= \pi_{\Sigma_i}(p_i(s))$, and $$\pi_{\Sigma_i}(\Sigma_i)\cap p({\mathbb{R}}/{\mathbb{Z}}\times [0,1])  = \pi_{\Sigma_i}(p_i({\mathbb{R}}/{\mathbb{Z}})) = p({\mathbb{R}}/{\mathbb{Z}}\times \lbrace i\rbrace),$$

    -   $c \in H_1 (W, {{\mathbb{Z}}_{2}})$.

    We will refer to $p({\mathbb{R}}/{\mathbb{Z}}\times [0,1]) $ as the vertical part of $\partial W$, and will denote it $\partial^{vert} W$.

    Two such 5-tuples $(W, \pi_{\Sigma_0},  \pi_{\Sigma_1}, p,  c)$ and $(W', \pi'_{\Sigma_0},  \pi'_{\Sigma_1}, p',  c')$ will be identified if there exists a diffeomorphism $\varphi \colon W \rightarrow W'$ compatible with the embeddings and preserving the classes.

-   composition of morphisms consists in gluing along the embeddings, and adding the homology classes.

We will not assign a Lagrangian correspondence to any cobordism, but only to certain elementary ones, in the sense of Morse theory. We will see that an arbitrary cobordism always decomposes into a finite number of such elementary cobordisms, hence giving rise to a sequence of Lagrangian correspondences, namely a morphism of ${\textbf{Symp}}$. The aim of this paragraph is to prepare the proof of the fact that this construction doesn’t depend on the decomposition of the cobordism. We will essentially adapt the results of [@connectedcerf] to the framework of ${\textbf{Cob}}$. Recall its principal one, which is false in dimension $1+1$:

\[GWWtheo\] Let $n\geq 2$,

1.  Every connected $(n+1)$-cobordism between connected $n$-manifolds admits 