Let $(Y,g_{Y})$ be a compact oriented Riemannian manifold of dimension $n$, $f:Y\to{{\bf R}}$ be a smooth Morse function. We will denote the set of critical points of $f$ by $Cr(f)$. If $x\in Cr(f)$, we will denote by $U_{x}$ (resp. $S_{x}$) the unstable (resp. stable) submanifolds associated with $x$. Namely, $U_{x}=\{y\in Y|\,lim_{t\to+\infty}e^{-t\,grad(f)}y=x\}$, and $S_{x}=\{y\in Y|\,lim_{t\to+\infty}e^{t\,grad(f)}y=x\}$.

Let $ind(x)$ be the Morse index of $x$, i.e. the negative rank of the quadratic form $({\partial}^{2}f)_{|T_{x}Y},x\in Cr(f)$. The manifolds $S_{x}$ and $U_{x}$ are diffeomorphic to open balls of dimensions $ind(x)$ and $n-ind(x)$ respectively. It follows that the cohomology of $S_{x}$ with compact support is a graded vector space with the only non-zero $1$-dimensional component in degree $ind(x)$: $H^{\ast}_{c}(S_{x})\simeq{{\bf Z}}[-ind(x)]$. A choice of generator defines an orientation of $S_{x}$. If the function $f$ satisfies Morse-Smale transversality condition, i.e. for any $x,y\in Cr(f)$ the manifolds $S_{x}$ and $U_{y}$ intersect transversally, then $Y=\sqcup_{x\in Cr(f)}S_{x}$ is a cell decomposition of $Y$. The cohomology $H^{\ast}(Y,{{\bf Z}})$ can be computed as the cohomology of the Morse complex $(M^{\ast}(Y,f),\partial)$, with the components $M^{i}(Y,f)=\sum_{x\in Cr(f),ind(x)=i}H^{i}_{c}(S_{x})$. Let us choose orientations of manifolds $S_{x}$ for all $x\in Cr(f)$. We endow $U_{x}$ with the dual orientations. The graded module $M^{i}(Y,f)$ can be identified with $\oplus_{0\leq i\leq n}{{\bf Z}}^{Cr_{i}}[-i]$ where $Cr_{i}$ is the set of critical points of $f$ of index $i$. The choice of orientations gives a basis $([x])_{x\in Cr(f)}$ of $M^{\ast}(Y,f)$.

The differential $\partial$ is the standard Morse differential:

where $({U}_{x}\cap{S}_{y})/{{\bf R}}$ an oriented $0$-dimensional manifold (a set of points with signs), and $deg(\cdot)\in{\bf Z}$ denotes the total number of points counted with signs. The action of ${{\bf R}}$ arises from the natural reparametrization $x\mapsto x+t$ of the gradient trajectories.

There is also a generalization $M^{\ast}(Y,f,\rho)$ of the Morse complex for a flat vector bundle $\rho$ (see [BZ], [HL]).

Here we will define the Morse category of smooth functions $M(Y)$ following [FuO]. It will be an $A_{\infty}$-pre-category over ${{\bf C}}$. Objects of $M(Y)$ are pairs $(f,\rho)$, where $f:Y\to{\bf R}$ is a smooth function, and $\rho$ is a local system of finite-dimensional complex vector spaces on $Y$. Before defining the transversality of objects, we will define the transversality of functions.

Suppose we are given a sequence of smooth functions $(f_{0},...,f_{k}),k\geq 2$ such that all $f_{i}-f_{j},i\neq j$ are Morse functions, and a sequence of critical points $x_{i}\in Cr(f_{i}-f_{i+1}),0\leq i\leq k-1,x_{k}\in Cr(f_{0}-f_{k})$. We will use oriented binary planar trees in order to describe certain moduli spaces associated with such sequences. Let us fix a planar trivalent tree $T$ with $k+1$ tails vertices. Among the tail vertices we choose one and call it the root vertex. Let us orient edges of $T$ along the shortest paths towards the root. Thus, $T$ becomes a binary tree considered as an oriented tree . We depict $T$ inside of the standard unit disc $D\subset{{\bf R}}^{2}$ in such a way that tail vertices of $T$ belong to $\partial D$, and connected components of $D\setminus T$ are cyclically numbered from $0$ to $k$ in the clockwise order. We assume that numbers attached to two regions near the root vertex are $0$ and $k$.

We define a gradient immersion of $T$ into $Y$ as a continuous map $j:T\to Y$ such that:

1) The restriction $j|_{e}$ is an orientation preserving homeomorphism of the edge $e$ onto an interval in the gradient line of $f_{l(e)}-f_{r(e)}$, where the label $l(e)$ (resp. $r(e)$) corresponds to the region of $D\setminus T$ which is left (resp. right) to $e$.

2) Each tail vertex $v$ is mapped to the point $x_{v}\in Cr(f_{l_{v}}-f_{r_{v}})$, where $l_{v}$ (resp. $r_{v}$) is the label of the region which is left (resp. right) to the only tail edge containing $v$.

We will need immersed binary trees (let us call them gradient trees) in order to define compositions and transversal sequences in the Morse $A_{\infty}$-pre-category. These structures can be defined in terms of certain varieties, which we are going to describe now.

Suppose that we are given a sequence of functions $(f_{0},...,f_{k}),k\geq 2$ and critical points $(x_{0},\dots,x_{k})$ as above, and a binary planar tree $T$. Let us consider the manifold $Y(T)=Y^{V_{i}(T)}$, where $V_{i}(T)$ is the set of internal vertices of $T$. We are going to define several submanifolds in $Y(T)$. For each tail vertex $v_{m},0\leq m\leq k-1$ we define $Z_{v_{m}}=\pi_{\hat{v}_{m}}^{-1}(U_{x_{m}})$ and for $m=k$ we define $Z_{v_{k}}=\pi_{\hat{v}_{k}}^{-1}(S_{x_{k}})$. Here $\hat{v}_{l}$ denotes the second endpoint of the edge of $T$ containing $v_{l}$, and $\pi_{v}:Y(T)\to Y$ is the canonical projection on the factor corresponding to $v\in V_{i}(T)$.

For pair $(f_{i},f_{j})$ we define a subset $Z_{i,j}\subset Y\times Y$, consisting of pairs $(y_{1},y_{2})$ such that $y_{1}\neq y_{2}$ and $y_{2}=e^{t\,grad(f_{i}-f_{j})}y_{1}$ for some $t>0$. Then $Z_{i,j}$ is a non-compact submanifold of $Y\times Y$.

An edge $e$ of $T$ we call internal if both endpoints of it are internal vertices. The set of internal edges we denote by $E_{i}(T)$. For each internal edge $e\in E_{i}(T)$, which separates two regions labeled by $l(e)$ (left) and $r(e)$ (right), we define a submanifold $Z_{e}=\pi_{e}^{-1}(Z_{l(e),r(e)})$, where $\pi_{e}:Y(T)\to Y\times Y$ is the natural projection.

It follows from the definitions that the space of gradient immersions of a given $T$ as above, up to homeomorphisms preserving tails, can be identified with ${\cal M}(T;f_{0},...,f_{k};x_{0},...,x_{k}):=(\cap_{0\leq m\leq k}Z_{v_{m}})\cap(\cap_{e\in E_{i}(T)}Z_{e})\subset Y^{V_{i}(T)}$.

It can be proven (see [Fu1]) that there exists a subset of second Baire category in $(C^{\infty}(Y))^{{\bf Z}}$ such that for any element $(f_{i})_{i\in{\bf Z}}$ of this set and for any strictly increasing sequence of integers $i_{0}<\dots<i_{k})$ and for any planar tree $T$ with $k+1$ tails, the sequence $(f_{i_{0}},\dots,f_{i_{k}})$ is $T$-transversal.

For any two transversal objects $W_{0}=(f_{0},\rho_{0})$ and $W_{1}=(f_{1},\rho_{1})$ we define the space of morphisms $Hom_{M(Y)}(W_{0},W_{1})$ as the Morse complex $M^{\ast}(Y,f_{0}-f_{1},\rho_{0}^{\ast}\otimes\rho_{1})$. Now we define the $A_{\infty}$-structure on $M(Y)$.

The map $m_{1}:Hom((f_{0},\rho_{0}),(f_{1},\rho_{1}))\to Hom((f_{0},\rho_{0}),(f_{1},\rho_{1}))[1]$ is the standard differential in the Morse-Smale complex. Higher compositions $m_{k}$ where $k\geq 2$ for transversal sequences of objects are linear maps

Each $m_{k}$ is defined as a sum $m_{k}=\sum\pm m_{k,T}$ where $T$ runs through the set of isomorphism classes of oriented binary planar trees with $(k+1)$ tails. Let us describe the summands $m_{k,T}$. For simplicity we will give the formulas in the case when all local systems are trivial of rank one.

Let us fix critical points $x_{i}\in Cr(f_{i}-f_{i+1}),0\leq i\leq k-1,y_{k}\in Cr(f_{0}-f_{k})$, such that $\sum_{0\leq i\leq k-1}ind(x_{i})=ind(y_{k})+2-k$, and orientations of manifolds $S_{x_{i}},0\leq i\leq k$. It follows from the definition of a transversal sequence that the moduli space of gradient trees ${\cal M}(T;f_{0},...,f_{k};y_{0},...,y_{k})$ is an oriented compact zero-dimensional manifold.

For local systems of higher ranks one proceeds as in the case of Fukaya categories, using flat connections in order to define an analog of the holonomy of local systems.

One can obtain slightly different formulas for $m_{k}$ in the following way. For any point $\gamma\in{\cal M}(T;f_{0},...,f_{k};y_{0},...,y_{k})$ we define the weight

Here $var_{\gamma}(f_{l(e)}-f_{r(e)})>0$ is a variation of $f_{l(e)}-f_{r(e)}$ along the gradient line $\gamma(e)$, which is defined such as follows: $var_{\gamma}(f_{l(e)}-f_{r(e)})=(f_{l(e)}-f_{r(e)})(y_{max})-(f_{l(e)}-f_{r(e)})(y_{min})$, where $y_{max}$ and $y_{min}$ are the endpoints of $\gamma(e)$, such that $(f_{l(e)}-f_{r(e)})(y_{max})-(f_{l(e)}-f_{r(e)})(y_{min})>0$. After extension of scalars to ${{\bf C}}_{\varepsilon}$ one can choose another basis in $Hom_{M(Y)}(W_{0},W_{1})$, namely $[y]_{new}=[y]exp({(f_{0}(y)-f_{1}(y))\over{\varepsilon}})$ for $y\in Cr(f_{0}-f_{1})$. Then the formulas for $m_{k}$ will be modified. The contribution of each $\gamma$ will be multiplied by $w_{\gamma}$. The formulas will be similar to those for the Fukaya-Oh category (see Section 5.2).

The other $A_{\infty}$-pre-category we are interested in will be a differential-graded category (dg-category for short). In other words, it is an $A_{\infty}$-category with strict identity morphisms and vanishing compositions $m_{n},n\geq 3$. We will call it de Rham category of $Y$ and denote by $DR(Y)$. Objects of $DR(Y)$ are same as for $M(Y)$. They are pairs $(f,\rho)$, where $f:Y\to{\bf R}$ is a smooth function and $\rho$ is a local system on $Y$. Morphisms are complexes defined by the formula

Notice that the space of morphisms does not depend on $f_{0}$ and $f_{1}$. The composition of morphisms is defined in the obvious way: in a local trivialization of $\rho_{0}$ and $\rho_{1}$ it is given by the product of matrices with the coefficients in $\Omega^{\ast}(Y)$.

Now we can formulate the main result of this section.

The proof of the theorem will occupy the rest of the section. First, we will discuss a version of formulas from “homological perturbation theory” (see [GS], [Me]). They will give an $A_{\infty}$-structure on a subcomplex of a dg-algebra. Then we will discuss an approach to the proof based on the ideas of [HL]. It seems plausible that an alternative proof (but, presumably, much more difficult) can be obtained within the framework of Witten complex, using methods of [BZ].

In this section we are going to restate in a convenient form some results from [GS] and [Me].

Let $(A,m_{n}),n\geq 1$ be a non-unital $A_{\infty}$-algebra, $\Pi:A\to A$ be an idempotent which commutes with the differential $d=m_{1}$. In other words, $\Pi$ is a linear map of degree zero such that $d\Pi=\Pi d,\Pi^{2}=\Pi$. Assume that we are given an homotopy $H:A\to A[-1]$, $1-\Pi=dH+Hd$. Let us denote the image of $\Pi$ by $B$. Then we have an embedding $i:B\to A$ and a projection $p:A\to B$, such that $\Pi=i\circ p$.

Let us introduce a sequence of linear operations $m_{n}^{B}:B^{\otimes n}\to B[2-n]$ in the following way:

a) $m_{1}^{B}:=d^{B}=p\circ m_{1}\circ i$;

b) $m_{2}^{B}=p\circ m_{2}\circ(i\otimes i)$;

c) $m_{n}^{B}=\sum_{T}\pm m_{n,T},n\geq 3$.

Here the summation is taken over all oriented planar trees $T$ with $n+1$ tails vertices (including the root vertex), such that the (oriented) valency $|v|$ (the number of ingoing edges) of every internal vertex of $T$ is at least $2$. In order to describe the linear map $m_{n,T}:B^{\otimes n}\to B[2-n]$ we need to make some preparations. Let us consider another tree $\bar{T}$ which is obtained from $T$ by the insertion of a new vertex into every internal edge. As a result, there will be two types of internal vertices in ${\bar{T}}$: the “old” vertices, which coincide with the internal vertices of $T$, and the “new” ones, which can be thought geometrically as the midpoints of the internal edges of $T$.

To every tail vertex of ${\bar{T}}$ we assign the embedding $i$. To every “old” vertex $v$ we assign $m_{k}$ with $k=|v|$. To every “new” vertex we assign the homotopy operator $H$. To the root we assign the projector $p$. Then moving along the tree down to the root one reads off the map $m_{n,T}$ as the composition of maps assigned to vertices of $\bar{T}$. Here is an example of $T$ and $\bar{T}$:

Proof. Clear. $\blacksquare$

Sketch of the proof. The proof is quite straightforward, so we just briefly show main steps of computations.

First, one observes that $p$ and $i$ are homomorphisms of complexes. In order to prove the theorem we will replace for a given $n\geq 2$ each summand $m_{n,T}$ by a different one, and then compute the result in two different ways. Let us consider a collection of trees $\{\bar{T}_{e}\}_{e\in E(\bar{T})}$ such that $\bar{T}_{e}$ is obtained from $\bar{T}$ in the following way:

a) we split the edge $e$ into two edges by inserting a new vertex $w_{e}$ inside $e$;

b) the remaining part of $\bar{T}$ is unchanged.

We assign $d=m_{1}$ to the vertex $w_{e}$ edge, and keep all other assignments untouched. In this way we obtain a map $m_{n,\bar{T}_{e}}:B^{\otimes n}\to B[3-n]$.

Let us consider the following sum (with appropriate signs):

We can compute it in two different ways: using the relation $1-\Pi=dH+Hd$, and using the formulas for $d(m_{j}),\,j\geq 2$ given by the $A_{\infty}$-structure on $A$. The case of the relation $1-\Pi=dH+Hd=:d(H)$ gives

where $m_{n}^{B,\Pi}$ is defined analogously to $m_{n}^{B}$, with the only difference that we assign to a new vertex operator $\Pi$ instead of $H$ for some edge $e\in E_{i}(T)$. Similarly, the summand $m_{n}^{B,1}$ is defined if we assign to a new vertex operator $1=id_{A}$ instead of $H$. Formulas for $d(m_{j})$ are quadratic expressions in $m_{l},\,l<j$. This gives us another identity

Thus we have $d(m_{n}^{B})=m_{n}^{B,\Pi}$, and it is exactly the $A_{\infty}$-constraint for the collection $(m_{n}^{B})_{n\geq 1}$. $\blacksquare$

Moreover, using similar technique, one can prove the following result.

For the convenience fo the reader we give an explicit formula for a canonical choice of $g$. The operator $g_{1}:B\to A$ is defined as the inclusion $i$. For $n\geq 2$ we define $g_{n}$ as the sum of terms $g_{n,T}$ over all planar trees $T$ with $n+1$ tails. Each term $g_{n,T}$ is similar to the term $m_{n,T}$ defined above, the only difference is that we insert operator $H$ instead of $p$ into the root vertex.

One can also construct an explicit $A_{\infty}$-quasi-isomorphism $A\to B$.

We would like to apply formulas for the $A_{\infty}$-structure on a subcomplex to the proof of the Theorem 2. In order to do that we need to identify the Morse complex with a direct summand of the de Rham complex. Our approach is based on the ideas of Harvey and Lawson (see [HL]).

Let $Y$ be a compact oriented smooth manifold, $dim\,Y=n$. The space of currents $D^{\prime}(Y)$ we will identify with the space of distribution-valued differential forms. Continuous linear operators $\Omega^{\ast}(Y)\to D^{\prime}(Y)$ are given by their Schwartz kernels, which are elements of $D^{\prime}(Y\times Y)$. Smoothening operators $D^{\prime}(Y)\to\Omega^{\ast}(Y)$ have kernels in $\Omega^{\ast}(Y\times Y)\subset D^{\prime}(Y\times Y)$.

With any oriented submanifold $Z\subset Y$ , $\dim\,Z=k$ of finite volume we associate a canonical current $[Z]$ of degree $n-k$ (namely, we can integrate smooth $k$-forms over $Z$).

Let $g_{Y}$ be a Riemannian metric on $Y$, and $f$ be a Morse-Smale function. The gradient flow $exp(t\,grad(f)),t\geq 0$ gives rise to a $1$-parameter semigroup acting on $\Omega^{\ast}(Y)$: $\psi^{t}(\alpha)=exp(t\,grad(f))_{\ast}(\alpha)$. Schwartz kernel of $\psi^{t}$ is $[G_{t}]$ where manifold $G_{t}\subset Y\times Y$ is given by $G_{t}:=graph(exp(t\,grad(f)))$. We also have the identity

where $H^{t}:\Omega^{\ast}(Y)\to\Omega^{\ast}(Y)\subset D^{\prime}(Y)$ is a linear operator of degree $-1$ defined by the distributional kernel $[Z_{t}]$, $Z_{t}:=\cup_{0\leq t^{\prime}\leq t}graph(exp(t^{\prime}\,grad(f)))$.

It is checked in [HL] that this picture has a limit (in certain sense) as $t\to+\infty$. Namely, there exist limits of currents $[G_{t}]$ and $[Z_{t}]$:

Linear operators $\psi^{\infty}$ (of degree zero) and $H^{\infty}$ (of degree $-1$), corresponding to these kernels, map $\Omega^{\ast}(Y)$ to $D^{\prime}(Y)$ and satisfy the identity

where $i:\Omega^{\ast}(Y)\to D^{\prime}(Y)$ is the natural inclusion. According to the de Rham theorem this inclusion is a quasi-isomorphism of complexes, therefore $\psi^{\infty}$ is. Morally, $\Pi_{\infty}:=\psi^{\infty}$ should be thought of as a projector. The image $\Pi_{\infty}(\Omega^{\ast}(Y))\subset D^{\prime}(Y)$ coincides with $\oplus_{x\in Cr(f)}{{\bf R}}\cdot[U_{x}]$. We have

Moreover, the operator $\Pi_{\infty}$ commutes with the differentials. Hence the complex $\Pi_{\infty}(\Omega^{\ast}(Y))$ is a finite-dimensional subcomplex of $D^{\prime}(Y)$ isomorphic to the Morse complex $M^{\ast}(Y,f)$. In fact it is quasi-isomorphic to both complexes $\Omega^{\ast}(Y)$ and $D^{\prime}(Y)$. In this way Harvey and Lawson prove that the de Rham cohomology is isomorphic to the cohomology of Morse complex.

In order to construct actual projectors and homotopies we will proceed as follows. Let $\rho_{\delta},\delta\to 0$ be a family of smooth closed differential $n$-forms on $Y\times Y$ such that $supp(\rho_{\delta})$ belongs to the open $\delta$-neighborhood $N_{\delta}$ of the diagonal $diag\subset Y\times Y$, and the cohomology class of $\rho_{\delta}$ in $H^{n}_{c}(N_{\delta},{{\bf R}})$ is the same as of $[diag]$.

We define $R_{\delta}:D^{\prime}(Y)\to\Omega^{\ast}(Y)$ as the integral operator given by the kernel $\rho_{\delta}$.

Proof . Part 1) follows from the fact that $\rho_{\delta}$ is a closed current. Part 2) follows from the fact that $R_{\delta}$ changes the supports of $Z_{i},i=1,2$ by $O(\delta)$. To prove part 3) one observes that the operators $id$ and $(R_{\delta})_{|\Omega^{\ast}(Y)}$ preserve the space of smooth forms $\Omega^{\ast}(Y)$, and $\rho_{\delta}$ is cohomologous to $[diag]$. $\blacksquare$

Let $x,y\in Cr(f)$ be two critical points of the same Morse index. Then $deg(S_{x}\cap U_{y})=\delta_{xy}$ (the Kronecker symbol). By the part 2) of the Lemma, for sufficiently small $\delta$ we obtain the identity

This implies the following result.

We define a homotopy operator $H_{\delta}:\Omega^{\ast}(Y)\to\Omega^{\ast}(Y)[-1]$ as an integral operator given by the kernel $(R_{\delta}\boxtimes R_{\delta})[Z_{\infty}]+(h_{\delta}\boxtimes h_{\delta})([diag]).$ (The last summand is well-defined because of the condition on the wave front of $h_{\delta}$). It is easy to check that the following identity holds:

Thus we have a family of homotopies and projectors parametrized by $\delta$.

There is a version of the previous construction, which will be useful in the next subsection. Namely, we start with a differential $n+1$-form $\rho$ on $Y\times Y\times(0,1)$ such that for the support of $supp(\rho)$ belongs to $\sqcup_{\delta>0}(N_{\delta},\delta)$ for all sufficiently small $\delta\in(0,1)$, and $\rho$ defines the same cohomology class in $H_{c}^{n}(Y\times Y\times(0,1))$ as $[diag]\times(0,1)$.

Let us consider now the spaces $\Omega_{0}^{\ast}(Y):=\varinjlim_{\delta\to 0}\Omega^{\ast}(Y\times(0,\delta))$ and $D_{0}^{\prime}(Y):=\varinjlim_{\delta\to 0}\Omega^{\ast}(0,\delta)\widehat{\otimes}D^{\prime}(Y).$ It is easy to see that both complexes $\Omega_{0}^{\ast}(Y)$ and $D_{0}^{\prime}(Y)$ are quasi-isomorphic to $\Omega^{\ast}(Y)$ .

We define a linear operator $R:D_{0}^{\prime}(Y)\to\Omega_{0}^{\ast}(Y)$ similarly to the definition of $R_{\delta}$. Then the Lemma and the Proposition hold with obvious changes. We will denote the corresponding objects by the same letters as before, skipping the subscript $\delta$ (like $H$ for the homotopy and $\Pi$ for the projector). Morally, they are obtained from the old objects by extending them as differential forms “in the direction of $\delta$”.

For simplicity we will assume that all local systems are trivial and have rank one. The general case is completely similar.

We are going to construct the following chain of $A_{\infty}$-equivalences connecting $DR(Y)$ and $M(Y)$:

Classes of objects of all these categories will be the same, and all functors will be identical on objects.

The $A_{\infty}$-pre-category $DR_{0}(Y)$ is in fact a dg-category, i.e. all sequences of objects are transversal, compositions $m_{k}$ vanish for $k\geq 3$ and it has strict identity morphisms. The space $Hom_{DR_{0}(Y)}(f_{0},f_{1})$ is defined as $\varinjlim_{\delta\to 0}\Omega^{\ast}(Y\times(0,\delta))=\Omega_{0}^{\ast}(Y).$ Clearly the space of morphisms does not depend on objects. Using the wedge product of differential forms we make $DR_{0}(Y)$ into a dg-category over the field ${\bf C}$. There is a natural functor $DR(Y)\to DR_{0}(Y)$, which is the identity map on objects. On morphisms it is the natural embedding of $\Omega^{\ast}(Y)$ as the subspace of forms on $Y\times(0,\delta)$, which are pullbacks of forms on $Y$. Clearly it establishes an equivalence of $A_{\infty}$-categories.

The $A_{\infty}$-pre-category $DR_{0}^{tr}(Y)$ is defined as the full subcategory of $DR_{0}(Y)$, and it differs from the latter only by the choice of transversal sequences. Namely, we use the same notion of transversality in $DR_{0}^{tr}(Y)$ as in the Morse category.

The next $A_{\infty}$-pre-category $DR_{0}^{tr,\Pi}(Y)$ is obtained from $DR_{0}^{tr}(Y)$ by applying homological perturbation theory. For any two transversal objects $f_{0},f_{1}$ of $DR_{0}^{tr}(Y)$ we define $Hom_{DR_{0}^{tr,\Pi}(Y)}(f_{0},f_{1})$ as $\Pi_{f_{0},f_{1}}(\Omega^{\ast}_{0}(Y))$. Here $\Pi_{f_{0},f_{1}}$ is the projector $\Pi$ corresponding to the Morse function $f_{0}-f_{1}$, it was described at the end of the previous subsection. We also have homotopies $H_{f_{0},f_{1}}$ associated with $f_{0}-f_{1}$. Then formulas of homological perturbation theory (summation over trees) give rise to an $A_{\infty}$-pre-category $DR_{0}^{tr,\Pi}(Y)$ and an equivalence $DR_{0}^{tr,\Pi}(Y)\to DR_{0}^{tr}(Y)$.

The last functor $\Psi:M(Y)\to DR_{0}^{tr,\Pi}(Y))$ will have no non-trivial higher components $\Psi_{n}$ for $n\geq 2$. The first component $\Psi_{1}$ of it is a linear map

for every transversal pair $(f_{0},f_{1})$. Recall that $Hom_{M(Y)}(f_{0},f_{1})$ has a basis $\{[x]\}$ labeled by critical points $x\in Cr(f_{0}-f_{1})$. We define $\Psi_{1}([x])$ as $R([S_{x}])$. It is clear that $\Psi_{1}$ gives a quasi-isomorphism of complexes for every transversal pair $(f_{0},f_{1})$.