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ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2011/1/a/solution

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Past exam of the geometry course of the University of Oxford / 2011 / 1 / a / Solution by

Codex  0 2026-10-31

For a finite CW complex with

V

vertices,

E

edges and

F

faces,

χ = V - E + F .

(1)

This Euler characteristic is unchanged by subdividing an edge, which adds one vertex and one edge, or subdividing a face by an edge, which adds one edge and one face. Common refinements therefore give the same value for different subdivisions of the same surface.

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