My current favorite frame for thinking about homology is as fixing Poincare’s initial conception of “counting submanifolds up to cobordism”. (I’ve learned this perspective from this excellent blog post, and I summarize my understanding below.)
In Analysis Situs, Poincare sought to count m-submanifolds of a given a n-manifold up to some equivalence relation—namely, being a boundary of some (m+1)-submanifold, i.e. cobordism. I personally buy cobordism as a concept that is as natural as homotopy for one to have come up with (unlike the initially-seemingly-unmotivated definition of “singular homology”), so I am sold on this as a starting point.
Formally, given a n-manifold X and m-submanifolds (disjoint) M1,…,Mm, being cobordant means there’s a (m+1)-submanifold W such that ∂W=M1⊔…⊔Mm. These may have an orientation, so we can write this relation as a formal sum ∑mi=1ciMi∼0 where ci=±1. Now, if there are many such (m+1)-submanifolds for which the Mi form a disjoint boundary, we can sum all of these formal sums together to get ∑mi=1aiMi∼0 where ai∈Z.
Now, this already looks a lot like homology! For example, above already implies Mi themselves have empty boundary (because manifold boundary of manifold boundary is empty, and Mi are disjoint). So if we consider two formal sums ∑mi=1aiMi and ∑mi=1biMi to be the same if ∑mi=1(ai−bi)Mi∼0, then 1) we are considering formal sums of Mi with empty boundary 2) up to being a boundary of a (m+1)-dimensional manifold. This sounds a lot like ker∂/im∂ - though note that Poincare apparently put none of this in a group theory language.
So Poincare’s “collection of m-submanifolds of X up to cobordism” is the analogue of Hm(X)!
But it turns out this construction doesn’t really work for some subtle issues (due to Heegaard). This led Poincare to a more combinatorial alternative to this cobordism idea that didn’t face these issues, which became the birth of the more modern notion of simplicial homology.
(The blog post then describes how Poincare’s initial vision of “counting submanifolds up to cobordism” can still be salvaged (which I plan to read more about in the future), but for my purpose of understanding the motivation behind homology, this is already very insightful!)
which explains the role that the resulting problem (representing homology class of manifolds by submanifolds/cobordisms) played in inspiring the work of René Thom on cobordism, stable homotopy theory, singularity theory...
Update to my last shortform on “Why Homology?”
My current favorite frame for thinking about homology is as fixing Poincare’s initial conception of “counting submanifolds up to cobordism”. (I’ve learned this perspective from this excellent blog post, and I summarize my understanding below.)
In Analysis Situs, Poincare sought to count m-submanifolds of a given a n-manifold up to some equivalence relation—namely, being a boundary of some (m+1)-submanifold, i.e. cobordism. I personally buy cobordism as a concept that is as natural as homotopy for one to have come up with (unlike the initially-seemingly-unmotivated definition of “singular homology”), so I am sold on this as a starting point.
Formally, given a n-manifold X and m-submanifolds (disjoint) M1,…,Mm, being cobordant means there’s a (m+1)-submanifold W such that ∂W=M1⊔…⊔Mm. These may have an orientation, so we can write this relation as a formal sum ∑mi=1ciMi∼0 where ci=±1. Now, if there are many such (m+1)-submanifolds for which the Mi form a disjoint boundary, we can sum all of these formal sums together to get ∑mi=1aiMi∼0 where ai∈Z.
Now, this already looks a lot like homology! For example, above already implies Mi themselves have empty boundary (because manifold boundary of manifold boundary is empty, and Mi are disjoint). So if we consider two formal sums ∑mi=1aiMi and ∑mi=1biMi to be the same if ∑mi=1(ai−bi)Mi∼0, then 1) we are considering formal sums of Mi with empty boundary 2) up to being a boundary of a (m+1)-dimensional manifold. This sounds a lot like ker∂/im∂ - though note that Poincare apparently put none of this in a group theory language.
So Poincare’s “collection of m-submanifolds of X up to cobordism” is the analogue of Hm(X)!
But it turns out this construction doesn’t really work for some subtle issues (due to Heegaard). This led Poincare to a more combinatorial alternative to this cobordism idea that didn’t face these issues, which became the birth of the more modern notion of simplicial homology.
(The blog post then describes how Poincare’s initial vision of “counting submanifolds up to cobordism” can still be salvaged (which I plan to read more about in the future), but for my purpose of understanding the motivation behind homology, this is already very insightful!)
You might enjoy
https://www.ams.org/journals/bull/2004-41-03/S0273-0979-04-01026-2/S0273-0979-04-01026-2.pdf
which explains the role that the resulting problem (representing homology class of manifolds by submanifolds/cobordisms) played in inspiring the work of René Thom on cobordism, stable homotopy theory, singularity theory...