Cutting a torus along a surface non-separating cycle

Question

Is it true that if we cut a torus along a surface non-separating cycle then it will form a cylinder? It's clearly true for a simple surface non-separating cycle that is like a ring on a donut. But what about more complex surface non-separating cycles with larger winding numbers?

Answer 1

Yes, this follows from the classification of surfaces (with boundary) and an Euler characteristic calculation.

If S is the surface obtained after cutting then then note that we have expressed the torus as a connect sum $T = A \# S$ (by taking a small neighbourhood around the curve you cut along). T has Euler characteristic $0$ and anulus (= cylinder) has Euler characteristic, 0. Hence, by the formula for the Euler characteristic of a connect sum $S$ has Euler characteristic $0$. It also has 2 boundary components and is orientable. Now the result follows from the classification of orientable surfaces boundary, i.e. there is a unique orientable surface with 2 boundary components and Euler characteristic $0$: the anulus (= cylinder).