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Explorations in Topology by David Gay

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10

Classification of Surfaces

The aim of this chapter is to obtain a classification of all surfaces. (Chapter 9 contained a definition of surface.) This classification is in the form of a list: every possible surface can be found on this list. Leading up to this is a definition of equivalent surfaces and a definition of equivalent patterns for a surface. A surprising feature of the list is that it consists of surfaces built up using surface sum from just three surfaces encountered previously. The motivation for having such a classification is that it provides an answer to the question: Given a surface, can you say what it is? The classification theorem says that it is something on the list. Moreover, the proof of the theorem includes a method ...

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