Learning algebraic topology, homotopy always felt like a very intuitive and natural sort of invariant to attach to a space whereas for homology I don't think I have anywhere as close of an intuitive handle or sense of naturality of this concept as I do for homotopy. So I tried to collect some frames / results for homology I've learned to see if it helps convince my intuition that this concept is indeed something natural in mathspace. I'd be very curious to know if there are any other frames or Deeper Answers to "Why homology?" I'm missing:
1. Measuring "holes" of a space
* Singular homology: This is the first example I encountered, which will serve as intuition / motivation for the later abstract definitions.
* Fixing some notations (feel free to skip this bullet point if you're familiar with the notations):
* Let's fix some space X, and recall our goal associating to that space an algebraic object invariant under homeomorphism / homotopy equivalence.
* First, a singular p-simplex is a map σ:Δp→X, intuitively representing a simplex living inside the space X. So there is a natural σ(i):Δp−1→X map which represents each of the i faces. Then, it is natural to consider the set {σ(i)}pi=0 as representing the "boundary" of the singular p-simplex σ.
* To make this last idea more precise, we define singular p-chain, which is a free abelian group generated from all the singular p-simplicies of a space, denoted Δp(X). In short, its elements look like (finite) formal sums ∑σ:Δp→Xnσσ. A singular p-simplex σ is naturally an element of this group via 1σ∈Δp(X).
* This construction, again, is motivated by the boundary idea earlier, since we now can define the boundary of a singular p-simplex σ as formal sum ∑pi=0σ(i)∈Δp−1(X).
* In fact, the boundary of a singular p-simplex σ is actually ∑pi=0(−1)iσ(i)∈Δp−1(X).
* Why? Intuition: if we draw these σ(i) of simple shapes like triangles (so σ:Δ2→X, hence σ(i):Δ1→X, which is identified with