Compact orientable surfaces are classified (up to homeomorphism) by its genus. The genus is just a natural number. Thus we can enumerate them: The genus zero surface is the sphere , the genus one surface is the torus , the genus two surface can be constructed by taking two toruses, cutting a little disc out of each torus, and then gluing the two boundaries that we created when cutting the holes. In this way we can construct the genus compact orientable surface for each . The operation of cutting a little disc out of two (connected) surfaces and gluing them by the boundary is called the connected sum. For two surfaces and their connected sum is usually denoted by . For we denote the genus surface by . When is zero we define the connected sum of zero surfaces to be the sphere .
Problem 1: Prove that covers for any .
The classification of compact non-orientable surfaces is similar. If is the projective plane then a compact non-orientable surface is homeomorphic to for some positive natural number .
Problem 2: Prove that covers for any positive natural .
Problem 3: Notice that is a compact surface. Describe it.
Problem 4: Prove that covers for any .
A nice corollary of this exercise is that, since composition of finite coverings is again a covering, the fundamental group of any compact surface is a subgroup of the fundamental group of , except maybe for and .