Intuition for Homology group

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.

• Maybe math.stackexchange.com/questions/40149/… helps. Commented Aug 4, 2021 at 15:36
• Spend some time reading Michael Henle's book. All will be clear. Commented Aug 4, 2021 at 15:48

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.