I want to prove the following:
For any three nonnegative integers v, e, f satisfying v − e + f = 2, there exists a CW-structure on $\mathbb{S}^2$ with v 0-cells (vertices), e 1-cells (edges), and f 2-cells (faces)
My idea is to split it into three cases:
$f=e+1$, $v=1$, in this case we construct the structure by attching $e$ loops to the vertex, filling them in and then mapping one circle and onto the loops and creating a gluing for the last face. I don't know how to write this formally or correctly this is just my intuition
$f=e$, $v=2$, in this case just attaching 1 loop to one vertex and other e-1 loops to the other vertex should give the wanted result. Again I am not sure if this is enough to prove anything.
$f>e$, $v>2$, in this case we can attach f of the edges to vertices to create loops and use the other v-2 edges to connect the v vertices without creating another face.
I don't know if anything here is correct or even intuitively right, I am very confused about this problem.