Unstable manifolds of a Morse function give a CW complex

Question

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)

Answer 1

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.

I will retract this for now. I do recall being told this, but I am not aware at this point in time where the gaps in his paper are, if any. (I do stand by my belief that a number papers in this area are incomplete.)

(2). The result you seek can be deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds. J. Topol. Anal. 2 (2010), no. 4, 469–526

An application of topological equivalence to Morse theory. arXiv:1102.2838

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea and Haller which also handles the case of a metric that is flat near the critical points. Lizhen tells me that the Burghelea-Haller paper is correct and one can deduce the desired result from their methods.

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)

Answer 2

A generic perturbation of the metric makes the flow Morse Smale, (stable and unstable manifolds meet transversally). In this situation, the unstable manifold do form a CW-complex. The unstable manifold of a critical point $x$ of index $k$ is an embedded disk of dimension equal to the Morse index; its closure is made adding a union of unstable manifolds of strictly less index. What is also true, and less obvious, is that the unstable manifold $W_x$ of a critical point $x$ also admits a "cell map", that is a homeomorphism from the open disk of dimension $i(x)$ to $W_x$ that extend continuously to the closures (i.e. from the closed disk to the closure of $W_x$), which makes the collections of the unstable manifolds a true CW-complex.

The first complete proof I think is in this paper:

L. Qin, On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds, J. Topol. Anal. 2 (2010), 469-526.

Answer 3

Let us show that there exists a metric for which stable and unstable manifolds of a given Morse function are transverse.

By Kupka-Smale theorem, a Morse function $f$ on a manifold with a Riemannian metric $m$ can be perturbed to become Morse-Smale, by genericity. The perturbed function $g$ has identical level-set foliation up to diffeomorphism so it is conjugate by a diffeomorphism $\phi$, more precisely $g = u \circ f \circ \phi$, where $u$ is an increasing diffeomorphism on the real line.

The stable and unstable manifolds of $u^{-1}\circ g$ for $m$ are transverse as composition by $u^{-1}$ preserves this property. Next we apply a global change of coordinate to both $u^{-1}\circ g$ and $m$ using $\phi^{-1}$. This will send $u^{-1}\circ g$ back to $f$ and $m$ to its pull-back by $\phi$.

Also this change of coordinate will send the stable/unstable manifolds of $u^{-1}\circ g$ for $m$ to the ones of $f$ for the pull-back of $m$. In particular the new stable/unstable manifolds are transverse. So the pull-back metric satisfies the transversality condition.

It remains to address the special Morse condition. It is not difficult to modify $m$ locally around critical points of $f$ so that the modified metric $m'$ is special Morse for $f$, using partitions of unity. We can then repeat the above construction with $m'$ instead of $m$.

While changes of coordinates preserve the special Morse condition, the perturbation of $f$ might destroy it if it affects neighborhoods of critical points. Fortunately the proof of the genericity of Morse-Smale condition produces a perturbation that does not affect those neighborhoods, so the above construction will satisfy the special Morse condition as well.