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11

# Classification (Part II), Existence and Four-Space

The main outcome of Chapter 10 was a list of all possible surfaces. This chapter builds on this in two ways. One way is to show that the list has no duplications. This is accomplished by showing that orientability and number of lakes are invariant under the equivalencies established for surfaces, that is, if surfaces S and S′ are equivalent and S is orientable with s lakes, then S′ is also orientable with s lakes. The second way this chapter builds on Chapter 10’s classification theorem is to deal with the question of existence. There is no question that many of the items on the list can actually be assembled in three-space. However, there are questions about the cross-cap and the Kline bottle. ...

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