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The following is probably well known, but I wasn't able to locate a reference in the literature.

  1. 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)?

  2. 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.

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6 Answers 6

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.

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My outline was pretty vague on the regular neighbourhoods part of the argument, so it's nice to have this reference. –  Ryan Budney Jan 11 '10 at 3:26

I'm confused about your 2nd question. Milnor's Morse theory does use a Riemann metric -- he's using the gradient flow. To define the gradient he needs an inner product on the tangent spaces. Without the gradient flow you don't have the cellular attaching maps.

Regarding your 1st question, there's something that's much better than a CW-structure. A Morse function gives a handle decomposition of the manifold. This can be used to talk about the smooth structure. A CW-decomposition is relatively degenerate in comparison. The handle decomposition is described in Milnor's h-cobordism notes.

Taking your 1st question more seriously, you run into technical problems. The gradient flows do not give you a CW-decomposition of the manifold -- for example consider Milnor's Morse Theory example of a torus with height function. The Morse function and its gradient flows gives you a genuine 1-skeleton (figure-8). But the attaching map for the 2-cell (to the figure-8) is not a continuous function if you use the gradient flow -- all points except for two go to the global minimum for the height function. This shows you the kind of problems you encounter if you want to produce a genuine CW-decomposition of the manifold.

So if you're not going to use solely the gradient flows to define the attaching maps for the proposed CW-decomposition, what do you allow? All smooth manifolds admit CW-decompositions so if you allow sufficient tweaking you can of course fix this construction but if you allow "too much" tweaking, the CW-decomposition won't be an invariant of the Morse function.

edit: Here is a way to tweak the process. The gradient flow does give you a genuine 1-skeleton. So take a regular neighbourhood of the 1-skeleton, and perturb the original vector field in this regular neighbourhood to point in towards the 1-skeleton. This makes the 2-cell attaching maps continuous (terminating in a finite amount of time). Then take a regular neighbourhood of the 2-skeleton, and perturb the vector field to point in towards the 2-skeleton. Again, you get flow lines terminating in finite-time so you get genuine 2-cell attaching maps. The problem with this is you're getting a CW-decomposition but it depends on more than the Morse function as you need to choose smooth regular neighbourhoods of the skeleta.

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Ryan, thanks! Yes, a metric is necessary. Will fix that. As to CW-structures on manifolds: how does one prove they exist without using, say, triangulations? I always assumed this can be done using Morse theory in some way ot another, but never bothered to find out, until now. As to how much tweaking should be allowed: I'm not sure myself; I was sort of hoping this had been discussed in the literature. –  algori Jan 11 '10 at 0:20
    
One way would be to follow the outline in my last paragraph, starting with "edit:". I imagine this has been discussed in the literature but I'm not sure where it would be. CW-structures on smooth manifolds aren't nearly as interesting or useful as handle decompositions so I think most people are happy to throw away CW-complexes for handle decompositions. Is this mostly a foundational issue for you or do you have a reason to favour CW-decompositions over handle decompositions of smooth manifolds? –  Ryan Budney Jan 11 '10 at 0:33
    
Thanks again, Ryan! No, this was out of pure curiosity, I don't have any serious reasons to consider CW-structures rather than handle decompositions. If one performs the construction you described, does one get the same cellular chain complex as in the procedure described in Milnor's book? –  algori Jan 11 '10 at 0:56

For 2, I'm going to make the simplifying assumption that $f$ is "weakly self-indexing", i.e. that if $c_1$ and $c_2$ are critical points with $ind(c_1)\geq ind(c_2)$ then $f(c_1)\geq f(c_2)$. This means that the cells are attached in the "right" order.

I claim that in this case the Morse homology complex of a Morse-Smale pair $(\rho,f)$ is isomorphic - not just quasi-isomorphic! - to the cellular homology complex of the handle decomposition. (As Ryan indicates, the latter involves $\rho$ too.)

The isomorphism sends a critical point of index $k$ to the $k$-cell given by its descending manifold. Each matrix entry in the Morse differential counts (with signs) gradient flow-lines from an index $k$ to an index $k-1$ critical point, or equivalently intersections between descending and ascending manifolds. The corresponding matrix entry in the cellular differential is the degree of the map $S^{k-1}\to S^{k-1}$ obtained from the attaching map by collapsing the $(k-2)$-skeleton to get a wedge sum of spheres, then projecting to one summand. But you can "see" the latter map by watching the downward gradient flow of points on the attaching sphere for some large fixed time; most points end up in the $(k-2)$-skeleton; the ones that don't are the ones which (approximately) flow towards an index $k-1$ critical point. This makes it a good exercise to equate matrix entries over $\mathbb{Z}/2$, and a more painful exercise to do it over $\mathbb{Z}$.

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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. –  algori Jan 11 '10 at 0:49
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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). –  Ryan Budney Jan 11 '10 at 1:02
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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. –  Ryan Budney Jan 11 '10 at 1:21
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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. –  algori Jan 11 '10 at 2:15
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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. –  Ryan Budney Jan 11 '10 at 2:31

You may also be interested in a preprint by Cohen, Jones, and Segal, Morse Theory and Classifying Spaces which uses a Morse function to construct a category out of the critical points and the flow lines. The classifying space of this category is homeomorphic to the original manifold (under certain conditions on the Morse function). The preprint can be found on the "papers" section of Ralph Cohen's homepage.

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Historically first reference (as I know) positively answering on your question 1 is the appendix by F. Laudenbach in the paper:

Bismut, Jean-Michel; Zhang, Weiping An extension of a theorem by Cheeger and Müller. (French summary) With an appendix by François Laudenbach. Astérisque No. 205 (1992),

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Seeing that algori is asking for a reference, I'd like to offer Liviu Nicolaescu's "Invitation to Morse Theory" as a superb modern treatment of the subject. I am fairly certain the result you are looking for is in there.

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