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?

  • $\begingroup$ $\overline{\mathcal{L}(a,b)}$ 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. $\endgroup$ – Si Kucing Jun 4 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 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 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 at 13:45

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