In their 2015 paper on random simplicial complexes, Costa and Farber casually mention without a proof that in triangulation of a closed surface, there is a formula $$\frac{\text{number of $1$-faces}}{\text{number of $2$-faces}} = \frac{3}{2}. $$ I was wondering if this is a well-known result and whether the following statements are true.
- in a triangulation of a surface, every $1$-simplex is contained in at most two $2$-simplexes.
- in a triangulation of a closed surface, every $1$-simplex is contained in exactly $2$-simplexes.
The former statement would imply the latter, but I don't know how to go on about proving (or disproving) the first statement.
In a 2-dimensional simplicial complex $X$, for every 1-simplex $E$, and for every point $p$ in the interior of $E$, the local homology $H_2(X,X-p)$ is isomorphic to $\mathbb Z^{k-1}$ where $k$ equals the number of 2-simplices that contain $E$.
But if $X$ is also a manifold then $H_2(X,X-p)$ is isomorphic to $\mathbb Z$ for all $p \in X$.
So every $1$-simplex is contained in exactly $k=2$ 2-simplices.