Intuition for Homology group

Question

I am reading homology from Hatcher but I am not getting what's homology group? I am calculating homology group of different spaces , but it's not clear to me what am I really doing? Why do one consider the singular n simplices and then by taking the free abelian group over those singular n simplices obtain a chain complex and from there we deduce homology groups of a space. But, I am wondering what's the motivation behind all of these , what's really going on?

1. I heard that Homology group of a space calculate holes in the space, but in what sense?
2. Secondly, I don't understand the reason to define the boundary map in that way, I mean why are we taking $(-1)^i$ there? what will be the problem if we don't care about the sign? Any example?
3. Also, when we are considering the free abelian group of the singular n-simplices what's the linear combination represent geometrically? what does it mean to multiply a map by an integer? Does it have any meaning?
4. Why do the boundary map is defined in such a way so that it satisfy the formula $d^2=0?$

I am really sorry if I couldn't formulate my queries well enough but as you can see I am totally confused and also I am reading this for the first time.

Please help me. Thank you.

Answer 1

My algebraic topology professor expressed the homology groups as "things which should be a boundary but aren't".

For example, consider a circle (or, if you're using simplicial homology, consider the simplicial complex of (the boundary of) a triangle). Clearly, the triangle "should be a boundary" - one can intuitively imagine that we could "fill in" the middle of the triangle, and then the triangle would be a boundary. But it is equally clear that it is not in fact a boundary, since we do not include the middle of the triangle in our complex. Thus, the triangle will generate non-trivial 1-dimensional homology.

Similarly, a sphere will generate 2-dimensional homology because it "should be" the boundary of a ball but isn't.

Things get more complicated when you consider an element of homology that has, say order 2. This is something which should be a boundary but isn't. But if you multiply it by 2, it is a boundary. This is how you get the homology of $\mathbb{R}P^n$ - you have things which should be boundaries but aren't, but if you add them to themselves, they become boundaries.