This does not any sense at all
If I take a torus, and take four points $x_1, x_2, y_1, y_2$ and identify $x_1$ with $y_1$ and $x_2$ with $y_2$, then I get a torus which two "singularities" you could say (or "connected holes")
A torus has one vertex, two edges and a surface. Such a torus with these four identified points has two vertices, two edges and one surface (NOT three vertices, as you can position the point of the original torus as one of those two connected points). So you should get 2 - 2 + 1 = 1
But that's wrong! And I don't know why
Apparently, you simply take here $\chi(T) - \chi(\text{two points}) = 0 - 2 = -2$
But why do we calculate it in this way? The only way I can interpret this equation above is because "you get 2 points out of 4, so 2 points get lost". But the exercise told us to construct a CW complex here!