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I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem.

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Basically we want to prove that the Morse homology is independent of the pseudo-gradient field and the morse function. Since the proof is a little bit long and I do not want to just summarize it, here is a link to the proof presented in the book.

http://www.math.stonybrook.edu/~sunscorch/quals/Major/Morse_Invariance.pdf

So, my confusion starts when the author proves that $(C_*(\tilde{F}|_{V \times A}),\partial _{\tilde{X}}) = (C_{*+1}(f_0), \partial_{X_0})$ (page 2 of pdf almost at the end of the page) and similarly for the other equality. After that he states that there are only two trajectories connecting critical points of $\tilde{F}$. So, I have three questions:

  • The first type of trajectories are the ones staying always in $A$, and I argued that because of $(C_*(\tilde{F}|_{V \times A}),\partial _{\tilde{X}}) = (C_{*+1}(f_0), \partial_{X_0})$. So, like analyzing the trajectories in such section is just as analyzing the trajectories of $X_0$, similarly for $X_1$. Is this the correct idea?
  • I do not get why there can only be trajectories from critical points of $f_0$ to those of $f_1$ (which are the second type of trajectories between critical points of $\tilde{F}$). My first thought about why we cannot have trajectories from critical points of $f_1$ to $f_0$ is because of how their indexes are related to the indexes of $\tilde{F}$ (almost at the beginning of page 2 of pdf) , but I am not sure why this would be true. Any suggestion?
  • My final question is in the definition of $\partial_{\tilde{X}}$. I do not know why in the first row, second column of its matrix representation, it is the zero matrix. Any ideas? (My hypothesis is that since we can only have trajectories as those described in the second bullet point, the number of trajectories from critical points of $f_1$ to those of $f_0$ and by definition of $\partial_{\tilde{X}}$, then that is why it is the zero matrix).

Again, I would really appreciate your help here. Maybe this is the easiest part of the proof and I am missing a basic fact, but I do not see it. Thanks in advance.

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For context, the Morse function $\widetilde{F} : V \times [-1/3, 4/3] \to \Bbb R$ is defined as $\widetilde{F} := F + g$ where $F : V \times [-1/3, 4/3] \to \Bbb R$ is a homotopy between $f_0$ and $f_1$ extended to a collar of $V \times I$ so that $F \equiv f_0$ throughout the time-interval $[-1/3, 1/3]$ and $F \equiv f_1$ throughout the time-interval $[2/3, 4/3]$, and $g : [-1/3, 4/3] \to \Bbb R$ is a Morse function such that $0$, $1$ are the maximum and minimum respectively, and moreover $g$ decreases fast enough in between such that $\partial_t F(x, t) + g'(t) < 0$ throughout $t \in (0, 1)$.

The condition $\partial_t F(x, t) + g'(t) < 0$ means $\partial_t \widetilde{F}(x, t) < 0$ for all $t \in (0, 1)$ as well; so the critical points of $\widetilde{F}$ can only occur on the slices $V \times \{0, 1\}$, around a neighborhood of which $\widetilde{F} \equiv f_0, f_1$ respectively. Loosely speaking, along the horizontal slices $V \times \{t\}$, $\widetilde{F}$ looks like the movie of the homotopy $F$, whereas along the vertical slices $\{v\} \times I$, $\widetilde{F}$ retains the cubic behavior of $g$. As a consequence, the critical set is $$\mathrm{Crit}(\widetilde{F}) = \mathrm{Crit}(f_0) \times \{0\} \sqcup \mathrm{Crit}(f_1) \times \{1\}$$

The pseudogradient vector field $\widetilde{X}$ is adapted to $\widetilde{F}$, so the critical points of $\widetilde{X}$ are the same as that of $\widetilde{F}$, which means that the heteroclinic flowlines have to begin and end at a pair of critical points of $\widetilde{F}$. Thus, there are three possibilities for a heteroclinic flowline $\gamma$ of $\widetilde{X}$:

$(1)$ $\gamma$ begins and ends at a pair of points in $\mathrm{Crit}(f_0) \times \{0\}$.

$(2)$ $\gamma$ begins and ends at a pair of points in $\mathrm{Crit}(f_1) \times \{1\}$.

$(3)$ $\gamma$ begins at a point in $\mathrm{Crit}(f_0) \times \{0\}$ and ends at a point in $\mathrm{Crit}(f_1) \times \{1\}$.

Since $\widetilde{X}$ is $C^1$-close to $X_0 + \nabla g$ and $X_1 + \nabla g$ on $V \times [-1/3, 1/3]$ and $V \times [2/3, 4/3]$ respectively, this forces the flowlines of the first two kinds stay inside $V \times [-1/3, 1/3]$ and $V \times [2/3, 4/3]$ respectively. The flowlines of the third kind crosses over from one side to the other.

The Morse-Smale complex decomposes as $C_k(V, \widetilde{F}, \widetilde{X}) = C_{k-1}(V, f_0, X_0) \oplus C_k(V, f_1, X_1)$ since critical points of index $k-1$ of $f_0$ gives rise to critical points of index $k$ of $\widetilde{F}$, as the descending flowline of $g$ emanating out of that point contributes to an extra dimension of unstable direction, whereas a critical point of index $k$ of $f_1$ remains a critical point of index $k$ of $\widetilde{F}$. The differentials $\partial_{\widetilde{X}}$ can therefore be written as a $2\times 2$ matrix, determined by the action on each component. The reason it is lower triangular is by our analysis of the possible flowlines above, any heteroclinic flowline starting at a point on $V \times \{1\}$ must end at a point on $V \times \{1\}$, therefore: $\partial_{\widetilde{X}} C_*(V, f_1, X_1) \subseteq C_{*-1}(V, f_1, X_1)$.


I think the above proof is interesting, but also a little bit of a trick which I don't understand very well. So let me say a few words on how I would intuit the invariance of Morse homology.

Given a Morse-Smale pair $(f, X)$ on a manifold $V$, one can write down a handlebody decomposition of $V$ corresponding to this pair. The Morse complex can equivalently be described as by writing down the chain groups $C_k$ generated by the $k$-handles of $V$, and the differential $\partial H$ given by linear combination of the $k-1$-handles $h_i$'s which $H$ is attached to, weighted by the intersection number of the attaching sphere of $H$ with that of the belt sphere of $h_i$.

According to the Morse-Cerf theorem, any pair of Morse functions $f_0, f_1 : V \to \Bbb R$ can be interpolated by a homotopy of functions which are Morse except at finitely many times, when it has at worst cubic singularities. In handle-theoretic language this says the handle decompositions obtained from $f_0$ and $f_1$ are related by handle-deaths, handle-births or handle-slides. Cancellation or creation of a pair of cancellable handles are simple homotopy equivalences of the corresponding Morse complexes, so this gives a proof of invariance.

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    $\begingroup$ Just a remark: the "continuation map" proofs are common in Floer theory, where Cerf theory is unavailable, so this is a good "warm-up" for those. $\endgroup$
    – Max
    May 15, 2021 at 16:34
  • $\begingroup$ @Max Thank you for the comment. Is it clear that there should be no correct theory of infinite-dimensional handlebody decompositions? For example, Morse theory on loopspaces gives the usual CW decomposition of $\Omega S^n$ with a single $kn$-cell for all $k \geq 0$. It is conceivable that in general one would need "finite-codimensional handles" as well. This is an idle question, not a serious one. Do you have a good reference for such invariance proofs in Floer theory? I would eventually like to read. $\endgroup$ May 15, 2021 at 16:47
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    $\begingroup$ @Baralaka Sen Morse theory still has finite index (the dimension of unstable submanifold), and so allows the usual handle attachment picture. In Floer theory both stable and unstable submanifolds are infinite-dimensional so there is no index, and only index difference still makes sense (via spectral flow/Conley-Zehnder index). Apparently, there is a more sophisticated view where Floer theory can be done via "semi-infinite cycles". Max Lipyanski worked on this before he left academia (arxiv.org/abs/0911.3714, arxiv.org/abs/1409.1126). $\endgroup$
    – Max
    May 15, 2021 at 23:54

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