> The crazy part of algebraic topology (to me) is that all these homology theories are isomorphic. This should not be crazy at all. The general recipe is that

> The crazy part of algebraic topology (to me) is that all these homology theories are isomorphic. This should not be crazy at all. The general recipe is that you take something for which there is a local solution, say in a ball or square, and then ask what are the obstructions that prevent you from patching the local solutions into a global solution. That is all that's going on with the ker/Im. For example, differential equations have local solutions (just the contraction mapping theorem). But can you patch them into global solutions? Locally spaces can be described as a ball or disk, but can you globally patch them together? So the obstructions end up measuring something topological. A series of local solutions on, say, a circle can patch together smoothly but when you wrap around to where you started you can be off - there is an obstruction. So the technique of finding local solutions can involve solving a differential equation or constructing a simplex but the obstructions of both have only to do with the consistency of patching these together, which is a topological question. As a historical note, mathematicians understood that counting the dimension of these obstructions reveals something about the underlying space, and there was hope that they would end up measuring finer structures -- e.g. the differential structure rather than just the topological structure. But the above discussion should at least motivate that this is unlikely. All the various *-homology theories ended up being mostly equivalent. It wasn't until the 1980s with the discovery of Simon Donaldson that true differential invariants were found, and the mechanism of finding these was to take the old homology invariants but apply them to an associated object whose own topology is shaped by the differential structure of the original space. For example, the moduli space of vector bundles associated to a space. Then the homological invariants of that space can detect differential invariants of the original space. There is a beautiful lecture series here: https://cmsa.fas.harvard.edu/donaldson-thomas-gromov-witten/ https://cmsa.fas.harvard.edu/donaldson-thomas-gromov-witten/