1 Introduction

Let \(A\) and \(B\) be augmented dgas over a commutative ring \(\Bbbk \). Recall that an \(A_{\infty }\) map \(f:A\Rightarrow B\) is a map of differential graded coalgebras \(\mathbf {B}A\rightarrow \mathbf {B}B\) between the bar constructions of \(A\) and \(B\). According to Stasheff–Halperin [14, Def. 8], the dga \(A\) is a strongly homotopy commutative (shc) algebra if

(i):

the multiplication map \(\mu _{A}:A\otimes A\rightarrow A\) extends to an \(A_{\infty }\) morphism

$$\begin{aligned} \Phi :A\otimes A \Rightarrow A \end{aligned}$$

in the sense that the base component \(\Phi _{(1)}:A\otimes A\rightarrow A\) of \(\Phi \) equals \(\mu \).

Munkholm [12, Def. 4.1] additionally requires the following:

(ii):

The map \(\eta _{A}:\Bbbk \rightarrow A\), \(1\mapsto 1\) is a unit for \(\Phi \), that is,

$$\begin{aligned} \Phi \circ (1_{A}\otimes \eta _{A}) = \Phi \circ (\eta _{A}\otimes 1_{A}) = 1_{A}. \end{aligned}$$
(iii):

The \(A_{\infty }\) map \(\Phi \) is homotopy associative, that is,

$$\begin{aligned} \Phi \circ (\Phi \otimes 1_{A}) \simeq \Phi \circ (1_{A}\otimes \Phi ):A\otimes A\otimes A\Rightarrow A. \end{aligned}$$
(iv):

The map \(\Phi \) is homotopy commutative, that is,

$$\begin{aligned} \Phi \circ T_{A,A} \simeq \Phi :A\otimes A\Rightarrow A. \end{aligned}$$

Note that we write \(1_{A}\) for the identity map of \(A\) and \(T_{A,A}:A\otimes A\rightarrow A\otimes A\) for the the transposition of factors. Also, the compositions and tensor products above are those of \(A_{\infty }\) maps.

Using acyclic models, Munkholm [12, Prop. 4.7] has constructed a natural shc structure on the normalized singular cochain complex \(C^{*}(X)\) of a space \(X\). Slightly earlier, Gugenheim–Munkholm [6] have given a recursive formula for \(\Phi \) in this case based on the Eilenberg–Zilber contraction, see Remark 4.2.

Singular cochain complexes are the main example of homotopy Gerstenhaber algebras (hgas) besides the Hochschild cochains of associative algebras. Recall that an hga is essentially an augmented dga \(A\) with maps

$$\begin{aligned} E_{k}:A\otimes A^{\otimes k}\rightarrow A \end{aligned}$$

for \(k\ge 1\) that induce a dga structure on the bar construction \(\mathbf {B}A\) compatible with the diagonal. See Sect. 3.1 for a reformulation in terms of the identities the operations \(E_{k}\) have to satisfy. Kadeishvili [8] has identified certain additional operations on an hga \(A\) that allow to define a \(\mathbin {\cup _1}\)-product on \(\mathbf {B}A\). We call such an hga extended, see Sect. 3.2. Singular cochain algebras are extended hgas.

As pointed out by Kadeishvili [9, Sec. 1.6], it follows from general considerations that any hga admits an shc structure in the sense of Stasheff–Halperin such that the composition of \(\mathbf {B}\Phi \) with the shuffle map,

$$\begin{aligned} \mathbf {B}A \otimes \mathbf {B}A {\mathop {\longrightarrow }\limits ^{\nabla }}\mathbf {B}(A\otimes A) \xrightarrow {\mathbf {B}\Phi } \mathbf {B}A, \end{aligned}$$
(1.1)

is homotopic to the product on \(\mathbf {B}A\) determined by the hga structure.

Our main result is the following. In the companion paper [4] we apply it to determine the cohomology rings of a large class of homogeneous spaces.

Theorem 1.1

Let \(A\) be an hga.

(i):

There is a canonical shc structure on \(A\) satisfying properties (i), (ii) and (iii) of the definition above.

(ii):

If \(A\) is extended, then the shc structure additionally satisfies property (iv).

(iii):

All structure maps commute with morphisms of (extended) hgas.

(iv):

The composition (1.1) is exactly the multiplication map on \(\mathbf {B}A\).

The \(A_{\infty }\) map \(\Phi \) is defined in Sect. 4 and the associativity and commutativity homotopies in Sects. 5 and 6. The verification that they satisfy the required identities is an elementary, but very lengthy computation that has been relegated to the appendices. The claimed naturality will be obvious from the construction. We conclude in Sect. 7 with results about \(A_{\infty }\) maps and shc maps from polynomial algebras to dgas and hgas, respectively.

2 Preliminaries

From now on, we call \(A_{\infty }\) maps strongly homotopy multiplicative (shm) because this terminology pairs better with “shc algebras”. We refer to the companion paper [4, Secs. 2-4] for our general conventions, the definitions of twisting cochains, twisting cochain homotopies and the corresponding families as well as those of shm maps and their homotopies, compositions and tensor products. We work over an arbitrary commutative ring \(\Bbbk \) with unit.

We recall from  [4, Sec. 2] our “\({\mathop {=}\limits ^{\varkappa }}\)” notation which alleviates us from explicitly specifying signs coming from the Koszul sign rule. For example, if we define a map \(F:A\otimes B\otimes C\rightarrow A'\otimes B'\) between complexes by

$$\begin{aligned} F(a,b,c)&{\mathop {=}\limits ^{\varkappa }}f(c) \otimes g(a,b), \end{aligned}$$
(2.1)

then we mean

$$\begin{aligned} F(a,b,c)&= (-1)^{(|a|+|b|+|g|)|c|}\, f(c) \otimes g(a,b). \end{aligned}$$
(2.2)

Composition of maps is distributed over tensor products. For instance,

$$\begin{aligned} G(a,b) {\mathop {=}\limits ^{\varkappa }}f_{1}(f_{2}(a))\otimes g_{1}(g_{2}(b)) \end{aligned}$$
(2.3)

defines the element

$$\begin{aligned} G = f_{1}\,f_{2} \otimes g_{1}\,g_{2} = (-1)^{|f_{2}||g_{1}|}\,(f_{1}\otimes g_{1})\,(f_{2}\otimes g_{2}) \end{aligned}$$
(2.4)

in the endomorphism operad.

3 Homotopy Gerstenhaber algebras

Homotopy Gerstenhaber algebras were introduced by Voronov–Gerstenhaber [15, §8]. For the convenience of the reader, we reproduce the definition of an (extended) homotopy Gerstenhaber algebra from [4, Sec. 6].

3.1 Definition of an hga

A homotopy Gerstenhaber algebra (homotopy G-algebra, hga) is an augmented dga \(A\) with certain operations

$$\begin{aligned} E_{k}:A \otimes A^{\otimes k}\rightarrow A, \qquad a\otimes b_{1}\otimes \dots \otimes b_{k}\mapsto E_{k}(a;b_{1},\dots ,b_{k}) \end{aligned}$$
(3.1)

of degree \(|E_{k}|=-k\) for \(k\ge 1\). It is often convenient to use the additional operation \(E_{0}=1_{A}\). These operations satisfy the following properties.

  • (i)   All \(E_{k}\) with \(k\ge 1\) take values in the augmentation ideal \({\bar{A}}\) and vanish if any argument is equal to \(1\).

  • (ii)   

    $$\begin{aligned} d(E_{k})(a;b_{\bullet })&{\mathop {=}\limits ^{\varkappa }}b_{1}\,E_{k-1}(a;b_{\bullet }) + \sum _{m=1}^{k-1}(-1)^{m}\,E_{k-1}(a;b_{\bullet },b_{m}b_{m+1},b_{\bullet }) \\&\qquad + (-1)^{k}\,E_{k-1}(a;b_{\bullet })\,b_{k}. \end{aligned}$$

    for all \(k\ge 1\) and all \(a\), \(b_{1}\), ..., \(b_{k}\in A\).

  • (iii)   

    $$\begin{aligned} E_{k}(a_{1}a_{2};b_{\bullet }) {\mathop {=}\limits ^{\varkappa }}\!\!\sum _{k_{1}+k_{2}=k}\!\! E_{k_{1}}(a_{1};b_{\bullet })\,E_{k_{2}}(a_{2};b_{\bullet }) \end{aligned}$$

    for \(k\ge 0\) and all \(a_{1}\), \(a_{2}\), \(b_{1}\), ..., \(b_{k}\in A\), where the sum is over all decompositions of \(k\) into two non-negative integers.

  • (iv)    

    $$\begin{aligned}&E_{l}(E_{k}(a;b_{\bullet });c_{\bullet }) \\&\quad {\mathop {=}\limits ^{\varkappa }}\sum _{\begin{array}{c} i_{1}+\dots +i_{k}+{} \\ j_{0}+\dots +j_{k}=l \end{array}} \!\!(-1)^{\varepsilon }\, E_{n}\bigl (a;\underbrace{c_{\bullet }}_{j_{0}},E_{i_{1}}(b_{1};c_{\bullet }),\underbrace{c_{\bullet }}_{j_{1}}, \dots ,\underbrace{c_{\bullet }}_{j_{k-1}},E_{i_{k}}(b_{k};c_{\bullet }),\underbrace{c_{\bullet }}_{j_{k}}\bigr ), \end{aligned}$$

    for all \(k\), \(l\ge 0\) and all \(a\), \(b_{1}\), ..., \(b_{k}\), \(c_{1}\), ..., \(c_{l}\in A\), where the sum is over all decompositions of \(l\) into \(2k+1\) non-negative integers,

    $$\begin{aligned} n = k + \sum _{t=0}^{k}j_{t} \qquad \text {and}\qquad \varepsilon = \sum _{s=1}^{k}i_{s}\Bigl (k+\sum _{t=s}^{k}j_{t}\Bigr ) + \sum _{t=1}^{k}t\,j_{t}. \end{aligned}$$
    (3.2)

A morphism of hgas is a morphism \(f:A\rightarrow B\) of augmented dgas that is compatible with the hga operations in the obvious way.

Remark 3.1

By the properties (iii) and (iv) as well as multilinearity, we can rewrite any expression formed within an hga as a linear combination of terms \(W\) such that no sums or scalar multiples occur inside \(W\) and such that the first argument of any operation \(E_{k}\) appearing in \(W\) is a single variable and neither a product nor another hga operation. When we speak of the terms appearing in some expression within an hga, we mean the terms appearing in such an expansion.

An expansion of this kind is actually unique and corresponds to a \(\Bbbk \)-basis for the operad \(F_{2}\mathcal {X}\) governing homotopy Gerstenhaber algebras, compare [10, Sec. 4] and [3, §1.6.6].

3.2 Extended hgas

In [8] Kadeishvili introduced the notion of an ‘extended hga’ as an hga \(A\) defined over \(\Bbbk ={\mathbb {Z}}_{2}\) that admits certain additional operations \(E^{i}_{kl}\). Based on this, he constructed \(\cup _{i}\)-products on \(\mathbf {B}A\) for all \(i\ge 1\). We will only need the family \(F_{kl} = E^{1}_{kl}\), but for coefficients in any \(\Bbbk \). We therefore define an hga to be extended if it comes with a family of operations

$$\begin{aligned} F_{kl}:A^{\otimes k}\otimes A^{\otimes l}\rightarrow A \end{aligned}$$
(3.3)

of degree \(|F_{kl}|=-(k+l)\) for \(k\), \(l\ge 1\), satisfying the following conditions: The values of all operations \(F_{kl}\) lie in the augmentation ideal \({\bar{A}}\) and vanish if any argument is equal to \(1\in A\). The differential of \(F_{kl}\) is given by

$$\begin{aligned} d(F_{kl})(a_{\bullet };b_{\bullet }) = A_{kl} + (-1)^{k}\,B_{kl} \end{aligned}$$
(3.4)

for all \(a_{1}\), ..., \(a_{k}\), \(b_{1}\), ..., \(b_{l}\in A\), where

$$\begin{aligned} A_{1l}&= E_{l}(a_{1};b_{\bullet }), \end{aligned}$$
(3.5)
$$\begin{aligned} A_{kl}&{\mathop {=}\limits ^{\varkappa }}a_{1}\,F_{k-1,l}(a_{\bullet };b_{\bullet }) + \sum _{i=1}^{k-1} (-1)^{i}\,F_{k-1,l}(a_{\bullet },a_{i}a_{i+1},a_{\bullet };b_{\bullet }) \nonumber \\&\quad + \sum _{j=1}^{l} (-1)^{k}\, F_{k-1,j}(a_{\bullet };b_{\bullet })\,E_{l-j}(a_{k};b_{\bullet }) \end{aligned}$$
(3.6)

for \(k\ge 2\), and

$$\begin{aligned} B_{k1}&{\mathop {=}\limits ^{\varkappa }}-E_{k}(b_{1};a_{\bullet }), \end{aligned}$$
(3.7)
$$\begin{aligned} B_{kl}&{\mathop {=}\limits ^{\varkappa }}\sum _{i=0}^{k-1} E_{i}(b_{1};a_{\bullet })\,F_{k-i,l-1}(a_{\bullet };b_{\bullet }) + \sum _{j=1}^{l-1} (-1)^{j}\,F_{k,l-1}(a_{\bullet };b_{\bullet },b_{j}b_{j+1},b_{\bullet }) \nonumber \\&\qquad + (-1)^{l}\,F_{k,l-1}(a_{\bullet };b_{\bullet })\,b_{l} \end{aligned}$$
(3.8)

for \(l\ge 2\), cf. [8, Def. 2].

The operation \(\mathbin {\cup _2}=-F_{11}\) is a \(\mathbin {\cup _2}\)-product for \(A\) in the sense that

$$\begin{aligned} d(\mathbin {\cup _2})(a;b) = a\mathbin {\cup _1}b +(-1)^{|a||b|}\,b\mathbin {\cup _1}a \end{aligned}$$
(3.9)

for all \(a\), \(b\in A\). This implies that the Gerstenhaber bracket in \(H^{*}(A)\) is trivial, compare [4, eq. (6.16)].

A morphism of extended hgas is a morphism of hgas that commutes with all operations \(F_{kl}\), \(k\), \(l\ge 1\).

Cochain algebras of simplicial sets, in particular singular cochain algebras of topological spaces, are naturally extended hgas, compare [4, Sec. 8.2].

4 The shm map

We define the family of maps

$$\begin{aligned} \Phi _{(n)}:(A\otimes A)^{\otimes n}\rightarrow A \end{aligned}$$
(4.1)

by

$$\begin{aligned} \Phi _{(n)}( a_{\bullet }\otimes b_{\bullet } ) {\mathop {=}\limits ^{\varkappa }}(-1)^{n-1} \!\!\! \sum _{\begin{array}{c} j_{1}+\dots +j_{n}\\ =n-1 \end{array}} \!\!\! E_{j_{1}}(a_{1};b_{\bullet })\cdots E_{j_{n}}(a_{n};b_{\bullet })\,b_{n} \end{aligned}$$
(4.2)

for \(n\ge 0\), where the sum is over all decompositions of \(n-1\) into \(n\) non-negative integers such that

$$\begin{aligned} \forall \; 1\le s\le n \qquad j_{1}+\dots +j_{s} < s. \end{aligned}$$
(4.3)

This condition means that the arguments of any term \(E_{j_{s}}(a_{s};\dots )\) are \(b\)-variables with indices strictly smaller than \(s\). It implies \(j_{1}=0\), so that each summand starts with the variable \(a_{1}\). Omitting the arguments \(a_{\bullet }\otimes b_{\bullet }=a_{1}\otimes b_{1}\), \(a_{2}\otimes b_{2}\), ..., the components of \(\Phi \) look as follows in small degrees:

$$\begin{aligned} \Phi _{(1)}&{\mathop {=}\limits ^{\varkappa }}a_{{1}}\, b_{{1}}, \end{aligned}$$
(4.4)
$$\begin{aligned} \Phi _{(2)}&{\mathop {=}\limits ^{\varkappa }}-a_{{1}}\, E_{{1}} ( a_{{2}};b_{{1}} ) \, b_{{2}}, \end{aligned}$$
(4.5)
$$\begin{aligned} \Phi _{(3)}&{\mathop {=}\limits ^{\varkappa }}a_{{1}}\, E_{{1}} ( a_{{2}};b_{{1}} ) \, E_{{1}} ( a_{{3}};b_{{2}} ) \, b_{{3}} + a_{{1}}\, a_{{2}}\, E_{{2}} ( a_{{3}};b_{{1}},b_{{2}} ) \, b_{{3}}. \end{aligned}$$
(4.6)

For \(n\ge 1\) the number of summands in \(\Phi _{(n)}\) is the Catalan number \(C_{n-1}\).

Proposition 4.1

The \(\Phi _{(n)}\) assemble to an shm map \(\Phi :A\otimes A\Rightarrow A\) that satisfies properties (i) and (ii) of the definition of an shc algebra.

Proof

The verification of property (i) is a direct computation, see Appendix A. For property (ii) we observe that the normalization condition for the hga operations implies that \(\Phi _{(n)}\) vanishes for \(n>1\) if all \(a_{i}\) or all \(b_{j}\) equal \(1\). Similarly, \(\Phi _{(n)}\) vanishes for \(n>1\) if \(a_{i}\otimes b_{i}=1\otimes 1\) for some \(i\), as required by the definition of a twisting family, see [4, eq. (3.4)]: For \(i<n\) we would have \(b_{i}=1\) as an argument to some \(E_{k}\)-term with \(k\ge 1\). For \(i=n\) the term \(E_{j_{n}}(a_{n};\dots )\) vanishes since \(a_{n}=1\) and \(j_{n}\ge 1\) as there is at least one more argument, namely \(b_{n-1}\). \(\square \)

Remark 4.2

Let \(X\) be a simplicial set. Gugenheim–Munkholm [6, Thm. 4.1\(_{*}\)] have given a recursive formula for an extension of the cup product to an shm map \(C^{*}(X)\otimes C^{*}(X)\Rightarrow C^{*}(X)\). It is based on the Eilenberg–Zilber contraction

(4.7)

Computer calculations suggest that this algorithm leads to our map \(\Phi \) if one uses a slight modification of the classical homotopy \(h:1_{C_{*}(X\times X)}\simeq \nabla AW\) defined by Eilenberg–Mac Lane. That the hga structure on \(1\)-reduced simplicial sets can be defined via the Gugenheim–Munkholm formula follows from combining results of Hess–Parent–Scott–Tonks [7, Sec. 5] and the author [5, App. A].

Proposition 4.3

The composition

$$\begin{aligned} \mathbf {B}A\otimes \mathbf {B}A {\mathop {\longrightarrow }\limits ^{\nabla }}\mathbf {B}(A\otimes A) \xrightarrow {\mathbf {B}\Phi } \mathbf {B}A \end{aligned}$$

coincides with the product on \(\mathbf {B}A\) given by the hga structure of \(A\).

Proof

We verify that the twisting cochain associated to the composition given above equals the twisting cochain \(\mathbf {E}\) corresponding to the multiplication on \(\mathbf {B}A\) as defined in [4, eq. (6.9)]. Let us consider the components

$$\begin{aligned} A^{\otimes k}\otimes A^{\otimes l} \xrightarrow {(\mathbf {s}^{-1})^{\otimes n}} \mathbf {B}_{k}A\otimes \mathbf {B}_{l}A {\mathop {\longrightarrow }\limits ^{\nabla }} \mathbf {B}_{k+l}(A\otimes A) {\mathop {\longrightarrow }\limits ^{\Phi }} A \end{aligned}$$
(4.8)

with \(k\), \(l\ge 0\) and \(n=k+l\).

A look at the formula for \(\Phi _{(1)}\) shows that (4.8) is the identity map of \(A\) if \((k,l)=(1,0)\) or \((0,1)\). Moreover, the map is zero if \(k\ne 1\) and \(l=0\), or if \(k=0\) and \(l\ne 1\) since for \(n\ge 2\) at least one term \(E_{m}\) with \(m\ge 1\) is contained in \(\Phi _{(n)}\) and this term vanishes if any argument equals \(1\). These cases are therefore verified.

Now assume \(k\), \(l\ge 1\), and let

(4.9)

be a term appearing in  such that is non-zero, where

(4.10)

Because \(b'_{1}\), ..., \(b'_{n-1}\) become arguments to \(E\) terms, they cannot equal \(1\). Hence \(a'_{1}=\dots =a'_{n-1}=1\) and \(i_{n-1}=1\) in (4.3). This implies

(4.11)

hence

(4.12)

It remains to verify that the sign is \(+1\) in the case \(k=1\). Write \(B=A\otimes A\) and

(4.13)

The summand of  that is not annihilated by \(\Phi \) is

(4.14)

where

(4.15)
(4.16)

It is mapped to

(4.17)

as desired. \(\square \)

Corollary 4.4

The following diagram commutes for any \(n\ge 0\):

figure a

Here we have written \(\nabla ^{[n]}\) and \(\mu ^{[n]}\) for the \(n\)-fold iterations of the shuffle map and the multiplication on \(A\), which are both associative. See [4, eq. (5.6)] for the definition of the iterations of the shm map \(\Phi \).

Proof

We proceed by induction. For \(n\le 1\) there is nothing to show, and the case \(n=2\) has been done above. For the induction step we observe that the parallelogram in the diagram

figure b

commutes by [4, Lemma 4.4] and therefore the outer triangle by induction. This establishes the claim for \(n+1\) and completes the proof. \(\square \)

5 Homotopy associativity

The goal of this section is to establish a homotopy \(h^{a}\) between the twisting cochains \(\Phi \circ (\Phi \otimes 1)\) and \(\Phi \circ (1\otimes \Phi )\). To state our definition, we need to introduce some terminology.

A \(c\)-product is a product of one or more variables \(c_{k}\); it is called proper if it has more than one factor. A \(b\)-term is a term of the form \(E_{m}(b_{j};\dots )\) with \(m\ge 0\) where all remaining arguments are \(c\)-products. A \(bc\)-product is a product of one or more \(b\)-terms, say ending with \(E_{m}(b_{j};\dots )\), followed by the variable \(c_{j}\). An \(a\)-term is a term of the form \(E_{m}(a_{i};\dots )\) with \(m\ge 0\) where all remaining arguments are \(b\)-terms, \(c\)-products or \(bc\)-products. An \(ab\)-product is a product of one or more \(a\)-terms and possibly \(b\)-terms that ends with an \(a\)-term. If we want to be more precise about the first variable of an \(E\)-term, we call it an \(a_{i}\)-term or a \(b_{j}\)-term.

To motivate our formula, we observe the following: By [4, eq. (4.4)] we have

$$\begin{aligned} (1\otimes \Phi )_{(n)}(a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet })=a_{1}\cdots a_{n}\otimes \Phi _{(n)}(b_{\bullet }\otimes c_{\bullet }). \end{aligned}$$
(5.1)

Note that \(\Phi _{(n)}(b_{\bullet }\otimes c_{\bullet })\) is a \(bc\)-product. To compute \((\Phi \circ (1\otimes \Phi ))_{(n)}(a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet })\) we use [4, eq. (3.20)]. Taking property (iii) of the definition of an hga into account, we see that we obtain a sum of terms \(\pm \,U\,V\) where \(V\) is a \(bc\)-product and \(U\) a product of \(a\)-terms having only \(bc\)-products as arguments. A similar argument, combined with the associativity condition (iv), shows that each term appearing in \((\Phi \circ (\Phi \otimes 1))_{(n)} (a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet })\) is of the form \(\pm \,U\,b_{n}\,W\) where \(U\) is an \(ab\)-product and \(W\) a \(c\)-product. Moreover no \(bc\)-products appear inside \(a\)-terms in this case, and the final variable of each \(c\)-product, say \(c_{j}\), corresponds to a \(b_{j}\)-term that appears as a factor of the top-level product and not as an argument to some \(a\)-term. The homotopy \(h^{a}\) has to interpolate between the two kinds of terms we have described.

We set \(h^{a}_{(0)}=\eta _{A}\). For \(n\ge 1\), we define

$$\begin{aligned} h^{a}_{(n)} {\mathop {=}\limits ^{\varkappa }}\sum (-1)^{\varepsilon }\,U\, V \end{aligned}$$
(5.2)

where the sum is over all \(ab\)-products \(U\) and all \(bc\)-products \(V\) satisfying the following conditions:

  • (i)   Each of the \(3n\) variables \(a_{1}\), ..., \(c_{n}\) appears exactly once in \(U\,V\). The \(a_{i}\)’s appear in ascending order, as do the \(b_{i}\)’s and the \(c_{i}\)’s. Moreover, \(a_{i}\) precedes \(b_{i}\) and \(b_{i}\) precedes \(c_{i}\) for each \(i\).

  • (ii)  The first argument to any \(E\)-term in \(U\,V\) has a larger index than the remaining arguments. (The definitions above imply that the first argument also has a smaller letter than the remaining arguments, where \(a<b<c\).)

  • (iii)   Any top-level \(b\)-term appearing in \(U\), say with first argument \(b_{i}\), comes right after the \(a\)-term with first argument \(a_{i}\).

  • (iv)   Define

    $$\begin{aligned} J_{a}&= \bigl \{\, j \bigm | b_{j} \text { appears in a } b\text {-term that is argument to an } a\text {-term}\,\bigr \}, \end{aligned}$$
    (5.3)
    $$\begin{aligned} J_{b}&= \bigl \{\, j \bigm | b_{j} \text { appears in a top-level } b\text {-term inside the } ab\text {-product}~U\,\bigr \}, \end{aligned}$$
    (5.4)
    $$\begin{aligned} J_{c}&= \bigl \{\, j \bigm | b_{j} \text { appears in a } bc\text {-product}\,\bigr \}. \end{aligned}$$
    (5.5)

    By construction, \(\{1,\dots ,n\}\) is the disjoint union of \(J_{a}\), \(J_{b}\) and \(J_{c}\). Note that \(J_{c}\) cannot be empty as \(V\) is a \(bc\)-product. We additionally require

    $$\begin{aligned}&J_{a}\ne \varnothing , \qquad \qquad J_{a} \cup J_{b} = \{\,1,\dots ,\nu \,\}, \end{aligned}$$
    (5.6)
    $$\begin{aligned}&J_{a}\setminus \{\nu \} = \bigl \{\, j \bigm | c_{j} \text { appears in a } c\text {-product, but not as last factor}\,\bigr \} \end{aligned}$$
    (5.7)

    where we have written \(\nu =\max J_{a}\). If \(c_{j}\) does not appear in a proper product, then it is considered to be the last factor of a \(c\)-product with a single factor. Consequently, we have \(n\ge 2\) and \(J_{c} = \{\nu +1,\dots ,n\}\).

The sign exponent \(\varepsilon \) in (5.2) is defined recursively. Write \(\mu =\min J_{a}\). If \(\mu =\nu \), that is, if \(J_{a}=\{\nu \}\), then

$$\begin{aligned} \varepsilon&= n + \text {contribution of the } b\text {-term}~E(b_{\nu };\dots ) \nonumber \\&\qquad + \text {contribution of each } bc\text {-product occurring inside an } a\text {-term}. \end{aligned}$$
(5.8)

The contributions are as follows: Consider a \(bc\)-product

$$\begin{aligned} E_{q_{j}}(b_{j};\dots )\cdots E_{q_{k}}(b_{k};\dots )\,c_{k} \end{aligned}$$
(5.9)

occurring in some term \(E_{p}(a_{i};\dots )\) as \(m\)-th argument (with \(a_{i}\) being at position \(0\) and the last argument at position \(p\)). The contribution of such a term is

$$\begin{aligned} (q_{j}+\dots +q_{k})(p-m+1). \end{aligned}$$
(5.10)

Note that \(q_{j}+\dots +q_{k}\) is the degree of the \(bc\)-product, considered as a function of its arguments, and \(p-m+1\) is the number of arguments of \(E_{p}(a_{i};\dots )\) from the \(bc\)-product (including) to the end. If \(E_{q}(b_{\nu };\dots )\) occurs in the term \(E_{p}(a_{i};\dots )\) as \(m\)-th argument, then its contribution is

$$\begin{aligned} \hat{\varepsilon } + m + q\,(p-m) \end{aligned}$$
(5.11)

where \(\hat{\varepsilon }\) is the degree of the terms preceding the \(E(a_{i};\dots )\)-term, again considered as a function of their arguments. For example, if \(U\,V\) is

$$\begin{aligned}&a_{1}\,b_{1}\,a_{2}\,E_{1}(b_{2};c_{1})\,a_{3}\,a_{4}\,a_{5}\,E_{2}\bigl (a_{6};E_{1}(b_{3};c_{2})\nonumber \\&\quad \cdot b_{4}\,E_{2}(b_{5};c_{3},c_{4})\,c_{5}\bigr )\,b_{6}\,c_{6}, \end{aligned}$$
(5.12)

then the contribution of the \(bc\)-product inside the \(a_{6}\)-term is \(2\cdot 1=2\), and that of the \(b_{3}\)-term is \(1+1+1\cdot 1=3\).

If \(J_{a}\) is not a singleton, then we compare (5.2) to a summand \((-1)^{\varepsilon '}\,U'\,V'\) with \(J_{a}'=J_{a}\setminus \{\mu \}\) and \(J_{b}'=J_{b}\cup \{\mu \}\). More precisely: We can write \(U\,V\) as

$$\begin{aligned}&E_{p_{1}}(a_{1};c_{\bullet })\,E_{q_{1}}(b_{1};c_{\bullet })\cdots E_{q_{\mu -1}}(b_{\mu -1},c_{\bullet })\,E_{p_{\mu }}(a_{\mu };c_{\bullet })\nonumber \\&\quad \cdot E_{p_{\mu +1}}(a_{\mu +1};c_{\bullet })\cdots E_{p_{i}}(a_{i}; c_{\bullet },E_{q}(b_{\mu };c_{\bullet }),\dots )\cdots c_{n} \end{aligned}$$
(5.13)

where the \(b_{\mu }\)-term is the \(m\)-th argument of the \(a_{i}\)-term. We define \(U'\,V'\) as

$$\begin{aligned}&E_{p_{1}}(a_{1};c_{\bullet })\,E_{q_{1}}(b_{1};c_{\bullet })\cdots E_{q_{\mu -1}}(b_{\mu -1},c_{\bullet })\,E_{p_{\mu }+\dots +p_{i-1} +m-1}(a_{\mu };c_{\bullet })\nonumber \\&\quad \cdot E_{q}(b_{\mu };c_{\bullet })\,a_{\mu +1}\cdots a_{i-1}\,E_{p_{i} -m}(a_{i};\dots )\cdots c_{n} \end{aligned}$$
(5.14)

where all \(c\)-variables \(c_{\bullet }\) appearing between \(a_{\mu +1}\) and \(b_{\mu }\) have been moved as additional arguments to the term \(E(a_{\mu };\dots )\). Moreover, the proper \(c\)-product starting with \(c_{\mu }\) is split into \(c_{\mu }\) and the remaining product. These two arguments replace the original \(c\)-product, wherever it appears in \(U\,V\). For example, if \(U\,V\) is

$$\begin{aligned} a_{1}\,b_{1}\,a_{2}\,b_{2}\,a_{3}\,E_{1}(a_{4};c_{1})\, E_{2}(a_{5};E_{1}(b_{3},c_{2}),b_{4})\,E_{1}(a_{6};b_{5})\, E_{1}(b_{6};c_{3}\,c_{4}\,c_{5})\,c_{6},\nonumber \\ \end{aligned}$$
(5.15)

then \(U'\,V'\) equals

$$\begin{aligned} a_{1}\,b_{1}\,a_{2}\,b_{2}\,E_{1}(a_{3};c_{1})\,E_{1}(b_{3}, c_{2})\,a_{4}\,E_{1}(a_{5};b_{4})\,E_{1}(a_{6};b_{5})\, E_{2}(b_{6};c_{3},c_{4}\,c_{5})\,c_{6}.\nonumber \\ \end{aligned}$$
(5.16)

The sign exponents \(\varepsilon \) and \(\varepsilon '\) are related by

$$\begin{aligned} \varepsilon ' - \varepsilon = {\tilde{\varepsilon }} + m + q\,(p_{i} -m) + \hat{\varepsilon }, \end{aligned}$$
(5.17)

where

$$\begin{aligned} {\tilde{\varepsilon }} = \sum _{s=1}^{\mu -1}(p_{s}+q_{s}) + \sum _{s=\mu }^{i-1}p_{s} \end{aligned}$$
(5.18)

is the degree of the expression preceding \(E(a_{i};\dots )\) as a function of its arguments, and \(\hat{\varepsilon }\) is the sign exponent for the summand of \(d(U'V')\) that recombines \(c_{\mu }\) and the following \(c\)-product to the original one. In the example (5.16) we have \(\hat{\varepsilon }=5\). Since \({\tilde{\varepsilon }}\) is a summand of \(\hat{\varepsilon }\), the difference \(\varepsilon -\varepsilon '\) is actually independent of \({\tilde{\varepsilon }}\).

Omitting the arguments, \(a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet }\), the components of \(h^{a}\) look as follows in small degrees. We also indicate the values of \(J_{a}\), \(J_{b}\) and \(J_{c}\) for each term. Note that the vanishing of \(h^{a}_{(1)}\) reflects the identity \((\Phi \circ (\Phi \otimes 1))_{(1)}=(\Phi \circ (1\otimes \Phi ))_{(1)}=\mu _{A}^{[3]}\).

(5.19)
(5.20)
(5.21)

Remark 5.1

The number of terms seems to grow rapidly with \(n\), by a factor close to \(10\) with each degree. There are no terms in \(h^{a}_{(1)}\), two terms in \(h^{a}_{(2)}\), \(25\) terms in \(h^{a}_{(3)}\), \(254\) terms in \(h^{a}_{(4)}\), \(2421\) terms in \(h^{a}_{(5)}\), \(22{,}522\) terms in \(h^{a}_{(6)}\), \(207{,}682\) terms in \(h^{a}_{(7)}\), \(1{,}911{,}954\) terms in \(h^{a}_{(8)}\) and \(17{,}635{,}830\) terms in \(h^{a}_{(9)}\).

Proposition 5.2

The maps \(h^{a}_{(n)}\) assemble to an shm homotopy \(h^{a}\) from \(\Phi \circ (\Phi \otimes 1)\) to \(\Phi \circ (1\otimes \Phi )\).

Proof

This is a very long direct computation, see Appendix B. We remark that this proof is the only place in this paper where we use the associativity condition (iv) for the hga structure, besides assuming that the product on \(\mathbf {B}A\) is associative.

Let us verify here the normalization condition [4, eq. (3.14)] for twisting homotopy families. Consider a term \(U\,V\) in (5.2) and assume that \(a_{i}=b_{i}=c_{i}=1\) for some \(i\).

If \(i\in J_{a}\), then the \(a\)-term containing \(b_{i}\) vanishes and therefore also \(U\,V\).

In the case \(i\in J_{b}\) the \(b_{i}\)-term is top-level and there is a \(c\)-product ending in \(c_{i}\). If \(i>1\) and \(b_{i-1}\) is not top-level, then it must be an argument to the \(a_{i}\)-term, which therefore vanishes. Otherwise, the \(c\)-product containing \(c_{i}\) is just \(c_{i}=1\) itself. Since it is argument to some \(a\)-term or \(b\)-term, the whole expression is again \(0\).

We finally consider the case \(i\in J_{c}\). Again, the \(c\)-product containing \(c_{i}\) is \(c_{i}=1\) itself. If the \(bc\)-term containing \(b_{i}\) does not end in \(c_{i}\), then \(c_{i}\) is argument to some later \(b\)-term in the same \(bc\)-product, so that \(U\,V\) vanishes. If the \(bc\)-product ends in \(c_{i}\), we look at \(b_{i-1}\). If it appears in the same \(bc\)-product, then \(c_{i-1}\) is an argument of the \(b_{i}\)-term. If \(i=\nu \) or if \(b_{i-1}\) is part of an earlier \(bc\)-product, then it appears inside the \(a_{i}\)-term, which once again forces \(U\,V\) to vanish. \(\square \)

Proposition 5.3

The associated coalgebra homotopy \(\mathbf {B}h^{a}\) vanishes on the image of the iterated shuffle map \(\nabla ^{[3]}:\mathbf {B}A\otimes \mathbf {B}A\otimes \mathbf {B}A\rightarrow \mathbf {B}(A\otimes A\otimes A)\).

Proof

Assume that \(h^{a}\) does not vanish on the term

$$\begin{aligned} \pm \bigl [ a'_{1}\otimes b'_{1}\otimes c'_{1}\bigm |\dots \bigm | a'_{n}\otimes b'_{n}\otimes c'_{n} \bigr ] \end{aligned}$$
(5.22)

appearing in

$$\begin{aligned} \nabla ^{[3]}\bigl ( [a_{1}|\dots |a_{k}]\otimes [b_{1}|\dots |b_{l}]\otimes [c_{1}|\dots |c_{m}]\bigr ) \end{aligned}$$
(5.23)

where \(n=k+l+m\ge 1\). By the definition of the shuffle map, two of the three factors are equal to \(1\) in each tensor product \(a'_{j}\otimes b'_{j}\otimes c'_{j}\). We must have \(b'_{j}\ne 1\) for \(j\in J_{a}\) because otherwise the \(b'_{j}\)-term would have value \(0\) or \(1\), in which case the surrounding \(a\)-term vanishes. This implies \(c'_{j}=1\) for all \(j\in J_{a}\), so that the \(c\)-product \({\tilde{c}}\) ending in \(c_{\nu }\) is \(1\). As a consequence, the \(E\)-term having \({\tilde{c}}\) as an argument vanishes. We conclude that for \(n\ge 1\) there is no term in the image of the shuffle map on which \(h^{a}\) assumes a non-zero value. In other words, \(\mathbf {B}h^{a}\,\nabla ^{[3]}=0\). \(\square \)

Remark 5.4

Corollary 4.4 applies in particular to \(\Phi ^{[3]}=\Phi \circ (\Phi \otimes 1)\), and a look at the proof shows that the conclusion equally holds for \(\Phi \circ (1\otimes \Phi )\). Together with Proposition 5.3, this implies (in a quite roundabout way) that condition (iv) of an hga structure indeed leads to an associative multiplication on the bar construction.

In the remainder of this section we collect some observations that will be used in Appendix B. We start by noting the following variant of property (ii) of the definition of an hga,

$$\begin{aligned} d(E_{p})\bigl (a;E_{q}(b;c_{\bullet }),c_{\bullet }\bigr ) {\mathop {=}\limits ^{\varkappa }}(-1)^{q(p-1)}\,E_{q}(b;c_{\bullet })\,E_{p-1}(a;c_{\bullet }) + \cdots \end{aligned}$$
(5.24)

valid for all \(k\), \(q\ge 0\) and all \(a\), \(b\), \(c_{\bullet }\in A\). This is a consequence of the convention (2.4) because we have permuted the operations \(E_{p-1}\) and \(E_{q}\). Similarly, property (iii) implies

$$\begin{aligned}&E_{k}\bigl (E_{p_{1}}(a_{1};b_{\bullet })\,E_{p_{2}}(a_{2};b_{\bullet });c_{\bullet }\bigr ) \nonumber \\&\quad {\mathop {=}\limits ^{\varkappa }}\sum _{k_{1}+k_{2}=k}\!\! (-1)^{p_{1}k_{2}}\, E_{k_{1}}\bigl (E_{p_{1}}(a_{1};b_{\bullet });c_{\bullet }\bigr )\, E_{k_{2}}\bigl (E_{p_{2}}(a_{2};b_{\bullet });c_{\bullet }\bigr ) \end{aligned}$$
(5.25)

for all \(k\), \(p_{1}\), \(p_{2}\ge 0\) and all \(a_{1}\), \(a_{2}\), \(b_{\bullet }\), \(c_{\bullet }\in A\).

Let us write \(\Phi '=\Phi \circ (\Phi \otimes 1)\). We note that each term appearing \(\Phi '(a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet })\) contains only one \(bc\)-product (at the very end) and that no two top-level \(b\)-terms are adjacent.

Lemma 5.5

Let \((-1)^{\varepsilon }\,W\) be a summand appearing in \(\Phi '_{(n)}(a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet })\). Assume that it has at least one \(b\)-term inside an \(a\)-term, and let \(\mu \) be the smallest index of such a \(b\)-variable. Then a term \(E_{q}(b_{\mu };\dots )\) appears as, say, the \(m\)-th argument  of a term \(E_{p_{r}}(a_{r};\dots )\), and \(c_{\mu }\) as the first variable inside a proper \(c\)-product \(c_{\mu }{\tilde{c}}\). Let \(W'\) be obtained from \(W\) in the same way as in (5.13), and let \(\varepsilon '\) be the sign exponent of \(W'\) in \(\Phi '_{(n)}(a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet })\). We have

$$\begin{aligned} \varepsilon '-\varepsilon \equiv {\tilde{\varepsilon }} + m + q\,(p_{i}-1) + \hat{\varepsilon } \pmod {2}. \end{aligned}$$

Here \({\tilde{\varepsilon }}\) is the degree of all \(E\)-operations in front of the \(a_{i}\)-term in \(W\), and \(\hat{\varepsilon }\) is the sign exponent of the part of the differential \(d(W')\) that recombines \(c_{\mu }\) and \({\tilde{c}}\) to \(c_{\mu }{\tilde{c}}\).

In other words, the rule used in the recursive sign definition (5.17) for \(h^{a}\) also applies to \(\Phi '\). Note that \(c_{\mu }\tilde{c}\) may be the trailing \(c\)-product in the case of \(\Phi '\), which is impossible for \(h^{a}\). This will be important for the pair 36.2.2. in Appendix B.2.

Proof

We can perform the modification \(W\rightarrow W'\) in steps: We move \(c\)-variables in front of the \(b_{\mu }\)-term to the preceding \(a\)-term (as in the pair 13. in Appendix B.1), we move the \(b_{\mu }\)-term to the preceding \(a\)-term (as in the pair 1.) or out of the current \(a\)-term if it is contained in the \(a_{\mu +1}\)-term (as in 2. and 4. –6.).

In each step, one could verify the signs directly, keeping track of

  • (1) the sign in the definition (4.2) of \(\Phi \),

  • (2) the sign given by [4, eq. (3.21)] that arises from the composition of the twisting cochains \(\Phi \) and \(\Phi \otimes 1\),

  • (3) the sign that accounts for distributing composition of maps over tensor products as in (2.4),

  • (4) the product of the signs that appears when the first argument in each term \(E_{i_{s}}(A;c_{\bullet })\) is split into its factors as in (5.25) where \(A\) is a summand of some \(\Phi _{(j_{t})}(a_{\bullet }\otimes b_{\bullet })\) given by (4.2), and

  • (5) the product of the signs appearing each time a term \(E_{l}(E_{j_{t}}(a_{t};b_{\bullet });c_{\bullet })\) is expanded according to the associativity rule (iv).

Alternatively, we can argue as follows: Since \(\Phi '\) is a twisting cochain, the corresponding family satisfies the defining equation [4, eq. (3.6)]. We observe that the terms \(W\) and \(W'\) share exactly one term in their differentials with respect to the \(\Bbbk \)-basis for \(F_{2}\mathcal {X}(n)\) described in Remark 3.1. Moreover, this common term is not produced by the differential of any other term appearing in \(\Phi '_{(n)}(a_{\bullet }\otimes b_{\bullet }\otimes c_{\bullet })\), nor by any term appearing on the right-hand side of the defining equation. Hence the two terms in question must cancel out, which leads to the claimed sign rule. \(\square \)

6 Homotopy commutativity

Let \(\Phi :A\otimes A\Rightarrow A\) be the shm map constructed in Sect. 4 or, more generally for the moment, any shm map extending the multiplication in \(A\) such that Proposition 4.3 holds. Then

$$\begin{aligned} \Phi _{(2)}(a\otimes 1,1\otimes b) + (-1)^{|a||b|}\,\Phi _{(2)}(1\otimes b,a\otimes 1) = a\mathbin {\cup _1}b \end{aligned}$$
(6.1)

for all \(a\), \(b\in A\), cf. [12, Prop. 4.8].Footnote 1 (For our \(\Phi \) this can be read off from (4.5).) Now assume that \(h\) is an shm homotopy from \(\Phi \) to \(\Phi \circ T\), as required by property (iv) of an shc algebra. A straightforward computation shows that

$$\begin{aligned} a\mathbin {\cup _2}b&= (-1)^{|a||b|}\,h_{(2)}(1\otimes b,a\otimes 1) - h_{(2)}(a\otimes 1,1\otimes b) \nonumber \\&\qquad + (-1)^{|a|}\, a\mathbin {\cup _1}h_{(1)}(1\otimes b) + h_{(1)}(a\otimes 1)\mathbin {\cup _1}b \end{aligned}$$
(6.2)

is a \(\mathbin {\cup _2}\)-product for \(A\) in the sense that it satisfies (3.9). As remarked in Sect. 3.2, a non-trivial Gerstenhaber bracket in \(H^{*}(A)\) is an obstruction to the existence of a \(\mathbin {\cup _2}\)-product and therefore to the homotopy commutativity of \(\Phi \).

In order to proceed, we put as an additional assumption in this section that \(A\) be extended. Let us define \(h^{c}_{(0)}=\eta _{A}\) and

$$\begin{aligned} h^{c}_{(n)}(a_{\bullet }\otimes b_{\bullet })&{\mathop {=}\limits ^{\varkappa }}\!\! \sum _{j_{1}+\dots +j_{n}=n}\!\!\nonumber \\&\quad E_{j_{1}}(a_{1};b_{\bullet })\cdots E_{j_{n}}(a_{n};b_{\bullet }) \nonumber \\&\quad - \sum E_{i_{1}}(b_{1};a_{\bullet })\cdots E_{i_{q}}(b_{q};a_{\bullet }) \, F_{kl}(a_{\bullet };b_{\bullet }) \, E_{j_{1}}(a_{q+1};b_{\bullet })\nonumber \\&\qquad \cdots E_{j_{p}}(a_{n};b_{\bullet }) \end{aligned}$$
(6.3)

for \(n\ge 1\). The first sum is over all decompositions of \(n\) into \(n\) non-negative integers. The second sum is over all positive integers \(p\), \(q\), \(k\), \(l\) and all non-negative integers \(i_{1}\), ..., \(i_{q}\), \(j_{1}\), ..., \(j_{p}\) such that

$$\begin{aligned} q+p= n, \quad \quad \quad \forall \; 1\le t\le q\qquad i_{1}+\dots +i_{t} < t\end{aligned}$$
(6.4)

and

$$\begin{aligned} i_{1}+\dots +i_{q}+k = q, \quad \quad \quad j_{1}+\dots +j_{p}+l = p. \end{aligned}$$
(6.5)

Omitting the argument \(a_{\bullet }\otimes b_{\bullet }\), the formula for \(h^{c}\) looks as follows in small degrees.

$$\begin{aligned} h^{c}_{(1)}&{\mathop {=}\limits ^{\varkappa }}E_{{1}} ( a_{{1}};b_{{1}} ), \end{aligned}$$
(6.6)
$$\begin{aligned} h^{c}_{(2)}&{\mathop {=}\limits ^{\varkappa }}a_{{1}}\, E_{{2}} ( a_{{2}};b_{{1}},b_{{2}} ) + E_{{1}} ( a_{{1}};b_{{1}} ) \, E_{{1}} ( a_{{2}};b_{{2}} ) + E_{{2}} ( a_{{1}};b_{{1}},b_{{2}} ) \, a_{{2}}\nonumber \\&\qquad - b_{{1}}\, F_{{1,1}} ( a_{{1}};b_{{2}} ) \, a_{{2}}, \end{aligned}$$
(6.7)
$$\begin{aligned} h^{c}_{(3)}&{\mathop {=}\limits ^{\varkappa }}a_{{1}}\, a_{{2}}\, E_{{3}} ( a_{{3}};b_{{1}},b_{{2}},b_{{3}} ) + a_{{1}}\, E_{{1}} ( a_{{2}};b_{{1}} ) \, E_{{2}} ( a_{{3}};b_{{2}},b_{{3}} ) \nonumber \\&\qquad + a_{{1}}\, E_{{2}} ( a_{{2}};b_{{1}},b_{{2}} ) \, E_{{1}} ( a_{{3}};b_{{3}} ) + a_{{1}}\, E_{{3}} ( a_{{2}};b_{{1}},b_{{2}},b_{{3}} ) \, a_{{3}} \nonumber \\&\qquad + E_{{1}} ( a_{{1}};b_{{1}} ) \, a_{{2}}\, E_{{2}} ( a_{{3}};b_{{2}},b_{{3}} ) + E_{{1}} ( a_{{1}};b_{{1}} ) \, E_{{1}} ( a_{{2}};b_{{2}} ) \, E_{{1}} ( a_{{3}};b_{{3}} ) \nonumber \\&\qquad + E_{{1}} ( a_{{1}};b_{{1}} ) \, E_{{2}} ( a_{{2}};b_{{2}},b_{{3}} ) \, a_{{3}} + E_{{2}} ( a_{{1}};b_{{1}},b_{{2}} ) \, a_{{2}}\, E_{{1}} ( a_{{3}};b_{{3}} ) \nonumber \\&\qquad + E_{{2}} ( a_{{1}};b_{{1}},b_{{2}} ) \, E_{{1}} ( a_{{2}};b_{{3}} ) \, a_{{3}} + E_{{3}} ( a_{{1}};b_{{1}},b_{{2}},b_{{3}} ) \, a_{{2}}\, a_{{3}} \nonumber \\&\qquad - b_{{1}}\, F_{{1,1}} ( a_{{1}};b_{{2}} ) \, a_{{2}}\, E_{{1}} ( a_{{3}};b_{{3}} ) - b_{{1}}\, F_{{1,1}} ( a_{{1}};b_{{2}} ) \, E_{{1}} ( a_{{2}};b_{{3}} ) \, a_{{3}} \nonumber \\&\qquad - b_{{1}}\, E_{{1}} ( b_{{2}};a_{{1}} ) \, F_{{1,1}} ( a_{{2}};b_{{3}} ) \, a_{{3}} - b_{{1}}\, b_{{2}}\, F_{{2,1}} ( a_{{1}},a_{{2}};b_{{3}} ) \, a_{{3}} \nonumber \\&\qquad - b_{{1}}\, F_{{1,2}} ( a_{{1}};b_{{2}},b_{{3}} ) \, a_{{2}}\, a_{{3}}. \end{aligned}$$
(6.8)

Proposition 6.1

Assume that the hga \(A\) is extended. The maps \(h^{c}_{(n)}\) assemble to an shm homotopy \(h^{c}\) from \(\Phi \circ T\) to \(\Phi \).

Note that \(h^{c}\circ T\) is a homotopy in the other direction.

Proof

This is a yet another lengthy direct verification, see Appendix C. It is helpful to observe the following: Consider a term appearing in the second sum of (6.3), and let \(1\le m\le n\). Then \(b_{m}\) appears in the leading group of \(E\)-terms if and only if \(a_{m}\) appears in the same group or in the \(F\)-term. Equivalently, \(a_{m}\) appears in the trailing group of \(E\)-terms if and only if \(b_{m}\) appears in the same group or in the \(F\)-term.

This in particular shows that \(h^{c}\) satisfies the normalization condition for twisting homotopy families because it is impossible for any term in the second sum and any \(m\) that \(b_{m}\) appears before the \(F\)-term and \(a_{m}\) after it. That the first sum vanishes if some \(b_{m}=1\) is clear. \(\square \)

Using the homotopy \(h^{c}\), we can generalize the formula for the \(\mathbin {\cup _1}\)-product on the bar construction given by Kadeishvili [8, Prop. 2] for \(\Bbbk ={\mathbb {Z}}_{2}\). Earlier, Baues [1, §2.9] obtained the dual formula for the cobar construction \(\Omega \,C(X)\) of a \(1\)-reduced simplicial set \(X\) and any \(\Bbbk \) (without using the surjection operad explicitly).

Corollary 6.2

The composition \(\mathbf {B}h^{c}\,\nabla _{A,A}\) is a coalgebra homotopy from the product with commuted factors to the regular product on \(\mathbf {B}A\). The associated twisting cochain homotopy \(\mathbf {F}\) is given by

$$\begin{aligned} \mathbf {F}\bigl ([a_{1}|\dots |a_{k}]\otimes [b_{1}|\dots |b_{l}]\bigr ) = {\left\{ \begin{array}{ll} 1 &{} \text {if } k=l=0, \\ {\mp } F_{kl}(a_{\bullet };b_{\bullet }) &{} \text {if } k\ge 1 \text { and } l\ge 1, \\ 0 &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

where the “\({\mp }\)” indicates a minus sign combined with the sign from [4, eq. (3.13)].

Proof

Assume \(n=k+l>0\) and consider a term

(6.9)

appearing in \(\nabla ([a_{1}|\dots |a_{k}]\otimes [b_{1}|\dots |b_{l}])\). Any such term is mapped to \(0\) by the first sum in the definition of \(h^{c}_{(n)}\). Assume that \(b'_{i}\) occurs in the leading group of \(E\)-terms in a non-zero contribution to the second sum. Then \(a'_{i}\) appears also in the leading group or in the \(F\)-term. The normalization conditions for the \(E\)-operations and the \(F\)-operations imply \(a'_{i}\ne 1\), so that we have \(b'_{i}=1\). By looking at the trailing group of \(E\)-terms, we similarly conclude \(a'_{i}=1\). Hence \(p=k\) and \(q=l\) in the definition of \(h^{c}_{(n)}\),

(6.10)

and all variables \(i_{t}\) and \(j_{s}\) are \(0\). \(\square \)

7 Polynomial algebras

Let \(A\) and \(B\) be augmented dgas, and let \(\mathfrak {b}\lhd B\) be a differential ideal. Recall from [4, eq. (3.12)] that an shm map \(f:A\Rightarrow B\) is called \(\mathfrak {b}\)-strict if we have \(f_{(n)}\equiv 0\pmod \mathfrak {b}\) for all components with \(n\ge 2\). Similarly, a homotopy \(h\) between \(f\) and another shm map \(g\) is called \(\mathfrak {b}\)-trivial if \(h_{(n)}\equiv 0\pmod \mathfrak {b}\) for \(n\ge 1\), see [4, eq. (3.18)].

We write \(\Bbbk [x]\) for a polynomial algebra on a generator \(x\) of even degree. We start with an analogue of the first part of [12, Prop. 6.2]. The second part will be addressed by Proposition 7.2.

Proposition 7.1

Let \(f\), \(g:\Bbbk [x]\Rightarrow A\) be \(\mathfrak {a}\)-strict shm maps. If there is a \(b\in \mathfrak {a}\) such that \(db=f_{(1)}(x)-g_{(1)}(x)\), then \(f\) and \(g\) are homotopic via an \(\mathfrak {a}\)-trivial homotopy.

Proof

We assume first that \(g\) is strict with . It is a direct calculation to verify that an \(\mathfrak {a}\)-trivial homotopy from \(f\) to \(g\) is given by the family

$$\begin{aligned} h_{(n)}(x^{k_{1}},\dots ,x^{k_{n}}) = (-1)^{n-1}\!\!\sum _{k'+k''=k_{n}-1}\!\! f_{(n+1)}(x^{k_{1}},\dots ,x^{k_{n-1}},x^{k'},x)\,a^{k''} \end{aligned}$$
(7.1)

for \(n\ge 1\), see Appendix D. The decomposition \(k'+k''=k_{n}-1\) is into non-negative integers. Note that \(h_{(n)}(x^{k_{\bullet }})\) vanishes for any \(n\ge 1\) if \(k_{n}\le 1\) or \(k_{m}=0\) for some \(m<n\). This in particular shows that \(h_{(n)}\) satisfies the normalization condition for twisting homotopy families.

As a consequence of this and [4, Lemma 2.2], we can assume that both \(f\) and \(g\) are strict in order to prove the general case. Then

$$\begin{aligned} h(x^{k}) = \!\! \sum _{k'+k''=k-1} \!\! f(x)^{k'} b\,g(x)^{k''} \end{aligned}$$
(7.2)

is an algebra homotopy from \(f\) to \(g\) (in the sense of [12, §1.11]) taking values in \(\mathfrak {a}\). It gives rise to an \(\mathfrak {a}\)-trivial shm homotopy \({\tilde{h}}\) from \(f\) to \(g\) with \({\tilde{h}}_{(1)}=h\) and \({\tilde{h}}_{(n)}=0\) for \(n\ge 2\). \(\square \)

The following is a variant of [12, Lemma 7.3] with an explicit homotopy. Recall that any hga is canonically an shc algebra in the sense of Stasheff–Halperin, and in the sense of Munkholm if it is extended. We refer to [4, Sec. 5] for the definition of an (\(\mathfrak {a}\)-natural) shc map.

Proposition 7.2

Let \(A\) be an hga such that all operations \(E_{k}\), \(k\ge 2\), take values in a common ideal \(\mathfrak {a}\lhd A\). Let \(a\in A\) be a cocycle of even degree and assume that there is a \(b\in \mathfrak {a}\) such that \(db=E_{1}(a;a)\). Then the dga map

$$\begin{aligned} f:\Bbbk [x] \rightarrow A, \quad x^{k}\mapsto a^{k} \end{aligned}$$

is an \(\mathfrak {a}\)-natural shc map.

Proof

We have to show that the dga map

(7.3)

is homotopic to the shm map

(7.4)

via an \(\mathfrak {a}\)-strict shm homotopy. Such a homotopy from the latter map to the former is given by the following twisting homotopy family, where we write \(x^{k_{\bullet }}\otimes x^{l_{\bullet }}\) for the sequence of arguments \(x^{k_{1}}\otimes x^{l_{1}}\), ..., \(x^{k_{n}}\otimes x^{l_{n}}\):

$$\begin{aligned} h_{(1)}(x^{k_{\bullet }}\otimes x^{l_{\bullet }})&= 0, \end{aligned}$$
(7.5)
$$\begin{aligned} h_{(2)}(x^{k_{\bullet }}\otimes x^{l_{\bullet }})&= \sum _{\begin{array}{c} l'+l''=\\ l_{1}-1 \end{array}} a^{k_{1}}\,E_{2}(a^{k_{2}};a^{l'},a)\,a^{l''+l_{2}} \nonumber \\&\quad - \sum _{\begin{array}{c} k'+k''=\\ k_{2}-1 \end{array}}\;\sum _{\begin{array}{c} l'+l''=\\ l_{1}-1 \end{array}} a^{k_{1}+k'+l'} b\,a^{k''+l''+l_{2}}, \end{aligned}$$
(7.6)
$$\begin{aligned} h_{(n)}(x^{k_{\bullet }}\otimes x^{l_{\bullet }})&= \sum \;\sum _{\begin{array}{c} l'+l''=\\ l_{n-1}-1 \end{array}} E_{i_{1}}(a^{k_{1}};a^{l_{\bullet }})\cdots E_{i_{n-1}}(a^{k_{n-1}};a^{l_{\bullet }}) \nonumber \\&\qquad \qquad {}\cdot E_{i_{n}+1}(a^{k_{n}};\dots ,a^{l_{n-2}},a^{l'},a)\,a^{l''+l_{n}} \nonumber \\&\quad \qquad ~~~~~- \sum \;\sum _{\begin{array}{c} k'+k''=\\ k_{n}-1 \end{array}}\;\sum _{\begin{array}{c} l'+l''=\\ l_{n-1}-1 \end{array}} E_{i_{1}}(a^{k_{1}};a^{l_{\bullet }})\cdots E_{i_{n-1}}(a^{k_{n-1}};a^{l_{\bullet }}) \nonumber \\&\qquad \qquad \qquad ~~~~~ {}\cdot a^{k'+l'} b\,a^{k''+l''+l_{n}}\nonumber \\&\quad - \sum \;\sum _{\begin{array}{c} k'+k''=\\ k_{n}-1 \end{array}}\;\sum _{\begin{array}{c} l'+l''=\\ l_{n-2}-1 \end{array}} E_{i_{1}}(a^{k_{1}};a^{l_{\bullet }})\cdots E_{i_{n-2}}(a^{k_{n-2}};a^{l_{\bullet }}) \nonumber \\&\qquad \qquad {}\cdot E_{i_{n-1}}(a^{k_{n-1}};\dots ,a^{l_{n-3}},a^{l'})\,a^{k'} b\,a^{k''+l''+l_{n-1}+l_{n}} \end{aligned}$$
(7.7)

for \(n\ge 3\). The first sum in the first group of (7.7) extends over all decompositions \(n-1=i_{1}+\dots +i_{n}\) into \(n\) non-negative integers such that

$$\begin{aligned} \forall \; 1\le s \le n\quad i_{1}+\dots +i_{s} < s. \end{aligned}$$
(7.8)

and the first sums in the other two groups of (7.7) similarly over all decompositions \(n-2=i_{1}+\dots +i_{n-1}\) into \(n-1\) non-negative integers such that

$$\begin{aligned} \forall \; 1\le s \le n-1\quad i_{1}+\dots +i_{s} < s. \end{aligned}$$
(7.9)

The decompositions into \(k'+k''\) and \(l'+l''\) also involve non-negative integers. Note that the formula for \(n=2\) is the same as the one for \(n\ge 3\) with the sum over \(l'+l''=l_{n-2}-1\) omitted. Also observe that \(b\) is of even degree \(2|a|-2\) and that \(i_{n}+1\ge 2\) in the first group of (7.7) so that \(h\) is indeed \(\mathfrak {a}\)-trivial. The verification of the homotopy property is a lengthy computation, see Appendix E.

That the normalization condition for twisting homotopy families is satisfied follows by direct inspection: Assume that \(k_{i}=l_{i}=0\) for some \(i\). It is clear that each term in the sum for \(n=2\) vanishes if \(k_{2}=0\) or \(l_{1}=0\). Similarly, each term in the first two sums for \(n\ge 3\) vanishes if \(k_{n}=0\) or \(l_{n-1}=0\) or \(l_{m}=0\) for \(m\le n-2\). For each term in the third sum we get this for \(k_{n}=0\) or \(l_{n-2}=0\) or \(l_{m}=0\) for \(m\le n-3\). In the case \(k_{n-1}=0\) we finally use that the corresponding \(E\)-term has another argument since \(i_{n-1}\ge 1\). \(\square \)

8 Concluding remark

It would certainly be desirable to have a more conceptual proof of Theorem 1.1 than our explicit construction. (Remark 4.2 might indicate a first step in this direction.) We point out, however, that the operad \(F_{2}\mathcal {X}\) governing hgas has non-trivial homology in the degrees we are interested in. In fact, since \(F_{2}\mathcal {X}(n)\) models the configuration space of \(n\) points in the plane, its homology \(H_{k}(F_{2}\mathcal {X}(n))\) is non-zero for all \(0\le k<n\), see the references given in Remark 3.1 as well as [13]. The Gerstenhaber bracket is a non-trivial element in \(H_{1}(F_{2}\mathcal {X}(n))\) for \(n\ge 2\), and its iterations give non-zero elements of higher degree.

A direct way to see this is the following: The Hochschild cohomology of an algebra \(A\) is an algebra over \(H(F_{2}\mathcal {X})\). If \(A\) is commutative, then \(HH^{0}(A)=A\), and \(HH^{1}(A)={{\,\mathrm{Der}\,}}(A)\) are the derivations of \(A\). The Gerstenhaber bracket of \(a\in A\) and \(D\in {{\,\mathrm{Der}\,}}(A)\) is given by \([D,a]=D(a)\in A\), cf.  [2, Props. 19, 22, Example 52]. If \(A=\Bbbk [x_{1},\dots ,x_{n}]\) is a polynomial algebra, then the expression in \(n\) elements from \(HH^{*}(A)\) involving a \(k\)-fold iterated bracket,

(8.1)

shows \(H_{k}(F_{2}\mathcal {X}(n))\ne 0\) for any \(0\le k< n\).