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.
For the sake of concreteness, whenever I say ‘space’ in this post, I mean connected CW complex.
We want to think about passing from the category of spaces to the homotopy category of spaces, but we would like to be as lazy as possible. We could identify any pair of maps which are homotopic, but it turns out we can get away with only identifying pairs of maps which are homotopic and are homotopy equivalences. To see this, notice that any homotopic maps factor through the maps , and and are homotopic and homotopy equivalences, so and also get identified.
This may seem like a rather silly point to make, but the advantage is that the latter class of maps is easier to characterize algebraically.
Whitehead’s Theorem. A map between spaces and is a homotopy equivalence if it induces an isomorphism on all homotopy groups. Furthermore, two such maps are homotopic if and only if they induce the same isomorphism.
Hence, to form the homotopy category, we can identify maps which induce the same isomorphism on homotopy groups. This is usually done by adding arrows to the category which are formal inverses to such classes of maps.
This leads naturally to the definition of rational homotopy. Given a space , we can throw away all the torsion information in by tensoring with ; ie, looking at the rational homotopy group . Inspired by Whitehead’s theorem, we say a map is a rational homotopy equivalence if it induces an isomorphism on all rational homotopy groups, and two such maps are rationally homotopic if they induce the same isomorphism. Thus, we can form the rational homotopy category by adding formal inverses to rational homotopy equivalences.
This is a coarser equivalence relation on spaces than you might think. To get a sense for what is going on here, think about any finite-sheeted covering map . This induces an isomorphism on all , and a surjection on with finite order kernel. Therefore, it is an isomorphism on all rational homotopy groups and hence a rational homotopy equivalence.
So rational homotopy is blind to quotienting by the free action of a finite group. This may seem weird, but in some ways it is desirable. Take, for instance, the study of hyperbolic Riemann surfaces. By the Riemann mapping theorem, these are all quotients of the hyperbolic disc by some subgroup of (a Fuchsian group). A useful relationship between two Fuchsian groups is that of commensurablity, that is, having intersection which is cofinite in each group. By the same argument as the previous paragraph, two Riemann surfaces with commensurable Fuchsian groups are rational homotopy equivalent (and the converse is also true).
Ok, so this is all well and good, except that homotopy groups are hard to calculate, and we don’t really know how much easier will be to compute. Thankfully, there is a theorem of Whitehead and Serre’s which allows us to avoid this entirely:
Theorem (Whitehead-Serre). A map induces an isomorphism on all rational homotopy groups (ie, is a rational homotopy equivalence) if and only if the map induced on all rational cohomology groups is an isomorphism.
Therefore, we can go back and redefine the rational homotopy category by inverting morphisms which induce an isomorphism on all rational cohomology groups.
Here’s where the algebra comes in. To any space we can assign its cochain complex . This determines a functor from the category of spaces to the category of differential graded algebras; specifically DGAs which are commutative and have no negative-degree terms. Let’s call these DGAs topological, and we can endow them with an equivalence relation by inverting maps which induce isomorphisms on cohomology (quasi-isomorphisms).
Then the rational homotopy category of spaces maps into the category of topological DGAs modulo quasi-isomorphism. What is even better is that if we restrict to simply connected, finite dimensional spaces , and topological DGAs which are finite dimensional in each degree and have zero first cohomology, then this functor is an equivalence of categories.
This allows us to abandon spaces entirely and work with DGAs. I find this personally very satisfying, since I already know several circumstances where it is useful to think of DGAs as almost like spaces, and this theorem lets me know exactly what is lost when thinking like this (its the information inaccessible to rational homotopy theory).
This also transitions into one of the most fun aspects of math (at least for me personally), which is taking an equivalence and seeing what natural constructions on one side look like on the other side. Topology is full of natural constructions like various topological products, suspensions, loop spaces, classifying spaces, fiber bundles, etc… Each of these becomes an interesting construction on DGAs which might otherwise seem bizarre and unmotivated. If the mood strikes me, I might write a follow-up post to these outlining some of these constructions.