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Orientability and Euler characteristic determine a nonempty compact connected boundaryless surface

Statement

Assume the Axiom of Choice (The Axiom of Choice). For nonempty compact connected boundaryless topological surfaces $M, N$ , the following are equivalent:

$M$ and $N$ are homeomorphic;
$M$ and $N$ have the same integral orientability and $χ (M) = χ (N)$ .

Euler characteristic alone is not asserted to classify surfaces.

Facts & Assumptions

Given: $M, N$ as in the statement.

[L1]

The classification theorem Classification of compact connected surfaces gives the sphere or a unique $g$ -handle model on the orientable side, with Euler characteristic $2 - 2 g$ for $g \geq 0$ , and a unique $k$ -crosscap model on the nonorientable side, with Euler characteristic $2 - k$ for $k \geq 1$ . Its canonical models are fixed polygonal quotients.

[L2]

A homeomorphism sends the local pair $(M, M ∖ {x})$ to the corresponding pair for $N$ and transports the continuous generating section of the integral orientation local system; it also induces isomorphisms on singular homology, preserving Euler characteristic as established in [L1] (R-orientation of a topological manifold, Euler characteristic of a finite CW complex).

Proof

1.1L1L2

If $h : M \to N$ is a homeomorphism, transport a local orientation section by $h_{* x} : H_{2} (M, M ∖ {x}; Z) \to H_{2} (N, N ∖ {h (x)}; Z)$ . The inverse homeomorphism transports it back, so integral orientability is equivalent for $M, N$ . The induced homology isomorphisms and Euler–Poincaré identity in [L1] give $χ (M) = χ (N)$ .

2.1L1∎

Conversely, suppose orientability and Euler characteristic agree. If both are orientable, [L1] gives genera $g_{M} = (2 - χ (M)) / 2 = (2 - χ (N)) / 2 = g_{N}$ , including the $g = 0$ sphere. If both are nonorientable, [L1] gives crosscap numbers $k_{M} = 2 - χ (M) = 2 - χ (N) = k_{N}$ . In either case [L1] makes both surfaces homeomorphic to the same fixed polygonal quotient, so composing one homeomorphism with the inverse of the other yields $M ≅ N$ . The only AC use is inherited from [L1].

Remarks

For example, the torus and Klein bottle both have Euler characteristic zero but differ in integral orientability; the biconditional therefore uses both invariants.

Depends on

Used by

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Sources

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