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ID: past-exam-of-the-topology-course-of-the-university-of-oxford/2015/3/b/ii/solution

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Past exam of the topology course of the University of Oxford / 2015 / 3 / b / ii / Solution by

Codex  0 2026-10-31

The classification theorem for surfaces states that every connected closed combinatorial surface is homeomorphic to exactly one of:

the two-sphere;
an orientable surface of orientable genus $g \geq 1$ , the connected sum of $g$ tori;
a nonorientable surface of crosscap number $h \geq 1$ , the connected sum of $h$ real projective planes.

The orientable family has Euler characteristic

2 - 2 g

; the nonorientable family has Euler characteristic

2 - h

. Orientability together with Euler characteristic distinguishes the entries.

Boundary words giving the required polygonal schemas are

S^{2} : Σ_{g} : N_{h} : a a^{- 1}, a_{1} b_{1} a_{1}^{- 1} b_{1}^{- 1} \dots a_{g} b_{g} a_{g}^{- 1} b_{g}^{- 1}, a_{1} a_{1} a_{2} a_{2} \dots a_{h} a_{h} .

(1)

A two-sided polygon here means a disk with its boundary split into two arcs; subdivision produces an ordinary polygon with more sides without changing the quotient. Alternatively the square word

a a^{- 1} b b^{- 1}

represents the sphere. Each four-letter block in the orientable word adds a torus handle; each repeated-letter block in the nonorientable word adds a projective-plane summand.

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