I’m asking this question because I’d like to understand better the neck-stretching argument in symplectic geometry and what kind of conclusions one might get out of it in my setting.

Assume that I’ve a Lefschetz fibration $E$ over $M$, where $M$ is an exact symplectic manifold (maybe with ends). Let $C$ be a contact type codimension $1$ submanifold of $M$. As far as I understood it, if I stretch the neck around $C$ I create a family of manifolds $M_t$, $t\in \Bbb R^+$ (with $M_0=M$) all symplectomorphic to $M$, “representing the stretching of the neck” around $C$ at time $t \geq 0$.

According to the thesis of Evans **[E]**, this process tend to a limit $M_{\infty}$ which is obtained from $M$ by cutting along $C$ and gluing the positive/negative part of the symplectisation of $C$ (in the thesis it’s denoted as $\Bbb S_{\pm}(C)$. I do believe that similarly this process should give rise to a family of Lefschetz fibrations $E_t$ over $M_t$ and a limiting fibration $E_{\infty}$ (I believe that we attach some trivial fibration over the ends we are introducing).

I’m interested in understanding the moduli space of pseudo-holomorphic sections of such fibration. Let us assume that I’ve chosen a generic compatible a.c. Structure $J$ and I remember that I don’t have bubbling due to my exactness assumption. Hence my stretching procedure gives rise to a family of manifolds $\mathcal{M}_t$. What can I say about this family of moduli spaces? How do I treat the limiting case $\mathcal{M}_{\infty}$? Is it somehow a boundary piece for the family of spaces parametrized by the positive reals?

I think I can come up with some naive reasoning supporting my claim, but since I’m trying to push something to the limit I’m expecting a lot of technicalities involved and I can’t appreciate them yet.

Since I’m rather new to the field, can someone points me if there are any obvious obstruction/difficulties in what I’m trying to understand and if there are any literature that Somehow sheds some light on what I’m trying to understand?

Thanks in advance for any comments!

**References**

**[E]** Jonathan David Evans - Symplectic topology of some Stein and rational surfaces