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  • $\begingroup$ Thanks, Tim! This is what I was hoping for in question 2. Is the weak self-indexing condition really necessary? The way I understand it, in general not every Morse function can be deformed into a function of the above type in the space of Morse functions. $\endgroup$ Commented Jan 11, 2010 at 0:49
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    $\begingroup$ Since $\mathbb R$ is contractible you can homotope any Morse function to any other. Since there are self-indexing Morse functions on a manifold, you can homotope your original Morse function to a self-indexing one. Cerf theory says the homotopy can be made to be Morse at all but finitely many times corresponding to certain cubic singularities where pairs of critical points of index $k$ and $k+1$ get created or destroyed. These correspond to elementary handle operatations (the kind you see in the proof of h-cobordism). $\endgroup$ Commented Jan 11, 2010 at 1:02
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    $\begingroup$ If it's not self-indexing you get a cell complex but not a CW-complex. i.e. the manifold $M$ is homotopy equivalent to a space which is constructed by inductively attaching cells (the cell dimensions do not need to be in increasing). Such things have the homotopy-type of CW-complexes by Whitehead's theorems on cell complexes. These are in Hatcher's Algebraic Topology text. $\endgroup$ Commented Jan 11, 2010 at 1:21
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    $\begingroup$ Ryan -- that's true, but if I understand correctly, we still get a CW-complex homotopy equivalent to $M$ with cells corresponding to the critical points of $f$ (essentially, by "pushing" the attaching maps out of too high dimensional cells). So the question of comparing the resulting cellular chain complex with the Morse complex still makes sense. Am I wrong? But there is no hope to obtain a genuine CW decomposition in this way, if the function is not self-indexing. $\endgroup$ Commented Jan 11, 2010 at 2:15
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    $\begingroup$ If your cells aren't attached in increasing order you have to do some work to construct a cellular chain complex -- a simple example would be to attach a 1-cell to $S^2$. What's the cellular chain complex? In this situation you can't define it to be $H_k(X^k,X^{k-1})$. You get a homotopy-equivalent CW-complex but then the cellular chain complex isn't well defined (at least, not from the Morse function). It's only defined up to chain equivalence coming from the choice of homotopy-equivalence with a CW-complex. $\endgroup$ Commented Jan 11, 2010 at 2:31