because ĥ ˆq ri ĝ ri = 0 for all r i 1 by Lemma 4.2. We fix n 1 k 2, k i 1 and r 1,..., r i 1 1 as in C. As follows from the previous observation, all terms in ˆq n i1 i=2 n i ĝ n are of the form δ k [[ˆψˆϕ]] r1 ĝ r1 [[ˆψˆϕ]] ri 1 ĝ ri 1 ĝ k i with representing a mapping s s the ˆq-kernels are given by 20. We can rewrite them in the form δ k [[ˆψˆϕ] r1 ĝ r1 [[ˆψˆϕ] ri 1 ĝ ri 1 [ n 1 δ n ĝ r1 i=2 ˆq n i1 n i δ i 1 n i u ĝ n ] ĝ k i, where n = n i k r 1 r i 1, n > 1. Applying the second point of Remark 4.3 to [[ˆψˆϕ] ĝ, the inducing hypothesis reduces the last display to 26 δ k ĥ ˆp r1 ĝ r1 ĥ ˆp ri 1 ĝ ri 1 ĥ ˆp n ĝ n ĝ k i

35 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 301 we write ĝ = ĥ ˆp 1 ĝ. By the first point of Remark 4.3, i=2 ˆq n i1 = Cn,r i>1 = Bn,k n n i ĝ n δ k ĥ ˆp r1 ĝ r1 ĥ ˆp ri 1 ĝ ri 1 ĥ ˆp ri ĝ ri ĝ k i δ k ĥ ˆp r1 ĝ r1 ĥ ˆp rk ĝ r k, so that for each term in the sum there exists u, r u > 1 they are of the form of terms in 26 with n > 1. Adding the remaining term δ n ĝ n and using the formula 13 for the ˆp-kernels, the proof concludes. Lemma 4.5. Let us assume 23, and also 27 ˆq n = δ n Ĥ s n ˆq n i1 i=2 to be true for all 2 m n. Then n i [[ˆψˆϕ] n ĥ ˆp n ĝ ˆf n = [[ˆψˆϕ] 1 δ n Ĥ s n [[ˆψˆϕ] n i1 i=2 n i Proof. By 24, we have for all m 2 Ĥ s n Ĥ s n. [[ˆψˆϕ] m = ĝ ˆf ˆq m Bm ĥ ˆp k ĝ ˆf ˆq r1, and [[ˆψˆϕ] 1 = ĝ ˆf. We can split [[ˆψˆϕ] 1 δ n Ĥ s n i=2 in two components and write [[ˆψˆϕ] n i1 n i Ĥ s n ĝ ˆf ˆq 1 δ n Ĥ s n ĝ ˆf ˆq n i1 i=2 i=2 Bn i1 n i n i Ĥ s n ĥ ˆp k ĝ ˆf ˆq r1 Ĥ s n.

36 302 J. KOPŘIA Because ˆq n = 1, we have 28 = ĝ ˆf ˆq n thanks to 27. As for the second component 29, consider k, r 1,..., r k Bn i1 for some i 2 with Bn i1 as in B. Then ĝ ĥ ˆp k ˆf ˆq r1 ĝ ˆf n i ˆq rk = Ĥ s n k ĥ ˆp k ĝ ˆf ˆq r1 ĝ ˆf r1 ĝ ˆf ˆq ru 1 ĝ [ ĝ ˆf ˆq ru r u 1 v=0. 1 v ˆf ru 1 ] δ i 1 ru 1 v Ĥ s ru 1i The reason for the appearance of such terms is that when r 2 and ĥ were in any other ˆq-kernel than δ i, we would get ˆq r gf h 1 which is trivial by Lemma 4.2. If r = 1, we get ĝ ˆf ˆq 1 ĥ = 0 because ˆq 1 = 1 and ˆfĥ = 0 by 23. Thus we have for i 2: Bn i1 ĝ ĥ ˆp k ˆf ˆq r1 ĝ ˆf n i ˆq rk = Bn i1 n i 1 r 1=1 ĥ ˆp k ĝ ˆf ˆq r1 Bn i1 u Ĥ s n ĥ ˆp uk ĝ ˆf ˆq 1 u ĝ ˆf ˆq r1 ĥ ˆp n r11 ĝ ˆf ˆq 1 n r1 ĝ ˆf ˆq r1, where ĝ ˆf ˆq r1 = ĝ ˆf ˆq r1 r 1 1 v=0 1 v δ i 1 r1 1 v Ĥ s r 1 1i. We notice ˆq m ĝ ˆf m 0 if and only if m = 1 cf., Lemma 4.2. Therefore, we expand the second contribution into 29 = i=2 Bn i1 i=2 n i 1 i=2 r 1=1 ĥ ˆp k ĝ ˆf ˆq r1 Bn i1 u ĥ ˆp uk ĝ ˆf ˆq 1 u ĝ ˆf ˆq r1 ĥ ˆp n r11 ĝ ˆf ˆq 1 n r1 ĝ ˆf ˆq r1

37 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 303 and for fixed k, i, r 2,..., r i sum up all terms of the form ĥ ˆp k ĝ ˆf ˆq 1 k i ĝ ˆf ˆq ĝ ˆf ˆq r2 ĝ ˆf ˆq ri in 29: ĥ ˆp k ĝ ˆf ˆq 1 k i r r 1=1 where r = n k i r 2 r i. We recall ĝ ˆf ˆq r1 = ĝ ˆf ˆq r1 and use 27 to get r1 1 1 u ĝ ˆf ˆq r1 ĝ ˆf ˆq r2 ĝ ˆf ˆq ri, δ r r 11 1 r1 1 u Ĥ s r 1 1i 30 ĥ ˆp k ĝ ˆf ˆq 1 k i ĝ ˆf ˆq r ĝ ˆf ˆq r2 ĝ ˆf ˆq ri. Clearly r > 1, and the summation over all terms in 29 leads to 29 = ĥ ˆp k ĝ ˆf ˆq r1 We conclude Bn,r 1>1 n 3 Bn u,r 1>1 ĥ ˆp k ĝ ˆf ˆq 1 u ĝ ˆf ˆq r2 ĥ ˆp n r1 ĝ ˆf ˆq 1 n r ĝ ˆf ˆq r. r=2 29 = Bn,k n ĥ ˆp k ĝ ˆf ˆq r1, because all terms are as those in 30 and there is always at least one u such that r u > 1 this is equivalent to r > 1 in 30. Recall we started with [[ˆψˆϕ] 1 δ n Ĥ s n [[ˆψˆϕ] n i1 and showed i= = ĝ ˆf ˆq n n i = Bn,k n Ĥ s n ĥ ˆp k ĝ ˆf ˆq r1. Taking into account the definition of [[ˆψˆϕ] n in 24, the desired conclusion follows immediately. Lemma 4.6. Let us assume 23 to be true. Then for all n 2 31 ˆq n = δ n Ĥ s n ˆq n i1 i=2 n i Ĥ s n.

38 304 J. KOPŘIA Proof. The proof is by the induction hypothesis on n. For n = 2, by 31 we have δ 2 ĝ ˆf ĥ δ 2ĥ 1 = δ 2 ĝ ˆf ĥ ĥ 1 which is certainly true. We assume the claim is true for all natural numbers greater than 1 and strictly less than n. Let us consider 2 j < n and k,i,r 1,..., r i 1, r i as given in Cn j 1. The same reasoning as in Lemma 4.5 leads to 32 δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri n j δ j 1 n j u Ĥ s n = δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri ĥ δ k ˆp r1 ĝ ˆf r1 [[ˆψˆϕ] r2 [[ˆψˆϕ] ri 1 ĥ ˆq ri ĥ δ k ˆp r1 ĝ ˆf r1 ĥ ˆp ri 2 ĝ ˆf ri 2 [[ˆψˆϕ] ri 1 ĥ ˆq ri δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 1 ĝ ˆf ri 1 ĥ ˆq ri δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 1 ĝ ˆf ri 1 ĥ ˆq ri Ĥ s k i. Hereby we expanded a general summand in the definition of ˆq n j1 as in 20, where [[ˆψˆϕ] rl ĥ ˆq ri ĥ ˆq ri r l 1 = [[ˆψˆϕ] r1 r = ĥ ˆq i 1 ri r = ĥ ˆq i 1 ri δ j 1 r l 1 u Ĥ s r i 1j, δ j 1 ri 1 u Ĥ s r i 1j, δ j 1 ri 1 u ĝ ˆf ri 1j. In the previous formulas there are no signs whatsoever, because ĥ ˆq ri pass through the terms of degree 0, and ĥ in Ĥ and δ j are of degree 1 and 1, respectively, so that their sign contributions cancel out. In the next few steps we show how the terms are organized: I. Let us choose k, i, r 1,..., r i given in Cn such that r i > 1, and sum up all terms of the form δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 1 ĝ ˆf ri 1 ĥ ˆq r out of the summation δ n Ĥ s n ˆq n i1 i=2 n i Ĥ s n

39 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 305 for all allowable r. We get where δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 1 ĝ ˆf r i ri 1 ĥ ˆq r, ĥ ˆq r = ĥ ˆq r r 1 Because r i < n, we get by the induction hypothesis r=1 δ r i r1 1 r 1 u Ĥ s r i. 33 δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 1 ĝ ˆf ri 1 ĥ ˆq ri If r i 1 > 1, we sum up all terms of the form δ k ĥ ˆp r1 ĝ ˆf r1 [[ˆψˆϕ] ĥ ˆq ri : with r ĥ δ k ˆp r1 ĝ ˆf i 1 r1 [[ˆψˆϕ] r ĥ ˆq ri, [[ˆψˆϕ] r = [[ˆψˆϕ] r r 1 r=1 δ r i 1 r1 1 r 1 u Ĥ s r i 1. Because r i < n, by the induction hypothesis is Lemma 4.5 fulfilled and the last display reduces to δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 2 ĝ ˆf ri 2 [ [[ˆψˆϕ] ri 1 ĥ ˆp ri 1 ĝ ˆf ri 1] ĥ ˆq ri The sum of the last display and 33 results in δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 2 ĝ ˆf ri 2 [[ˆψˆϕ] ri 1 ĥ ˆq ri which is the same expression as for r i 1 = 1 because [[ˆψˆϕ] ri 1 = ĥ ˆp ri 1 ĝ.., ˆf ri 1 in this case. Repeating this procedure, we arrive at δ k [[ˆψˆϕ]] r1 [[ˆψˆϕ]] ri 1 ĥ ˆq ri. We summarize the previous considerations: for k, i, r 1,..., r i as in Cn such that r i > 1, we have δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri = δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ri 1 ĝ ˆf r i ri 1 ĥ ˆq r r=1 r=1 r δ k ĥ ˆp r1 ĝ ˆf i 1 r1 [[ˆψˆϕ] ri i ĥ ˆq ri

40 306 J. KOPŘIA 34 δ k ĥ ˆp r1 ĝ ˆf r1 r 2 r=1 [[ˆψˆϕ] ri i ĥ ˆq ri δ k r 1 r=1 [[ˆψˆϕ] r [[ˆψˆϕ] r [[ˆψˆϕ] r2 [[ˆψˆϕ] ri i ĥ ˆq ri. II. Let us choose k, i, r 1,..., r i as in Cn such that i > 1, r i = 1, and there exists 1 u i 1 such that r u > 1 and r u1 = = r i 1 = 1. Then δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ru ĝ ˆf i u1 ĥ 1 k i r 1 = δ k [[ˆψˆϕ] r [[ˆψˆϕ] r2 [[ˆψˆϕ] ru ĝ ˆf i u1 ĥ 1 k i 35 r=1 δ k ĥ ˆp r1 ĝ ˆf r1 r 2 r=1 ĝ ˆf i u1 ĥ 1 k i [[ˆψˆϕ] r [[ˆψˆϕ] ru δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ru 1 ĝ ˆf ru 1 ĝ ˆf i u1 ĥ 1 k i δ k ĥ ˆp r1 ĝ ˆf r1 ĥ ˆp ru 1 ĝ ˆf ru 1 ĝ ˆf i u1 ĥ 1 k i r u r=1 r u r=1 [[ˆψˆϕ] r ĥ ˆq r with ĥ ˆq r = ĥ ˆq r r 1 v=0 1 v δ r u r1 1 r 1 v ĝ ˆf ru. By Lemma 4.4 r u r=1 ĥ ˆq r = ĥ ˆp ru ĝ ˆf ru, which can be justified in the same way as in the first step I. We expand all terms in the summation denoted 32 ˆq n i1 i=2 n i Ĥ s n

41 ON THE HOMOTOPY TRANSFER OF A STRUCTURES 307 and use 34 a 35 to rewrite terms in the definition of ˆq n : ˆq n i1 i=2 Certainly, n i = Cn,k n δ n Ĥs n = Cn,k=n Ĥs n δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri. δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri, which together with 20 completes the proof. Remark 4.7. Adopting slight changes in the proofs, our claims can be reformulated as follows: 36 Lemma 4.4: On the assumption 23 holds for all n 2 Bn ĥ ˆp r1 ĝ r1 ĥ ˆp rk ĝ r k = ĝ n i=2 Cn i1 n i [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri ĝ n, where we write ĥ ˆp r1 ĝ r1 ĥ ˆp rk instead of δ k ĥ ˆp r1 ĝ r1 ĥ ˆp rk and [[ˆψˆϕ]] r1 [[ˆψˆϕ]] ri 1 ĥ ˆq ri instead of δ k [[ˆψˆϕ] r1 [[ˆψˆϕ] ri 1 ĥ ˆq ri ; Lemma 4.5: On the assumption 23 holds for all n 2 ˆf ˆq n Bn ˆf ˆq r1 ˆf ˆq rk ˆf n = ˆf δ n Ĥ s n 37 ˆf ˆq n i1 i=2 n i Bn i1 Ĥ s n, ˆf ˆq r1 ˆf ˆq rk where we write ˆf ˆq r1 ˆf ˆq rk instead of ĥ ˆp k ĝ ˆf ˆq r1 ĝ ˆf ˆq rk ;