A surface is a two-dimensional manifold. The classification of compact surfaces was “known,” in some sense, by the end of the nineteenth century. Möbius  and Jordan  offered proofs (for orientable surfaces in ℝ3) in the 1860’s. Möbius’ paper is quite interesting; in fact he used a Morse-theoretic approach similar to the one presented in this chapter. The main interest in Jordan’s attempt is in showing how the work of an outstanding mathematician can appear nonsensical a century later.
KeywordsBoundary Component Euler Characteristic Compact Surface Morse Function Klein Bottle
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