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I am reading Morse Homology from the book by Audin and Damian. I have read the proof about $\overline{\mathcal{L}(a,b)}$ (space of broken trajectories connecting two critical points) being a manifold with boundary when $ind(a)-ind(b)=2$ , but I don't get why $\mathcal{L}(a,b)$ is dense in $\overline{\mathcal{L}(a,b)}$. Actually, I don't even get why $\mathcal{L}(a,c) \times \mathcal{L}(c,b) \neq \emptyset$ implies $\mathcal{L}(a,b) \neq \emptyset$ and I was wondering how we could extend this argument in general and prove the fact that $\overline{\mathcal{L}(a,b)}$ is a manifold with corner with $dim=ind(a)-ind(b)-1$ .

Do you know other references on this subject?

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  • $\begingroup$ $\overline{\mathcal{L}(a,b)}$ is by construction is the compactification of $\mathcal{L}(a,b)$ so $\mathcal{L}(a,b)$ must be dense. Where do you see $\mathcal{L}(a,c) \times \mathcal{L}(c,b) \neq \emptyset$ implies $\mathcal{L}(a,b) \neq \emptyset$ ? By the way it's nicer if at least you provide an image of a specific page of the book (if you don't want to or cannot write up the whole thing). Generally people may not familiar with the notation. $\endgroup$ Jun 4, 2020 at 23:45
  • $\begingroup$ This argument is in page 61 to 67 of the book. $\overline{\mathcal{L}(a,b)}$ is the space of broken trajectories and it is not implicit in the definition that this space is a compactification of $\mathcal{L}(a,b)$. You need to prove $\mathcal{L}(a,b)$ is dense. See proposition 3.2.6 in the book. $\endgroup$
    – ali_ns
    Jun 5, 2020 at 8:10
  • $\begingroup$ Well, $\overline{\mathcal{L}(a,b)}$ is defined to be the compactification of $\mathcal{L}(a,b)$ with respect to the $C^{\infty}_{loc}$-topology, in particular $\overline{\mathcal{L}(a,b)} \supset \mathcal{L}(a,b)$. There is something in Morse/Floer theory called "gluing", which basically asserts that every broken trajectory is attained as the $C^{\infty}_{Loc}$-limit of a sequence of genuine trajectories. Thus $\mathcal{L}(a,c)\times \mathcal{L}(c,b) \neq \emptyset$ implies $\mathcal{L}(a,b) \neq \emptyset$. $\endgroup$
    – noctusraid
    Jun 30, 2020 at 13:42
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    $\begingroup$ As for references, there are many: Schwarz's book on Morse Homology (although that might be too hardcore), Salamon's Floer Theory notes contain a subchapter on Morse homology. Ritter's notes (here: people.maths.ox.ac.uk/ritter/morse-cambridge/full.pdf) might be a good place to start. $\endgroup$
    – noctusraid
    Jun 30, 2020 at 13:45

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Proposition 3.2.8 in Audin-Damian is what you're looking for. For any element $\lambda = (\lambda_1,\lambda_2) \in \mathcal{L}(a,c) \times \mathcal{L}(c,b)$ there is a continuous map $\psi:[0,\delta) \to \overline{\mathcal{L}}(a,b)$ (for some $\delta>0$) such that $\psi(0) = \lambda$ and $\psi(s) \in \mathcal{L}(a,b)$. So you get density since $\mathcal{L}(a,b) \owns \psi(1/n) \to_{n \to \infty} \psi(0) = \lambda \in \mathcal{L}(a,c) \times \mathcal{L}(c,b)$. On the other hand, if there is any element $ \lambda$ as above, then $\psi(\delta/2) \in \mathcal{L}(a,b)$, proving $\mathcal{L}(a,b)$ is nonempty.

Also, if you accept the claim that $\overline{\mathcal{L}}(a,b)$ is a 1-dimensional manifold with boundary such that its interior is $\mathcal{L}(a,b)$, then both of your statements come from standard facts about manifolds: if $M$ is an $n$-dimensional ($n \geq 1$) manifold with boundary $\partial M$, then $\partial M$ is contained in the closure of $\mathrm{Int}\,M$ inside $M$, and also $\partial M \neq \varnothing \implies M \neq \varnothing$. (Hint for proving that: firstly, $\mathbb{R}^{n-1}\times \{0\}$ is in the closure of $\mathbb{R}^{n-1} \times (0,\infty)$ inside $\mathbb{R}^n$; secondly, any open set containing an element of $\mathbb{R}^{n-1} \times \{0\}$ contains an element of $\mathbb{R}^{n-1} \times (0,\infty)$).

For higher difference of indices, the compactified moduli space is also a manifold with corners, in some sense, but the proof is much more involved. It is proven by Wehrheim: https://arxiv.org/abs/1205.0713 . But it is quite subtle: only after requiring some technical conditions there is a canonical structure of a manifold with corners on that space.

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