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 $\delta$”.