Trying to understand Lens Spaces

Question

I am new to Lens Spaces and I am trying to construct some easy ones to understand better. So let's take $L(3,1)$.

Using the alternative definition proposed on the Wikipedia page I am doing as follows:

1. Drawing a triangle $[a_0,a_1,a_2]$.
2. Put two points $n$ and $s$ (north and south pole) above and below the triangle
3. Connect $n$ and $s$ to the three vertices
4. Do the following identifications: $n \leftrightarrow s$, $a_0 \leftrightarrow a_1$, $a_1 \leftrightarrow a_2$, $a_2 \leftrightarrow a_0$.

Now I want to count how many simplices I am left with after the identification. I believe I have:

• two $0$-simplices
• four $1$-simplices (3 given from the initial triangle edges and 1 coming from the fact that I identify all external edges)
• four $2$-simplices (1 given by the initial triangle and 3 coming from the gluings)
• two $3$-simplices

Is this correct?

Thanks in advance for your help :)