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

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

Codex  0 2026-10-31

A finite subdivision into vertices, edges and polygonal faces defines the Euler characteristic

χ (M) = V - E + F .

(1)

Count cells after all identifications. Subdividing an edge adds one vertex and one edge; splitting a face adds one edge and one face. Each operation leaves

V - E + F

unchanged. Comparing subdivisions through a common refinement proves independence of the subdivision.

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