Before we can start some basic definitions to come into the topic:
- Suppose $T$ is the standard closed triangles, the convex hull of the three basic vectors inside $\Bbb{R}^3$. Consider $T$ is the space with the subspace topology of $\Bbb{R}^3$. Look up to the disjoint union $\coprod{T}$ of $f$ triangles (with $f\in2\Bbb{Z}$). We define a gluing pattern of $\coprod{T}$ as a pairing of the $3f$ edges by indicating all edges with some other (no edge is unlabelled).
Thus we get a equivalence relation on $\coprod{T}$. We can get a quotient space $F$ and a quotientmap $\pi:\coprod{T}\rightarrow F$. We got also the theorem that says that such a $F$ is a closed manifold! Some more definitions:
- A cone is a surface with boundary made by gluing $d$ faces arranged around a common vertex, for some $d\geq3$. Of course it is homeomorphic to the closed disc, and its boundary (a $d$-sided polygon) is homeomorphic to the circle. Note that a cone must have at least three sides.
- A closed surface $F$ made by gluing triangles is called a $closed$ $combinatorial$ surface if the union of the (closed) faces incident at any vertex of $F$, thought of as a subspace of $F$, is a cone (centred on that vertex). (On a surface with boundary, the corresponding definition is to require this condition at all vertices not in the boundary.).
Now I want to prove the following:
Suppose a combinatorial surface. The map $\pi:\coprod{T}\rightarrow F$ restricted to any closed face is an injection (so no face is glued to itself). Also any two distinct closed faces of $F$ meet in either a single common edge, a single common vertex, or are disjoint.
I understand the first part of the question but i have no idea how to prove such a result. Should one begin with supposing $\pi(x)=\pi(y)$ to prove the injectivity? What means the second part and how to prove that??
Thank you very much!