Euler Characteristic of General Surface

Theorem

Let $S$ be a surface.

Let $\map \chi S$ denote the Euler characteristic of $S$.

Then:

$\map \chi S = 2 - 2 h - c$

where:

$h$ is the number of handles

$c$ is the number of cross-caps

that make up $S$.

Proof

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Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler characteristic