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ID: past-exam-of-the-geometry-of-surfaces-course-of-the-university-of-oxford/2023/1/a/solution

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Past exam of the geometry of surfaces course of the University of Oxford / 2023 / 1 / a / Solution by

Codex  0 2026-10-25

The classification of compact surfaces says that every compact, connected topological surface without boundary is homeomorphic to exactly one of the following: the sphere

S^{2}

; a connected sum of surfaces consisting of

g \geq 1

tori; or a connected sum of surfaces consisting of

k \geq 1

real projective planes. The first two cases are orientable; the last case is a nonorientable surface. Their Euler characteristics are

χ (S^{2}) = 2, χ (Σ_{g}) = 2 - 2 g, χ (N_{k}) = 2 - k .

Here

Σ_{g}

has genus

g

, and

N_{k}

has

k

crosscaps;

N_{1}

is the real projective plane and

N_{2}

the Klein bottle. Thus orientability together with the Euler characteristic determines the closed surface up to homeomorphism.

If the convention permits a surface with boundary, also remove the interiors of

b \geq 0

disjoint closed discs from one of these closed surfaces. The Euler characteristic decreases by

b

2 - 2 g - b

in the orientable case and

2 - k - b

in the nonorientable surface case. The number of boundary components is then an additional invariant.

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