 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2009/a/1/iv/solution

A square model of the torus has one vertex, two edges and one face, so its Euler characteristic is zero. The connected-sum formula gives

χ (#^{g} T^{2}) = g \cdot 0 - 2 (g - 1) = 2 - 2 g .

The real projective plane has a bigon model with one vertex, one edge and one face, hence characteristic one. Therefore

χ (#^{h} RP^{2}) = h - 2 (h - 1) = 2 - h .

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