There are two definitons of genus of a connected, orientable surface. The first one defines genus as an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. The second one — as the number of handles on it.
Question. How to prove that they are equivalent?
So far, I have tried to prove that from the second definition follows the first one. I understood, that cutting of closed curve leaves manifold connected if this curve is homeomorphic to $a$ cycle (defined in the attached picture) of one of the torus (I consider handles as torus with holes sticked to the sphere). For the case of curve lying on sphere, by the Jordan Curve Theorem (any simple closed curve separates $S^2$ into two connected components) we obtain that such curves divide surface into two connected components . I don't understand how to deal with the arbitrary closed curve on $M_g$, e.g., if it goes along several handles in different ways. I have thought about splitting the manifold into sphere and handles, thus, splitting the curve into smaller parts, and dealing with each part separately, but this doesn't seem to lead me anywhere.
