Intuitively, I understand what it means to take a surface and cut it along a curve. However, I don't feel that I have a comfy, rigorous definition of what this actually means. The book A Primer on Mapping Class Groups by Farb and Margalit use the following definition:
Given a simple closed curve $\alpha$ in a surface $S$, the surface obtained by cutting $S$ along $\alpha$ is a compact surface $S_{\alpha}$ equipped with a homeomorphism $h$ between two of its boundary components so that
the quotient $S_{\alpha}/(x\sim h(x))$ is homeomorphic to $S$, and
the image of these distinguished boundary components under this quotient map is $\alpha$.
Something about this definition feels lacking to me. Are there other equivalent definitions out there, or should I learn to work with this definition better?
Yes, of course:
Suppose that $\alpha$ is a smooth simple loop on $S$; equip $S$ with a complete Riemannian metric $g$. Then $S- \alpha$ has an incomplete Riemannian metric obtained by the restriction of $g$. Take $S'$ to be the completion of $(S- \alpha,g)$. This is your surface with boundary obtained by splitting of $S$ along $\alpha$.
Instead of using Riemannian geometry one can use PL topology. Triangulate $S$ so that $\alpha$ is a subcomplex. Let $N(\alpha)$ be the (closed) regular neighborhood of $\alpha$ in $S$. Then $S'$ is the complement $S- int(N(\alpha))$.
You can combine 1 and 2 by assuming that $\alpha$ is smooth and remove from $S$ an open tubular neighborhood ${N}^\circ(\alpha)$ of $\alpha$.