 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2010/a/1/iii/c/solution

ab a^{- 1} b

gives the Klein bottle. The side identifications produce one vertex, two edges and one face. Consequently

χ (K) = 1 - 2 + 1 = 0.

Unlike the torus, this surface is nonorientable; its equal Euler characteristic alone does not distinguish them.

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