The following is probably well known, but I wasn't able to locate a reference in the literature.
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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)?
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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.
Best Answer
The result you are looking for is Theorem 4.18 in "An Introduction to Morse Theory" by Yukio Matsumoto, published by AMS in 2002 (translated from Japanese). The connections between Morse functions, handle structures, and CW complex structures are all explained here. Mapping cylinders play a key role in the proof of the theorem, which is similar in spirit to what Ryan outlined in his answer. This chapter of the book also covers the connection to chain complexes, the Morse inequalities, and Poincaré duality. It looks like a nice exposition, though I haven't tried reading it closely.