A coauthor of mine and I want to use the following innocent looking statement in a forthcoming paper:

**Statement.** Let $M^{2n}$ be a compact manifold and let $f$ be a Morse function with critical points of even indices. Then for *some* choice of a metric $g$ on $M^{2n}$, the closure of each unstable manifold in $M^{2n}$ is a cycle of dimension equal to the index of the corresponding critical point.

I thought naively that such a statement should be contained in some classical book (*but the answers given below indicate that this might be not the case*).

Here is an idea of how to deduce the statement from the literature. Let's take the paper of Francois Laudenbach

http://www.numdam.org/article/AST_1992__205__219_0.pdf

and look into Remark 3. This remark claims something much stronger, namely that even without assumption on even indices the union of unstable manifolds give a structure of a CW complex on $M$ *in case* there exists a metric $g$ on $M$ such that the gradient flow satisfies the Morse-Smale condition and additionally the gradient vector field is *Special Morse* (i.e. looks like $\sum_i{\pm}x_i\frac{\partial}{\partial x_i}$).

Unfortunately, it is not stated in this paper whether such a metric $g$ always exists (*added: according to John and Alessia this is very simple*)

**Question.** Is there a reference or short proof for the above **Statement**? Or maybe one can say that a metric satisfying Morse-Smale condition and Special Morse condition always exists?

**Added.** I would like to thank John, Pietro and Alesia for answers. I still hope that the exact **Statement** that I want might be from 20th century, not 21st. Indeed, suppose that all the indices are even, and $g$ is Morse-Smale. Then for each unstable cell $W$ the set $\bar W\setminus W$ has Hausdorf dimension at most $\dim W-2$. Should not this give a well-defined cycle in $M^{2n}$?

**Question 2** I don't quite understand what is *Morse Homology*, but should not the above **Statement** be a trivial part of this theory?

(what about this preprint: https://arxiv.org/pdf/math/9905152.pdf ? looks relevant)