I tend to think of homotopy theory a little bit like ‘The One That Got Away’ from mathematics as a whole. Its full of wistful fantasies about how awesome it would have been if things could only have worked out. Imagine if homotopy groups of spaces and homotopy classes of maps were as easy to compute as homology groups… we’d be teaching undergrads about Postnikov towers and topology might very well end up a subset of group theory.
Instead, homotopy theory is a hopeless, incalculable mess in all but the trivial cases… that bitch. The canonical example here is the Toda’s Table of the Homotopy Groups of Spheres. That’s right, even the simplest imaginable case - homotopy classes of maps between spheres – is a wildly unpredictable mess with only a handful of a structure theorems.
So homology and cohomology theories rule the day; not as powerful as homotopy groups, but infinitely more tractable. However, recently I’ve become somewhat enamored of a weaker form of homotopy which is just weak enough where you can actually say things: Rational Homotopy Theory. The general idea is to simply ignore any information coming from torsion homotopy groups. After all, all the hideousness in Toda’s table is finite groups; we know the infinite homotopy groups, and they represent reasonably interesting phenomena.
The main upshot of this is that all the information of a space (up to rational homotopy) can be packaged in a differential graded algebra. Rational homotopy equivalence becomes quasi-isomorphisms, and so the question of whether two spaces are rational homotopic is very reasonable. With some mild restrictions, it can be shown that spaces up to rational homotopy are the same as DGAs up to quasi-isomorphism. This opens the door for recasting much of topology as purely algebraic constructions.