Now that we’ve all gotten over the excitement surrounding the new iPad, I wanted to talk about something else which I actually find very exciting (unlike the iPad). This semester Jacob Lurie is giving a course on Chromatic Homotopy Theory. This is a beautiful picture which relates algebraic topology and algebraic geometry. Hopefully with Jacob at the helm we’ll also see the derived/higher categorical perspective creeping in. This seems like a great opportunity the learn this material “in my heart”, as my old undergraduate advisor used to say.
And with most of our principal bloggers distracted by MathOverflow, it also seems like a good time to experiment with new media. So here’s the plan so far:
The offshoot is that today I want to talk a little about chromatic homotopy and about the Atiyah-Hirzebruch Spectral Sequence.
But basically what you will see is that chromatic homotopy theory is about constructing a dictionary between two a priori radically different things. On one side we have the stable homotopy category with topological objects like commutative ring spectra. One the other side we have the algebraic geometry of (the stack of) formal group laws.
The basic connection between these two things goes back to the work of Quillen on complex orientable cohomology theories.
Chern Classes and Formal Group Laws
So what is a “complex orientable” cohomology theory you ask? What is a “cohomology theory” for that matter? A (generalized) cohomology theory is a sequence of functors
from topological spaces to abelian groups which satisfies all of the usual axioms of cohomology except the dimension axiom: can be non-zero. The main structure that really makes this a cohomology theory and “computable” (Ha!) is that we have Mayer-Viatoris long exact sequences (and hence suspension isomorphisms). The Atiyah-Hirzebruch spectral sequence provides a method for computing for reasonable spaces X, like . We’ll talk about that shortly.
Most of the cohomology theories we will be concerned with are going to be multiplicative. This means that we have natural “cup products” which turn into a graded ring.
Definition: A complex oriented cohomology theory is a multiplicative cohomology theory E equipped with an isomorphism of graded rings
What does this mean? Well as we know is the classifying space of complex line bundles and it’s cohomology gives us characteristic classes. This means that for a complex oriented cohomology theory we have a distinguished characteristic class in E-cohomology for complex line bundles (given by the generator t).
For any complex line bundle L over a space X, we have a first Chern class
obtained by pulling back the class t.
Even better, the splitting principle allows us to bootstrap our way up to get higher Chern classes for all complex vector bundles. So another way to describe complex oriented cohomology theories is that they have Chern classes for complex vector bundles. (This explains their name).
The connection to formal groups comes about when we start tensoring line bundles. For a complex oriented cohomology theory we get an isomorphism
On this space there are three interesting line bundles. There are the line bundles given by the two projections (these have Chern classes u and v). Then there is the tensor product of these two line bundles. It has a Chern class which is a formal power series
The punchline is that because tensoring line bundles is commutative and associative and since there is a unit (all up to isomorphism) the power series f(u,v) must satisfy certain properties. It must be a formal group law. If we don’t want to choose an specific isomorphism
then we still get a formal group. Maybe we’ll talk more about this another time.
This can really be seen as part of the general idea of algebraic topology. What we are doing is taking things which live in the world of topology (in this case multiplicative cohomology theories (ie. commutative ring spectra)) and extracting algebraic invariants. In this case the algebraic invariants are just more complicated than the easy ones we learn in the first semester of an algebraic topology course. They are formal groups.
One exciting thing is that sometimes this is a complete invariant. Even better, more complicated structure like the geometry of the stack of formal groups is reflected in the category of spectra. This is what chromatic homotopy theory is all about.
So now this raises the question: For which multiplicative cohomology theories is it possible to choose an isomorphism
This is where the Atiyah-Hirzebruch spectral sequence makes an appearance.
The Atiyah-Hirzebruch Spectral Sequence
The AHSS was probably invented by Whitehead, but not published. Here is the statement we are shooting for:
Theorem: Let X be a CW-complex and E a cohomology theory. Then there is a spectral sequence
What is this theorem telling us? Well it is saying that there is a way to approximate the E-cohomology of a space X by the ordinary cohomology of that space. For some spaces, like , we know the ordinary cohomology very well. For example this gives the following corollary:
Corollary: If E is even, i.e. if q is odd, then E is complex orientable.
More generally being complex orientable is the same as saying that the AHSS degenerates.
As far as spectral sequences go, the Atiyah-Hirzebruch spectral sequence is pretty easy to construct. Probably the best way is to use the method of exact couples introduced by Massey. This is because the AHSS has many generalizations and exact couples make those generalizations more accessible.
An exact couple (A,C, f,g,h) is a triangle of maps
which is exact at each spot. Here A and C are objects in some abelian category.
For the usual story of spectral sequences, this abelian category is the category of double complexes of R-modules for some ring R. However the method of exact couples works more generally. For example the abelian category of double complexes of Mackey functors is important for constructing spectral sequences in equivariant cohomology. Anyway, I digress.
Out of an exact couple we can extract a new exact couple, (A’, C’, f’, g’, h’). I won’t describe this. You can look at the wikipedia page or in the appendix I added to the lecture notes. This new exact couple is the next page of the spectral sequence. By iterating this process we get the whole spectral sequence. The objects C are what we usually think of as the pages of the spectral sequence, the rest of the structure lets us construct the differentials, etc.
Okay, so how do we build the AHSS?
Well if we are given a CW-complex structure for X, then we get a natural filtration of X by the skeleta:
Meanwhile, if E is a cohomology theory, then for (CW) pairs (X,A) we get relative cohomology groups . Moreover, these fit into long exact sequences, just as for ordinary cohomology.
Now we can build our exact couple as follows:
and we have
consists of the ordinary CW-cochains of X with coefficients in . This is the -term. With more work one can identify the -term of the spectral sequence with the ordinary cohomology of this chain complex. That’s it!