Triangulation of compact surfaces

A triangulation of a compact surface S is a finite family of closed subsets \{ T_1, T_2, \dotsc, T_m\} that cover S, and a family of homeomorphisms \varphi_k: \mathcal{T} \to T_k, where \mathcal{T} \subset \mathbb{R}^2 is a proper triangle in the plane. We say that the sets T_k are triangles as well, and the images by \varphi_k of a vertex (resp. edge) of the triangle \mathcal{T} is also called a vertex (resp. edge). We impose one condition: Given two different triangles T_k and T_j, 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 \mathbb{S}_2, 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 \mathbb{T} given by the quotient space of the square \square_2 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 P_1. We construct a triangulation by placing two more vertices on the horizontal borders, P_2, P_3, two more vertices in the vertical borders, P_4, P_5,, four more vertices inside the square, P_6, P_7, P_8, P_9,, and joining all of them with edges, as shown in the image below:

This gives the following triangles:

\begin{array}{ccc}P_1P_2P_5&P_2P_5P_6&P_2P_3P_6\\P_3P_6P_7&P_1P_3P_6&P_1P_5P_7\\P_4P_5P_6&P_4P_6P_8&P_6P_7P_8\\P_7P_8P_9&P_4P_7P_9&P_4P_5P_7\\P_1P_2P_4&P_2P_4P_8&P_2P_3P_8\\P_3P_8P_9&P_3P_4P_9&P_1P_3P_4\end{array}

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 v 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.
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