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I have heard a statement in the following direction. Given a compact Riemannian manifold $M$ with a Morse function on it possibly satisfying some extra assumptions. Then this data induces a CW-structure on $M$: to each critical point corresponds a cell - its unstable submanifold. The main difficulty is to construct a compactification of these cells.

I am looking for a reference to a precise statement of the result, possibly with some details of the construction.

At the moment I am less interested in a detailed proof.

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    $\begingroup$ This is answered in mathoverflow.net/questions/11375/…, see in particular, Hatcher's answer. Voting to close as a duplicate. $\endgroup$ – Igor Belegradek Apr 9 at 12:24
  • $\begingroup$ @IgorBelegradek: Thanks for that reference: I have posted my answer there and also voted to close this question. $\endgroup$ – Thomas Rot Apr 9 at 14:08
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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.

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