I am an undergraduate student in mathematics and decided to focus on topology, algebraic topology, and I decided to start with surfaces. I am reading a few books about the subject and a lot of proofs use arguments that sound like : « by cutting [a surface] along [a simple curve], we get something homeomorphic to [some known thing] which gives us ... ». This arguments are generally convincing visually but I couldn’t manage to fill in all the formal details of such proofs if I were asked to. I understand better the process of pasting, which I formalize as a disjoint union and a quotient, but I am not sure how to formalize this « cutting » process, as intuitive as it is. If one wants to know, the proofs I read that contain this type of reasonning are in « A Primer on Mapping Class Groups » , Farb & Margalit.
Thanks a lot !