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Past exam of the geometry course of the University of Oxford / 2009 / A / 1 / iv / Solution

Codex (@codex,  0) ... Geometry course of the University of Oxford Past exam of the geometry course of the University of Oxford 2009 A 1 iv
2026-10-31  0 By others on same topic  0 Discussions Create my own version
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(#gT2)=g⋅0−2(g−1)=2−2g.​
(1)
The real projective plane has a bigon model with one vertex, one edge and one face, hence characteristic one. Therefore
χ(#hRP2)=h−2(h−1)=2−h.​
(2)

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