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:

$$\partial([x])=\sum_{y\in Cr(f),ind(y)=ind(x)+1}deg(({U}_{x}\cap{S}_{y})/{{\bf R}})\cdot[y],$$

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]).