An absolutely central concept/tool in modern mathematics is that of (co)homology. In one of its earliest incarnations, the singular (co)homology of a topological space, it consists of a sequence of groups that measure the number of ‘holes’ in a space of various dimensions. This can take an inscrutable geometric shape, and boil it down to a more understandable sequence of groups.
One of the strengths, but often confusing aspects, of (co)homology is that it has a dizzying number of different definitions, which are often equivalent but in a highly non-obvious way.
For example, the singular (co)homology of a topological space M is defined by mapping a bunch of test spaces (n-simplices, eg. points, intervals, triangles, tetrahedra…) into M, cooking up a chain complex out of this, and then applying an algebraic machine to it.
An alternate definition, the de Rham cohomology, involves looking at `generalized functions’ (i.e. differential forms) on the manifold which satisfy a partial differential equation.
Amazingly, these two definitions produce isomorphic cohomology groups, a theorem known as de Rham’s theorem. It tells us that studying the differential equations which can exist on a space turns out to tell us a lot about its topology. This is a deep and central theme in geometry, going back as far as the Gauss-Bonet theorem, and culminating in the famous Atiyah-Singer index theorem.
The map in the de Rham isomorphism is actually quite straightforward to construct. Differential forms are precisely the gadgets that we can integrate. And so given a test space Y mapping into M, we can simply integrate our differential form over Y.
Okay, so far so good. What I really want to talk about today is a very cool result of Gugenheim that I’ve finally started to understand.
To explain this result, I first need to pose a puzzle. And for this, I need to tell you a bit more about cohomology. The singular and de Rham cohomology groups aren’t simply given by a sequence of groups, there’s more structure lurking. If we add up all the cohomology groups then we get a graded vector space, and on this space, we have a product. So the cohomology groups are actually graded algebras!
Furthermore, the de Rham isomorphism is actually an algebra map! Okay, but now here’s the puzzle. The de Rham isomorphism is ‘defined on the chain level’. This means that it is defined between the chain complexes before we take cohomology. Now the chain complexes also carry the structure of an algebra. But the product on the de Rham complex is commutative, whereas the product on the singular chains is non-commutative. So the *chain level* de Rham map is not a map of algebras.
So why in the world is the map between the cohomologies an algebra map? Where is this coming from on `the chain level’? This is what is addressed in Gugenheim’s theorem.
This is what he explains: let’s consider the de Rham map between chains, and compare the map applied to a product, and the product of the map applied to two elements. These aren’t the same. But there is a map which gives a `chain homotopy’ between these two things! So it is an algebra map up to homotopy!
Now if we want everything to work out, then this new map (i.e. h above) should also satisfy an analog of the homomorphism equation. But this fails in general 😦 No problem, there is a third map that replaces the desired equality with a chain homotopy. You can probably see where this is going: we need to produce a sequence of maps, and these all satisfy a complicated set of equations telling us that at each stage we have an equation that is satisfied ‘up to homotopy’. This is a structure that is well known (to those who know these sorts of things). It’s called an A-infinity map, and it’s exactly the sort of thing that, after taking cohomology, implies that the de Rham isomorphism preserves the product.
The proof of this theorem (at least the idea) is actually quite simple and beautiful, so I want to say a few words about it. There are basically three main points, and I’ll go over them in turn.
The first is quite simple, we just need to reformulate the definition of an A-infinity map in a more user-friendly way. This is achieved by the bar complex. The bar complex of a differential graded algebra A is, first of all, the vector space obtained by shifting the degrees of A by 1, and then taking all the tensor powers and adding them up. But this vector space is also equipped with a differential that encodes both the original differential on A and the multiplication. Now that we’ve done this, we can say that an A-infinity map between two differential graded algebras is just a chain map between their respective bar complexes. (In fact, I think it’s actually enough to map from the bar complex of the de Rham complex to the singular cochain complex).
The second insight is really cool and goes back to some work of Adams and Chen. This is to interpret the bar complex geometrically.
Given a space M, we can construct a new space PM consisting of all paths in M. It’s conveniently called the path space. Inside of this space, we have the subset of all paths whose endpoints are a given fixed point of M. This is the space of based loops OM. The insight is that the cohomology of the bar complex computes the cohomology of the based loop space! So amazingly, in order to properly compare the de Rham and singular cohomology of a space, we need to go through its loop space.
The way to actually implement this is through Chen’s iterated integral procedure. This is a bit technical, but I’ve drawn some pictures to help make it more intuitive. First, it requires us to think of an n-simplex as a sequence of n decreasing points in the unit interval. Here’s a picture in dimension 2:
As a result of this description, if we’re given a point in an n-simplex and a path in M, then we can evaluate the path at the n distinct times, and so get a point in the n-fold product of M.
So given a collection of differential forms on M (which gives an element of the bar complex), we pull them back to the different factors of M, multiply (i.e. wedge) them together, pull them back to the product of a simplex and the path space, and then integrate over the simplex. Whew!
This gives the map from the bar complex to the forms on the path space, and we can further restrict to the based loop space. If we write out this map concretely, we see that it has the form of an iterated sequence of integrals, and looks something like this:
This takes us part of the way. We’ve taken forms in the bar complex, and produced some forms on the based loop space.
The final step is to take a form on the based loop space and to produce a singular cochain on M (i.e. a functional on the singular chains of M).
So let A be a form on the based loop space, and let f: Y to M be an n-simplex. Well, an obvious first step would be to take the space of paths in Y, PY, which has an obvious induced map to PM. And so, making sure that the endpoints are fixed, we take our form A, pull it back to PY and integrate. Okay, so maybe this isn’t so easy, because we have no idea what PY looks like. But here’s the final insight: PY has the structure of a cube of dimension n-1. Namely, we have a canonical map I^(n-1) -> PY which somehow `hits everything’. Here’s a picture in 2-dimensions.
So finally, we can pull A back to the cube and then integrate. And this gives the map in Gugenheim’s theorem!
Here are a couple of nice references if you want to look into more details:
Francis Bischoff 