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Euler characteristic of an orientable compact surface

Statement

Assume the Axiom of Choice (The Axiom of Choice). A nonempty compact connected orientable boundaryless topological surface $S$ has a unique genus $g \in Z_{\geq 0}$ and Euler characteristic $χ (S) = 2 - 2 g$ . The sphere has $g = 0$ , represented by a genuine paired $a a^{- 1}$ digon. In particular, $χ (S)$ is even and at most $2$ .

Facts & Assumptions

Given: $S$ as in the statement.

[L1]

Classification of compact connected surfaces identifies every nonempty compact connected orientable boundaryless surface with either the sphere or the explicit $g$ -torus sum for some $g \geq 1$ . Its canonical finite CW cell counts are respectively $(2, 1, 1)$ and $(1, 2 g, 1)$ , and it proves uniqueness of the model integer. Its only AC use is inherited from finite triangulation.

[L2]

For a finite CW complex the Euler characteristic is the alternating cell count (Euler characteristic of a finite CW complex).

Proof

1.1L1L2

Apply [L1]. In the sphere case define $g = 0$ ; its paired digon gives $χ (S) = 2 - 1 + 1 = 2 = 2 - 2 g$ by [L2]. In the remaining orientable case [L1] supplies the $g$ -fold torus word, with $g \geq 1$ , one vertex, $2 g$ paired edges and one face. Hence $χ (S) = 1 - 2 g + 1 = 2 - 2 g$ by [L2].

2.1L1step 1.1∎

The formula gives $g = (2 - χ (S)) / 2$ . Thus no second nonnegative integer can be the genus of the same surface; this agrees with the uniqueness in [L1]. Because $g \geq 0$ is integral, $χ (S)$ is even and at most $2$ . The AC assumption enters only through [L1], not through this arithmetic.

Remarks

The empty reduced word is only notation for the genus-zero terminal case; the geometric sphere model remains the paired digon.

Depends on

Used by

Genus and Euler characteristic of a compact Riemann surface Definition

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gallier and Xu, A Guide to the Classification Theorem for Compact Surfaces (standard reference, not scraped)