So I have a (compact) surface $\Sigma$ and two open disks on the surface call them $A$ and $A'$ such that the intersection contains a simple curve $P$. What I want to do is construct an isotopy between $A$ and $A'$ which leaves $P$ fixed.
I'll let you know what I have so far, but maybe there is a cleaner way to do this. In the intersection there is an open set, U, around $P$ which is homeomorphic to the disk, (simply take an $\varepsilon$ neighborhood of $P$). I want to stretch $U - P$ via some isotopy $F$ to the remainder of $A$ and then another one $G$ which takes $U-P$ to the remainder of $A'$. I'm pretty sure that would be enough but I'm not sure how to work that bit of detail out though.
Once we have those two I can compose those two isotopies (well one with it's inverse) to get what I want.
Any hints to my missing part?
Edit: Corrected an error.