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4.5.1 Surfaces with holes

Using this result, we can obtain the Euler characteristic of a surface with any number of holes by successively inserting the holes one at a time. For example, since a closed disc has Euler characteristic 1, it follows that a closed disc with 1 hole has Euler characteristic 0, a disc with 2 holes has Euler characteristic −1, and so on.

Problem 22

Use the above approach to determine the Euler characteristic of a torus with k holes.


Since a torus has Euler characteristic 0, it follows from Theorem 10 that a torus with 1 hole has Euler characteristic −1, a torus with 2 holes has Euler characteristic −2, and so on. In general, a torus with k holes has Euler characteristic −k.

Theorem 11

Inserting k holes into a surface reduces its Euler characteristic by k.


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