Morse complexes and multiplicative structures

Abstract

In this article we lay out the details of Fukaya’s $A_\infty$-structure of the Morse complexes of manifold possibly with non-empty boundary. We show that these $A_\infty$-structures are homotopically independent of the made choices. We emphasize the transversality arguments that make some fiber products smooth.

Appendices

Appendix A: Complements on the submanifolds with $C^1$ conic singularities

Definition A.1

Let K be a compact submanifold of M with $C^1$ conic singularities. The k-skeleton $K^{[k]}$ of K is the union of the strata of K which are of codimension at least $n-k$ in M.

Lemma A.2

Let A and B be two compact submanifolds of M with $C^1$ conic singularities. Then, for a generic diffeomorphism g of M the image g(A) is transverse to B, meaning that each stratum of g(A) is transverse to every stratum of B. Moreover, this transversality is fulfilled in an open set of the $C^1$ topology of $Di\!f\!f(M)$.

Proof

Thanks to the group action, it is enough to prove this statement near the Identity of M. Assume the skeleton $A^{[k]}$ is already transverse to B. So, near any point $x\in A^{[k]}$, each stratum of A is transverse to B. The latter directly follows from the conic transverse structure.

Let S be an open $(k+1)$-stratum of A; it is transverse to B outside some compact set $C\subset S$. By the very first transversality theorem of Thom [24], an arbitrarily small ambient isotopy supported in a neighborhood of C makes S successively transverse to the 0-skeleton, the 1-skeleton, and so on, until being transverse to B. Moreover, these smooth approximations of $Id_M$ which fulfils the above requirement form a $C^1$ open set. This double induction gives the desired genericity, including the $C^1$ openness of transversality. $\square$

Lemma A.3

Let A and B be two submanifolds with conic singularities which are mutually transverse. Then their union $A\cup B$ is a submanifold with $C^1$ conic singularities. Its strata are of one of the following forms where S is a stratum of A and $\Sigma$ is a stratum of B: $S\cap \Sigma$ or $S{\smallsetminus } \Sigma$ or $\Sigma {\smallsetminus } S$.

Proof

The only matter is about the structure at points in $A\cap B$. Set $\Lambda =S\cap \Sigma$. For $x\in \Lambda$, let $\Theta _{B, \Sigma , x}$ be the transverse conique structure to $\Sigma$ in M induced by B on the normal fiber $\nu _x(\Sigma ,M)$. By transversality, we have the equality $\nu _x(\Sigma ,M)=\nu _x(\Lambda , S)$. So, $\nu _x(\Lambda , S)$ is equipped with $\Theta _{B,\Sigma , x}$. Similarly we have the transverse conic structure $\Theta _{A,S,x}$ induced on the normal fiber $\nu _x(\Lambda , \Sigma )$.

These two transverse structures can be trivialized over a small open neighborhood of x in $\Lambda$. Since the two bundles $\nu (\Lambda , S)$ and $\nu (\Lambda , \Sigma$) are complementary in $\nu (\Lambda ,M)$, the two trivializations are independent. Therefore, they can be realized by a same $C^1$ diffeomorphism of M. Hence, the conic structure induced by $A\cup B$ on the normal fiber $\nu _x(\Lambda , M)$ is the join (in the sense of the piecewise linear topology) $\Theta _{A,S,x}*\Theta _{B,\Sigma , x}$. $\square$

The next lemma and its corollary can be proved in the same way.

Lemma A.4

Let, for $i=1,2$, $A_i\subset M^{\times k_i}$ be a compact submanifold with $C^1$ conic singularities. Then $A_1\times A_2\subset M^{\times (k_1+k_2)}$ is so.

Corollary A.5

In the same product setting as in Lemma A.4, let $p_i: M^{\times k_i}\rightarrow M$ be a projection to one factor of the product. If the restrictions $p_1\vert A_1$ and $p_2\vert A_2$ are transverse, then the fiber product over M of these two maps is a compact submanifold $M^{\times (k_1+k_2-1)}$ with $C^1$ conic singularities.

Appendix B: Some applications of Sard’s theorem to immediate transversality

Let $S_1$ and $S_2$ be two smooth²⁹ submanifolds of positive codimension in $\mathbb {R}^n$, possibly equal. One defines the space of secants from $S_1$ to $S_2$ by

$$\begin{aligned} Sec_{S_1,S_2} := \{(u,x)\in \mathbf {\mathbb {R}}^n\times \mathbb {R}^n\mid x\in S_1 \text { and }x+u\in S_2\} \end{aligned}$$

(B.1)

Let $\pi : Sec_{S_1,S_2}\rightarrow \mathbf {\mathbb {R}}^n$ denote the projection $(u,x)\mapsto u$.

If $f_1(x)= 0\ (\text {resp. } f_2(x)= 0)$ are two local systems of regular equations defining $S_1$ (resp. $S_2$), the—potential—tangent space to $Sec_{S_1,S_2}$ at (u, x) is defined by the linearized system

$$\begin{aligned} \left\{ \begin{array}{l} Df_1(x)\cdot \delta x=0\\ Df_2(x+u)\cdot (\delta x+\delta u)=0. \end{array} \right. \end{aligned}$$

(B.2)

This system is of maximal rank and hence $Sec_{S_1,S_2}$ is a smooth submanifold.

Proposition B.1

For almost every $u\in \mathbf {\mathbb {R}}^n{\smallsetminus } \{0\}$³⁰ and every $x\in \pi ^{-1}(u)$, the two tangent vector spaces $T_xS_1$ and $T_{x+u}S_2$ are not coplanar in the sense that there is no hyperplane containing both of them.

Note that when $\mathrm{codim}\,S_1+\mathrm{codim}\,S_2>n$ and $\pi ^{-1}(u)\ne \emptyset$ coplanarity is automatic.

Proof

Thanks to the smoothness assumption Sard’s Theorem is available. It tells us that almost every u is a regular value of $\pi$ (possibly with an empty inverse image). But an easy argument shows that u is a critical value of $\pi$ if and only if there exists $x\in \pi ^{-1}(u)$ such that the tangent spaces $T_xS_1$ and $T_{x+u}S_2$ are coplanar.

Indeed, in case of coplanarity, there is a non-zero linear form L vanishing on $\ker Df_1(x)$ and $\ker Df_2(x+u)$. For $(\delta u, \delta x)$ solution of (B.2), we have $L(\delta x)=0\text { and } L(\delta x+\delta u)=0$. Then $\delta u$ is forced to belong to $\ker L$ and hence $D\pi (u,x)$ is not surjective.

Conversely, if for every $x\in \pi ^{-1}(u)$ the tangent spaces $T_xS_1$ and $T_{x+u}S_2$ are not coplanar the matrix of $\left( \begin{array}{c} Df_1(x) \\ Df_2(x+u) \end{array}\right)$ is of maximal rank. Then, for every $\delta u$ one can solve the linear system

$$\begin{aligned} \left\{ \begin{array}{l} Df_1(x)\cdot \delta x= 0\\ Df_2(x+u)\cdot \delta x=-Df_2(x+u)\cdot \delta u, \end{array} \right. \end{aligned}$$

(B.3)

and hence, u is a regular value of $\pi$. $\square$

Here is the corollary we are interested in: non-coplanarity is a criterion for a translation flow to be of immediate tranversality (Definition 3.4).

Corollary B.2

(Non-coplanarity criterion) Let $S\subset {\mathbb {S}}^{n-1}$ be a smooth compact submanifold with $C^1$ conic singularities³¹ in the unit $(n-1)$-sphere. Let C be the cone based on S with the origin O as a vertex. Then, there exists some residual set $R\subset \mathbf {\mathbb {R}}^n$, actually an open and dense subset, such that u belonging to R is equivalent to each of the following properties:

• 1. The translated cone $C+u$ is transverse to C.

• 2. The translation flow generated by u is a flow of immediate transversality to C.

Proof

We first show the two items are equivalent. Let $(S_1,S_2)$ be a pair of strata from the cone C (that is, punctured cones based on strata in S.) If $S_1+u$ is not transverse to $S_2$ at a then $S_1+tu$ is not transverse to $S_2$ at ta; indeed, the tangent spaces to $S_1$ and $S_2$ are constant along each generating line. Therefore, non-transversality is preserved along a positive translation semi-flow.

In Proposition B.1, we have checked that the critical set of $\pi : Sec_{S_1,S_2}\rightarrow \mathbf {\mathbb {R}}^n$ is the set of pairs (u, x) displaying coplanarity of $T_xS_1$ and $T_{x+u} S_2$. So, $crit (\pi )$ is a cone; it is closed in $(\mathbf {\mathbb {R}}^n{\setminus } \{0\})\times S_1$ as usual for a critical set. One also controls its closure as explained below.

It is easy to describe $\{0\}\times S_1$ as a part of the closure of $crit (\pi )$ in $\mathbf {\mathbb {R}}^n\times S_1$. Indeed, for $(u_j,x_j)$ tending to $(0,x_0)$ with $x_0\in S_1$, the ray $\mathbb {R}_+x_j$ goes to the ray $\mathbb {R}_+x_0$. A pair (x, y) of points staying on the same ray makes $(y-x,x)\in crit(\pi )$. Therefore, the renormalized sequence $(u_j/\Vert u_j\Vert ,x_j/\Vert u_j\Vert )$ is asymptotic to this part of $crit(\pi )$ which is isomorphic to $\mathbb {R}_+\times S_1$.

Let us now consider the case of $(x_j)$ going to $x_0$ in another stratum $S_0$ of C; this stratum lies in the closure of $S_1$. Up to a subsequence, the sequence of tangent spaces $T_{x_j} S_1$ has a limit which contains $T_{x_0}S_0$; this is Whitney Condition A which holds since the singularities are $C^1$ conic. Let ${\bar{\pi }}: \mathbf {\mathbb {R}}^n\times {\bar{S}}_1\rightarrow \mathbf {\mathbb {R}}^n$ denote the extension of $\pi$ to the closure of its domain. If $(u_j, x_j)\in crit(\pi )$ for every j—that is some coplanarity—this condition A implies $\lim _j (u_j, x_j)$ in $\mathbf {\mathbb {R}}^n\times S_0$ is a critical point of ${\bar{\pi }}$.

The cone $crit({\bar{\pi }})$ is a subcone of the cone based on $S\times S$. Since S is compact $crit({\bar{\pi }})$ has a compact base. The same holds for the set of critical values in $\mathbf {\mathbb {R}}^n$. Then the set $R_{S_1,S_2}$ of regular values of ${\bar{\pi }}$ is open; moreover it is dense, as stated in Proposition B.1. The desired R is the finite intersection of $R_{S_1,S_2}$ over all pairs of strata. $\square$

B.3

Product family of cones³² This consists of the product $V\times (\mathbb {B}^{n-k}, Q)$ where Q is a cone in $\mathbb {B}^{n-k}$ as in the previous corollary and V is a compact k-dimensional manifold. A translation flow $(u^t)$ is generated by a smooth section $u: V\rightarrow V\times \mathbf {\mathbb {R}}^{n-k}$, that is a translation vector u(p) in each fiber $\{p\}\times \mathbb {B}^{n-k}$, depending smoothly on p. The flow acts on the fiber over p by the formula

$$\begin{aligned} u^t(p, x)=\bigl (p, x+t u(p)\bigr ). \end{aligned}$$

(B.4)

The germ of this flow is said to be of immediate transversality to $V\times Q$ if $u^t(V\times Q)$ is transverse to $V\times Q$ for every small positive t.

Since Q is a cone and only translations are involved, the flow $(u^t)$ is of immediate transversality if and only if $u^\theta (V\times Q)$ is transverse to $V\times Q$ for some $\theta >0$. Explicitely, this reads by saying that for every $p\in V$ one of the following two properties holds:

$$\begin{aligned} \left\{ \begin{aligned}&\text { -- The translation }u^\theta (p)\text { maps }\{p\}\times Q\text { transversely to itself in } \{p\}\times \mathbb {B}^{n-k}.\\&\text { -- For every hyperplane }H\text { in }\mathbb {B}^{n-k}\text { bitangent to }Q\text { at some points }x\text { and }x+u^\theta (p),\\&\quad \text { the operator }\partial _V u^\theta \vert _{p}\text { maps the tangent space }T_{p}V\times \{0\}\text { transversely to the}\\&\quad \text { codimension-one space }T_pV\times H. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(B.5)

Property (B.5) is open in the $C^1$ topology of sections (see Corollary B.2). Using the same idea as in Proposition 3.1, namely applying Sard’s theorem to a finite dimensional family of sections of $V\times \mathbf {\mathbb {R}}^{n-k}$ which is submersive on each fiber, one gets that immediate transversality is generic. More precisely, we have the following.

Proposition B.4

Regarding the smooth sections $V\rightarrow V\times \mathbf {\mathbb {R}}^{n-k}$ as generators of (germs of) fiberwise translation flows on $V\times \mathbb {B}^{n-k}$, the set of those which generate immediate transversality to $V\times Q$ is open and dense in the $C^1$ topology of smooth sections. For short, these sections are said to be generic. Moreover, the following relative version holds: every germ of generic section along boundary $\partial V$ extends to a generic section over V.

Proof

For the relative version, the given germ extends arbitrarily to ${\tilde{\sigma }}:V\rightarrow V\times \mathbf {\mathbb {R}}^{n-k}$. Then, ${\tilde{\sigma }}$ has a generic approximation $\sigma$ which can be connected to ${\tilde{\sigma }}\vert \partial V$ among the generic germs thanks to openness. $\square$

The only remaining issue is to make coexist Proposition B.4 and Corollary B.2 when a stratum of a manifold with conic singularities enters the n-ball about a 0-stratum. Here is the main concept related to this question.

B.5

The reduced translation flow  In the setting of Corollary B.2, we consider a k-dimensional stratum $S_k$ of the cone C, $k>0$, and a compact subdomain ${{\underline{S}}}_{\,k}$ (Sect. 3.5 (2).) By definition of conic singularities, C induces a conic bundle over ${{\underline{S}}}_{\,k}$. Namely, there exists a tube $N_k$, which is a trivial $(n-k)$-disc bundle over ${{\underline{S}}}_{\,k}$ whose fibers $N_{k,x}, \ x\in {{\underline{S}}}_{\,k}$, are planar in the unit ball $\mathbb {B}^n$. The fibers $C\cap N_{k,x}$ are conic and form a trivial cone subbundle of $N_k$.

Let u be the generator of a (germ of) translation flow in $\mathbb {B}^n$. For every $x\in {{\underline{S}}}_{\,k}$ and every $y\in N_{k,x}$ one uses the splitting of the tangent space

$$\begin{aligned} T_y\mathbb {B}^n= T_x{{\underline{S}}}_{\,k}\oplus T_yN_{k,x} . \end{aligned}$$

(B.6)

Here, the tangent space $T_x{{\underline{S}}}_{\,k}$ is carried to y by parallelism with respect to the ambient affine structure of $\mathbb {B}^n$ and $N_{k,x}$ is thought of as spanning an $(n-k)$-dimensional affine subspace in $\mathbb {R}^n$. The splitting decomposes the vector u into horizontal and vertical components at x, that is:

$$\begin{aligned} u= u_\mathfrak {h} ^k(x) \oplus u_\mathfrak {v}^k(y) \end{aligned}$$

(B.7)

with $u_\mathfrak {h} ^k(x)\in T_x{{\underline{S}}}_{\,k}$ and $u_\mathfrak {v}^k(y)\in T_yN_{k,x}$. Note that this splitting is constant along the fiber $N_{k,x}$, that is, independent of y. The vertical component $x\mapsto u_\mathfrak {v}^k(x)$ is a section of $N_k$ which is termed the reduction of u to $N_k$.

B.6

Reducing process  Let $\partial {\underline{S}}_{\,k}$ denote the frontier of ${\underline{S}}_{\,k}$ in $int(\mathbb {B}^n)$. Fix also an interior collar neighborhood $W_k$ of $\partial {\underline{S}}_{\,k}$ in ${\underline{S}}_{\,k}$ and let $E_k$ denote the part of $N_k$ over $W_k$. Without loss of generality we may assume $E_k\subset \mathbb {B}^n$. Let $\mu :W_k\rightarrow [0,1]$ be a smooth function equal to 1 near $\partial {\underline{S}}_{\,k}$ and 0 near the opposite face of $W_k$. This $\mu$ is lifted to $E_k$ as a constant function in each fiber $N_{k,x}$. The lifted $\mu$ is still noted $\mu$ and called a balancing function.

The balanced reducing process consists of replacing the constant vector field u on $E_k$ by the vector field

$$\begin{aligned} u_{\mu }^k(x):= \mu (x)u_{\mathfrak {h}}^k(x) +u_{\mathfrak {v}}^k(x). \end{aligned}$$

(B.8)

It is constant in each fiber $E_{k,x}$. Note that $u_\mu ^k$ is equal to u in the part of $N_k$ over a small neighborhood of $\partial {\underline{S}}_{\,k}$. Such a vector field also reads

$$\begin{aligned} u_\mu ^k= \mu \,u+ (1-\mu )u_{\mathfrak {v}}^k. \end{aligned}$$

This vector field is termed the balanced reduction of u.

Fig. 9

Two sectional views of the tubes $N_j$ and $N_k$ in $\mathbb {B}^n$

B.7

Skew associativity formula   For $j<k$, let $S_j$ and ${\underline{S}}_{\,j}$ be a j-dimensional stratum of the cone $C\subset \mathbb {B}^n$ and its compact subdomain; and let $\lambda : {\underline{S}}_{\,j}\rightarrow [0,1]$ be a balancing function for $S_j$. The position of $S_{\,j}$ with respect to $S_{\,k}$ is specified in Sect. 3.5 (see Fig. 9). If $N_{j,x}$ is a fiber of $N_j$, with $x\in {\underline{S}}_{\,j}$, and y is a point in $N_{j,x}\cap W_k$, the fiber $N_{k,y}$ is an affine subspace of $N_{j,x}$. Then, the reducing process with respect to stratum $S_k\cap N_{j,x}$ may be applied to the translation vector $u^j_\mathfrak {v}$ in the $(n-j)$-ball $N_{j,x}$. One gets the next fomula along the fiber $N_{k,y}\subset N_{j,x}$:

$$\begin{aligned}&u^k_\mathfrak {v}= \left( u^j_\mathfrak {v}\right) ^k_\mathfrak {v} \quad \text {and}\quad u^k_\mathfrak {h}= u^j_\mathfrak {h}+ \left( u^j_\mathfrak {h}\right) ^k_\mathfrak {h} \text { that is,}\nonumber \\&u^k_\mu = \mu \lambda \, u^j_\mathfrak {h} +\mu \left( u^j_\mathfrak {h}\right) ^k_\mathfrak {h} + \left( u^j_\mathfrak {v}\right) ^k_\mathfrak {v}. \end{aligned}$$

(B.9)

Note these formulas hold regardless of the functions $\lambda$ and $\mu$. The subscript $\mathfrak {h}$ has two different meanings: one stands for parallelism to $T_xS_j$ and the second one for parallelism in $N_{j,x}$ to $T_y\bigl (S_k\cap N_{j,x}\bigr )$.

There are analogous formulas associated with a sequence of strata $S_{j_1}, S_{j_2}, ..., S_{j_r}$ when each is in the closure of the next one.

Proposition B.8

In the setting of Corollary B.2 of a stratified cone $C\in \mathbb {B}^n$, it is assumed that the conic transverse structure to each stratum has a global trivialization.³³ If u generates a flow of immediate transversality to C then we have:

• 1. The reduction $u^k_\mathfrak {v}$ of u to $N_k$ generates a flow of immediate transversality to $C\cap N_k$.

• 2. The flow generated by the balanced reduction of u to $N_k$ is of immediate transversality to $C\cap E_k$.

Proof

The matter deals with bi-1-jets of C (or pairs of tangent planes to C.) This allows one to linearize the considered vector field at any desired point without changing the problem.

On the linear disc bundle $N_k$ we have two linear connections $h_0$ and $h_1$ (seen as plane distributions complementary to the fibers): $h_0$ is parallel to $T_xS_k$ along the fiber $N_{k,x}$ for every $x\in S_k$; and $h_1$ is given by the assumed global trivialization of $N_k$. The difference between them, seen as a vertical deviation, is measured by a 1-form $\omega$ on $S_k$ valued in the vector space of linear endomorphisms of the vector bundle spanned by $N_k$.

By assumption, the vector u generates a translation flow of immediate transversality to C. Let $\alpha$ be the minimum angle between $T_y C$ and a hyperplane containing $T_{y+tu}C$ for every $y\in C$ and small positive t. The lowest bound of this angle is positive by assumption on u; it is independent of t since C is a cone and it is a minimum since C has a compact base.

Claim. If $h_0=h_1$, then the statement holds.

Indeed, by the above assumption the distribution $h_0$ is tangent to C. Then, transversality to C translates to the vertical component of the flow of u.

By an order-one Taylor expansion at $y\in C$, an “infinitesimal contact”, namely coplanarity of $D_y(u^k_\mathfrak {v})(T_yC)$ to $T_yC$, implies at most transversality to C with an arbitrarily small angle for some small $t>0$, contradicting $\alpha >0$. This proves (1) in this setting. If (2) fails, it should fail infinitesimally which is impossible by (1). $\square$

Let $y\in N_{k,x}$ and let $y+tu^k_\mathfrak {v} (x)$ be the vertically displaced point for a small time t; suppose both points belong to C. The planes $h_1(y)$ and $h_1(y+tu^k_\mathfrak {v} (x))$ are both tangent to C but could be not parallel anymore. Nevertheless, thanks to the 1-form $\omega$ which measures the “difference $h_1-h_0$”, one computes that the angle between $h_1(y)$ and $h_1(y+tu^k_\mathfrak {v} (x))$, the latter being translated to y, is a O(t). Therefore, if $t>0$ is sufficiently small, this angle is negligible with respect to $\alpha$. So, the reasoning for the claim still holds. $\square$

Appendix C: Basics on homotopical algebras

In this appendix we review the basic terminology and result of the theory of $A_\infty$-algebras. We refer the reader to Kenji Lefèvre–Hasegawa’s thesis [16] for a comprehensive treatment. However, here we use the sign convention introduced [12].

Here, $\mathbf {k}$ is a unitary ring.

Definition C.1

An $A_p$-algebra is a $\mathbf {k}$-module equipped with a collection of $\mathbf {k}$-module maps $m_i:A^{\otimes i} \rightarrow A$, $1 \le i\le p$ , of degree $2-i$ satisfying the identities

$$\begin{aligned} \sum _{j+k+l=i} (-1)^{j+kl}m_{j+l+1}(1^{\otimes j} \otimes m_k\otimes 1^{\otimes l})=0 \end{aligned}$$

(C.1)

for all $p \ge i\ge 1$.

Similarly an $A_\infty$-algebra is a graded $\mathbf {k}$-module A together with a collection of $\mathbf {k}$-module maps $m_i:A^{\otimes i} \rightarrow A$, $i\ge 1$ , of degree $2-i$ such for all p , $(A, \{m_i\}_{1\le i\le p})$ is an $A_p$-algebra.

Remark C.2

According to the sign convention in [16] one should put $(-1)^{jk+l}$ instead of $(-1)^{j+kl}$. It turns out that these two definitions are equivalent. Indeed if $(m_1, m_2,\ldots )$ is an $A_\infty$-structure according to the sign convention of [16], then $(m_1, (-1)^{2\atopwithdelims ()2 }m_2,\ldots (-1)^{i\atopwithdelims ()2 }m_i,\ldots )$ is an $A_\infty$-structure by our sign convention. The sign conventions in [16] is justified by the cobar construction. The signs in [12] correspond to that of the opposite algebra in [16].

Let $(A, \{m_i\}_{1\le i\le p})$ and $(A', \{m'_i\}_{1\le i\le p})$ be two $A_p$-algebras. An $A_p$-morphism from $(A, \{m_i\}_{1\le i\le p})$ to $(A', \{m'_i\}_{1\le i\le p})$ consists of a collection of maps $f_i: A^{\otimes i}\rightarrow A'$, $1 \le i\le p$, with the $|f_i|=1-i$ satisfying the conditions

$$\begin{aligned} \sum _{j+k+l=i} (-1)^{{j+kl} }f_{j+l+1}\ (1^{\otimes j} \otimes m_k\otimes 1^{\otimes l})=\sum _{k=1}^i \sum _{i_1+\cdots +i_k=i} (-1)^{\epsilon _{i_1, \ldots , i_k}}m'_k( f_{i_1} \otimes \cdots \otimes f_{i_k})\nonumber \\ \end{aligned}$$

(C.2)

where $\epsilon _{i_1, \ldots , i_k} = \sum _{j=1}^k (k-j)(i_j-1)$.

Remark C.3

If we follow the sign convention of [16], then equation of C.2 transforms into

$$\begin{aligned} \sum _{j+k+l=i} (-1)^{l+jk} f_{j+l+1}\ (1^{\otimes j} \otimes m_k\otimes 1^{\otimes l}) =\sum _{k=1}^i \sum _{i_1+\cdots +i_k=i} (-1)^{\epsilon _{i_1, \ldots , i_k}}m'_k( f_{i_1} \otimes \cdots \otimes f_{i_k})\nonumber \\ \end{aligned}$$

(C.3)

where $\epsilon _{i_1, \ldots , i_k} = \sum _{j} ((1-i_j) \sum _{1\le k\le j} i_ k)$.

If $(m_i)$ and $(f_i)$ satisfy the equation (C.2), then $(m_1, (-1)^{2\atopwithdelims ()2 }m_2,\ldots , (-1)^{i\atopwithdelims ()2 }m_i,\ldots )$ and

$$\begin{aligned} \left( f_1, (-1)^{2\atopwithdelims ()2 }f_2,\ldots (-1)^{i\atopwithdelims ()2 }f_i,\ldots \right) \end{aligned}$$

satisfies (C.3).

A collection of $\mathbf {k}$-module maps $f=\{f_i\}_{i\ge 1}: A^{\otimes i} \rightarrow A'$ is said to be a morphism of $A_\infty$-algebras if for all p, $\{f_i\}_{ 1\le i\le p}$ is a morphism of $A_p$-algebras.

An $A_\infty$-morphism $f=\{f_i\}_{i\ge 1}$ is said to be a quasi-isomorphism if the cochain complex map $f_1$ is a quasi-isomorphism.

Definition C.4

Let A and $A'$ be two $A_\infty$-algebras with the corresponding differentials D and $D'$ on the bar constructions BA and $BA'$. Suppose that $f=\{f_i\}, g=\{ g_i\}: A\rightarrow A'$ are two $A_\infty$-morphisms and F and G are the coalgebra morphisms corresponding to f and g. Then a homotopy between f and g is a (F, G)-coderivation $H: BA\rightarrow BA'$ such that

$$\begin{aligned} F-G= D'H -HD. \end{aligned}$$

(C.4)

Theorem C.5

(Prouté [21], see also [16])  We suppose that $\mathbf {k}$ is a field. Then we have:

• 1. For connected $A_\infty$-algebras, homotopy is an equivalence relation (Theorem 4.27).

• 2. A quasi-isomorphism of $A_\infty$-algebras is a homotopy equivalence (Theorem 4.24).