Triangulation of compact surfaces

A triangulation of a compact surface is a finite family of closed subsets that cover and a family of homeomorphisms , where is a proper triangle in the plane. We say that the sets are triangles as well, and the images by of a vertex (resp. edge) of the triangle is also called a vertex (resp. edge). We impose one condition: Given two different triangles and their intersection must be either void, or a common vertex, or a common edge.

Since the (surface of a) tetrahedron is homeomorphic to the sphere we may consider this as a valid triangulation for the latter. This triangulation is formed basically by four triangles, four vertices and six edges.

The following example shows a triangulation of a torus, performed on the representation of given by the quotient space of the square by the proper identification in the border. Note that in this representation, all of the vertices of the square are actually the same vertex, which we denote We construct a triangulation by placing two more vertices on the horizontal borders, two more vertices in the vertical borders, , four more vertices inside the square, , and joining all of them with edges, as shown in the image below:

This gives the following triangles:

Two important observations about triangulations of any compact surface:

• Each edge belongs to exactly two triangles. This is due to the fact that the surfaces have at every point, a neighborhood that is homeomorphic to an open ball.
• Given a vertex in a proper triangulation, it is possible to sort all the triangles that share than vertex, say clock or counterclockwise, in such a way that two consecutive triangles share a common edge. This is a direct consequence of the previous observation, and the fact that the union of all the triangles sharing a common vertex must be a connected set.