Covering Surfaces

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 S^2, the genus one surface is the torus T^2, 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 n compact orientable surface for each n. 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 A and B their connected sum is usually denoted by A \# B. For n \geq 0 we denote the genus n surface by n T^2 = T^2 \# \ldots \# T^2. When n is zero we define the connected sum of zero surfaces to be the sphere S^2.

Problem 1: Prove that n T^2 covers 2 T^2 for any n>1.

The classification of compact non-orientable surfaces is similar. If P^2 is the projective plane then a compact non-orientable surface is homeomorphic to n P^2 for some positive natural number n.

Problem 2: Prove that (n-1)T^2 covers n P^2 for any positive natural n.

Problem 3: Notice that T^2 \# P^2 is a compact surface. Describe it.

Problem 4: Prove that n P^2 covers 3 P^2 for any n\geq 3.

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 3 P^2, except maybe for T^2 and 2 P^2.

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Covering Surfaces

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