# Morse Homology and The Space of Broken Trajectories

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?

• $\overline{\mathcal{L}(a,b)}$ 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. Jun 4, 2020 at 23:45
• 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. Jun 5, 2020 at 8:10
• 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 $\mathcal{L}(a,b) \neq \emptyset$. Jun 30, 2020 at 13:42
• 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. Jun 30, 2020 at 13:45

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 $$\mathbb{R}^{n-1} \times (0,\infty)$$).