# How to prove that the complex in Morse homology is isomorphic to the one in cellular homology

Since every stable submanifold with orientation in Morse homology is actually a cell in cellular homology, it suffices to prove the two boundary map coincide. Intuitively one may accept it is true by noticing the degree of a map between two cpt manifold is the number of points(with orientation) in a generic fiber, because in cellular homology the degree of certain map becomes the fiber number which is closely related to the flow line number in Morse homology. However, it is rough. There are lots of technical problem for eg. the orientation, and why the fiber of critical point is generic.

Is there any reference directly proving in this way? Or could anybody provide me some hint on how to do the technical work (Idea will suffice. I think I know some basic differential topology)?

It doesn't matter whether the Morse function gives a CW-decomposition or a CW-cplx homotopy type.(But I still believe for a self-indexed Morse function it does give a CW-decomposition. Ahh probably these terms involved are not standard ones. By self-indexed Morse function I mean f(p)=Ind(p) for a critical point $p\in M$.) Anyway, a critical point is associated with a cell in the homotopy-equivalent CW-cplx, which generates the chain complex in the cellular homology. Recall that the chain complex of Morse homology associated to a self-indexed Morse function is a graded free module generated by critical points with "orientation", with differential determined by counting the intersection number of some pair of stable and unstable submanifolds.

Every graded part of above two complex coincide(generated by critical points with some kind of "orientation" specified), and we have reasons to believe the two differential might be the same(Try applying the homotopy-equivalent then taking quotients and see the degree of certain map, which in the sense of fiber number is very likely to be the same as the intersection number I mentioned in Morse homology. Indeed I'm omitting something, but I think they are too standard.) I'm asking a way to identify the two differentials rigorously. Any further question?

It seems I should explain more things. At first I thought what I was asking might be a standard result, and these might suffice to remind you. Probably I was wrong.

M be the compact manifold. Given a Morse-Smale pair (f,g) with f self-indexing.(I looked up Milnor's book realizing "self-indexing" is the right term.) The Morse homology with respect to (f,g) (denoted by $C^M_*$) and the cellular homology of the CW-cplx associated to f (denoted by $C^{CW}_*$) are mentioned. It's clear every graded part of two chains coincide. Now checking the differentials. $p_i$ and $q_j$ will denote either critical points of index k and k-1 resp. or the corresponding k and k-1 cells in the CW-cplx corresponding to f resp.(I suppose the context is clear, and it shouldn't involve ambiguity)

$d_k:C^{CW}_k\rightarrow C^{CW}_{k-1}$ is defined by $d_k(p_i)=\sum_j deg(\Phi_j)q_j$ where $\Phi_j:S^{k-1}\rightarrow X^{k-1}\rightarrow S^{k-1}$ with the first map the corresponding attaching map, last map the quotient map killing everything outside $q_j$.

$d_k:C^{M}_k\rightarrow C^{M}_{k-1}$ is defined by $d_k(p_i)=\sum_j \#(w^-_{p_i}\cap w^+_{q_j})q_j$ where $w^-_{p_i}$, $w^+_{q_j}$ are corresponding stable and unstable submanifold resp.

Orientation matters above are assumed to be taken care of.

To prove they have the same coefficient, it suffices to prove $deg(\Phi_j)=\#(w^-_{p_i}\cap w^+_{q_j})$. Now one must identify the $f_j$ in order to calculate.

What I'd like to prove is the above equation. Below is why I thought there might be a proof just in a direct way.

There is a htpy-equivalence(I suppose one can actually choose a deformation retract. Am I right?) $h_{n-1}:M_{n-\epsilon}\rightarrow X^{n-1}$ the n-1 skeleton of the CW cplx associated to f. $M_{n-\epsilon}\cup_{f_i} e^n_i$ is homotopy-equiv to $X^{n-1}\cup_{h_{n-1}\circ f_i} e^n_i$(As in Milnor's book near the $p_i$ under the Morse coordinate, attaching along $f_i$ is just adding a disc between two hyperbola). $h_{n-1}\circ f_i$ should be the attaching map. We have to identify its degree after quotient. Since the $\#(w^-_{p_i}\cap w^+_{q_j})$ is the same as the number of flow lines between $p_i$ and $q_j$(in some sense), intuitively by tracing along $h_{n-1}$ I can see the fiber over a certain point of $h_{n-1}\circ f_i$ is indeed the flow line number. But I didn't take care of the orientation, and I cannot conclude what is the degree by just observing one fiber.

(I looked at my original question again and found there is some ambiguity indeed. Thanks for helping me clarify this.)

I happen to see a related question on MO, though still no answer to mine is found. http://mathoverflow.net/questions/11375/cw-structures-and-morse-functions-a-reference-request. Actually I was asking exactly for the hint of the last sentence of Tim Perutz's answer, the "more painful exercise to do it over Z".

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Your question is too vague. Precisely which two complexes are you trying to relate? – Ryan Budney Jul 31 '11 at 7:07
Budney, I think for a compact manifold and a self-indexed Morse function f there is only a fixed way to obtain the chain complex of Morse homology. And since the CW decomposition follows from f, we have also a fixed way to obtain the chain complex of cellular homology. Sorry but where did my question become vague? – Honglu Jul 31 '11 at 12:57
I'm still not clear what you're talking about. Morse functions do not (easily) give CW-decompositions of spaces, they give homotopy-equivalences to CW-complexes. So either you're being sloppy or you're not saying what it is you're trying to do. Please explicitly mention what it is you're doing, because you have not. – Ryan Budney Aug 1 '11 at 5:50
I added more details in the end of the question. Anything still unclear? – Honglu Aug 1 '11 at 6:43
Well...I think I added all the detail... – Honglu Aug 2 '11 at 3:36