I'm reading "Morse Theory and Floer Homology" by Audin-Damian and got stuck in the proof that the Morse differential indeed induces a homology (i.e. $\partial^2=0$). This is basically the compactification theorem of broken trajectories, Section 3.2 in the book. What I can't see is how Proposition 3.2.8 implies Theorem 3.2.7. Here below the details.

Theorem 3.2.7: If Ind(a) = Ind(b) + 2, then $\bar{\mathcal{L}}(a, b)$ is a compact manifold of dimension 1 with boundary.

Proposition 3.2.8: Let $f$ be a Morse function on a compact manifold $M$ and with pseudo-gradient $X$ satisfying the Smale condition. Let $a,c,b\in crit(f)$ with index $k-1,k,k+1$ respectively and $\lambda_1\in\mathcal{L}(a,c)$ and $\lambda_2\in\mathcal{L}(c,b)$. Then there is a open $O_{(\lambda_1,\lambda_2)}\subseteq\overline{\mathcal{L}}(a,b)$ and a continuous injection $\psi:[0,\delta)\rightarrow O_{(\lambda_1,\lambda_2)}$ differentiable on the interior and such that \begin{align*} \begin{cases} \psi(0)=(\lambda_1,\lambda_2)\in \overline{\mathcal{L}}(a,b)\\ \psi(s)\in \mathcal{L}(a,b)\ \forall s>0 \end{cases}. \end{align*}

I suppose that this embedding should allow us to show that the dimension of $\overline{\mathcal{L}}(a,b)$ is the same as ${\mathcal{L}}(a,b)$ which we already know, but I can't recall a theorem that states that or see a way to show it.

Thank you in advance


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