Is homology unnecessary what computing the number of holes in a space?

Question

Why can't the number of holes simply be determined by dividing the number of distinct edges with the number of distinct vertices? For example a torus defined by equatig opposite sides of a sheet of paper and equating appropriate points, one gets that there are only 2 distinct sides and no Boundaries making its 1st homology ZxZ . So a torus has 2 normal holes meaning there are on a torus 2 distinct closed loops up to homotopy.

Answer 1

I'll answer the question in your title which as I take it asks

Is homology unnecessary in computing the number of holes in a topological space?

Well the point of homology theory is to find computable invariants for a topological space $X$, those being the $n$-th homology groups $H_n(X)$.

If you have $2$-dimensional topological manifolds (called surfaces), that are either orientable or non-orientable, counting their "holes", which would mean determining their genus does give you one particular invariant via the Euler-characteristic which is $2-2g$ for orientable surfaces and $2-g$ for non-orientable, connected closed surfaces.

But the Euler-Characteristic can actually be defined in terms of homology, because for any finite CW Complex $X$, the Euler characteristic $\chi(X)$ is $\chi(X) = \sum_n (-1)^n \operatorname{rank}(H_n(X))$.

The take away from this is that homology gives you invariants that are far more general and powerful than "counting the number of holes" in topological spaces.