TITLE: CW-structures and Morse functions: a reference request
QUESTION [19 upvotes]: The following is probably well known, but I wasn't able to locate a reference in the literature.

Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a Riemannian metric on $M$. As explained in Milnor's Morse theory and many other sources, starting from $f$ and $\rho$ we can construct a CW-complex $M'$ homotopy equivalent to $M$. However, it seems natural to ask whether $f$ gives a CW-structure on $M$ itself, say, such that the corresponding cellular chain complex is isomorphic to the cellular chain complex of $M'$. Is there a reference for that (preferably, one that contains detailed proofs)?
For a generic choice of the couple $(\rho,f)$ one can construct a chain complex (which I believe is called the Morse complex and) which computes the homology of $M$. What is the standard reference for that? This is implicitly done in Milnor's h-cobordism book, chapter 7. Is it true that the Morse complex is isomorphic to the cellular chain complex of $M'$ from question 1?

upd: the original version of the posting contained some very wrong claims and had to be rewritten.
upd1: restored part of question 2 from the original posting. I deleted it thinking it would be trivial, but it seems that it isn't.

REPLY [3 votes]: I have not read it (It is on my ever-growing todo list), but the paper
Qin, Lizhen(1-WYNS)
On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds.
J. Topol. Anal. 2 (2010), no. 4, 469–526.
58E05 (37D15 57R19)

should contain the proofs of what you want. From the Mathscinet review of D. Hurtubise

This paper contains precise statements and careful proofs of several
  essential results that are fundamental to the moduli space approach to
  Morse theory. Most of the results in this paper have appeared and/or
  been used in other papers, but this is the first self-contained
  reference that provides clear and complete proofs of all of the
  following: (1) the smooth structures on the compactified spaces that
  arise from the gradient flow of a Morse-Smale function, (2)
  orientation formulas for the strata of the compactified spaces, and
  (3) the CW structure determined by the unstable manifolds of a
  Morse-Smale function. The results are proved for a Morse function on a
  complete Hilbert manifold that satisfies the Palais-Smale Condition
  (C) and has finite index at each critical point (the CF case).

Of course this also proves the CW structure in the finite dimensional case.