In fact I haven't studied manifold theory yet. Only I know is fundamental concepts of the differential geometry. According to the Wikipedia, It said meaning of the genus is the number of holes of a surface. Other people said "handle" instead of "hole"
link : https://en.wikipedia.org/wiki/Genus_(mathematics)
The $\chi(M) $ is the Euler-Poincare characteristic of a surface $M$ (I.e. $\chi(M) =V-E+F$) in this post. According to the Gauss bonnet theory, $\chi(M) = 2(1-h)$ [$M$ is a closed surface].
I have a really serious problems for finding the $\chi(M)$ using the genus.
First example) As you can see there is a single hole on a sphere. Just relying intuitively thinking, the number of the handle, $h$, would be $1$. Then, $\chi(M) = 2(1-h)=0$. But $\chi(M) =1$ When I finding $V,E$ and $F$. What is missing point for this?
Second example) There are $4$-holes on the sphere(below picture). You would easily check there are 2 holes side of sphere. But there are also holes at the south and north poles. Finding the V,E and F I got the result $\chi(M)=-2$. But In my thought $h=4$, So $\chi(M) = 2(1-h) =-6$ [Surely the answer is $-2$ not –$6$]. I'm very confused.
As you can expected, I have a problem finding the number of genus(Or I don't know exact meaning of it.) What is the exact meaning of the genus? It looks like having many flaw just relying "the number of holes" How can I count genus for any surfaces?
Best Answer
"Genus" is not really a well-defined concept except for a closed connected surface; "closed" means in particular without boundary, so strictly speaking it excludes your examples, which have boundaries given by the edges of the holes.
What is well-defined in much greater generality is first the Euler characteristic and second the Betti numbers or more abstractly the homology groups. Your first example, the punctured sphere, has Betti numbers $b_0 = 1, b_1 = 0, b_2 = 0$ and so it has Euler characteristic $\chi = 1$; it is not a closed connected surface, whose Euler characteristic is always even. So it does not have a genus at all.
Your second example, the sphere with $4$ punctures, has Betti numbers $b_0 = 1, b_1 = 3, b_2 = 0$ and so it has Euler characteristic $\chi = 1 - 3 = -2$. It, again, is not a closed connected surface so it does not have a genus.
A closed connected orientable surface of genus $g$ has Betti numbers $b_0 = 1, b_1 = 2g, b_2 = 1$ (note the $b_2 = 1$ here which does not appear in your examples) and so has Euler characteristic $2 - 2g$. But as we see with the second example, there are other surfaces which have the same Euler characteristic as such a surface but are not closed.
Some people will tell you that the Betti number $b_i$ counts the "number of $i$-dimensional holes" but personally I think this idea is more confusing than enlightening and I do not recommend thinking about things this way.