The Genus Theorem

The Genus Theorem is a fundamental result in topology that relates the Euler characteristic of a surface to its genus. Specifically, the theorem states that for any closed orientable surface, the Euler characteristic is equal to 2-2g, where g is the genus of the surface.

The Genus Theorem

To prove the Genus Theorem, we first need to define some basic concepts. A closed surface is a two-dimensional manifold that is compact and without boundary. An orientable surface is one that can be given a consistent orientation, such that a continuous curve traced along the surface always returns to its starting point with the same or opposite orientation. The genus of a surface is a topological invariant that measures the number of "handles" or "holes" in the surface.

The Euler characteristic of a surface is defined as the alternating sum of its Betti numbers:

χ = b0 - b1 + b2 - ...,

where b0 is the number of connected components, b1 is the number of independent loops, and b2 is the number of independent surfaces. For a closed surface, we have b0 = 1 and b2 = 1, since there is only one connected component and one independent surface. Therefore, the Euler characteristic simplifies to

χ = 1 - b1.

Next, we need to show that for any closed orientable surface with genus g, the number of independent loops is 2g. To do this, we use the fact that any closed orientable surface can be obtained by gluing together a certain number of tori, each with one or more holes. A torus with one hole has one independent loop, while a torus with n holes has 2n independent loops. Therefore, a closed orientable surface with genus g can be obtained by gluing together g tori, each with one or more holes. This gives a total of 2g independent loops.

Substituting b1 = 2g into the formula for the Euler characteristic, we obtain

χ = 1 - 2g = 2 - 2g,

which is the desired result.

Therefore, we have proved the Genus Theorem by showing that the Euler characteristic of a closed orientable surface is equal to 2-2g, where g is the genus of the surface.