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4.6 The Classification Theorem

In this subsection we state the Classification Theorem for surfaces, which classifies a surface in terms of its boundary number β, its orientability number ω and its Euler characteristic χ, each of which is a topological invariant – it is preserved under homeomorphisms.

Let us remind ourselves of these three numbers.

  • A surface may or may not have a boundary, and, if it does, then the boundary has finitely many disjoint pieces. The number of these boundary components is the boundary number β of the surface. (See Section 2.2.)

  • Some surfaces are orientable and some are not. The orientability number ω of a surface is 0 if the surface is orientable and 1 if it is non-orientable. (See Section 3.2.)

  • Surfaces can be subdivided, and for a given surface the Euler characteristic χ = VE + F is independent of the subdivision used to compute it, and is therefore a property of the surface itself. (See Section 4.3.)

These three numbers β, ω and χ are collectively called the characteristic numbers of the surface.

Now, all three characteristic numbers are necessary for classifying surfaces since, for example:

  • a sphere and a torus both have β = 0 and ω = 0;

  • a torus and a Klein bottle both have β = 0 and χ = 0;

  • a torus and a cylinder both have ω = 0 and χ = 0.

It also turns out that these three numbers are sufficient to classify surfaces, and so we have the central result of this block.

Theorem 14: Classification Theorem for surfaces

Two surfaces are homeomorphic if and only if they have the same values for the characteristic numbers β, ω and χ.


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