Chromatic Stable Homotopy Theory and the AHSS

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:

  • During lectures I’m going to be live-TeXing notes, which I’ll flush out and post to my website. (Jacob’s also posting his own notes!)
  • In addition, I’ll try to post blog articles (like this one) about related topics or topics I find interesting/confusing.
  • There might be a little MO action thrown in for fun.
  • The offshoot is that today I want to talk a little about chromatic homotopy and about the Atiyah-Hirzebruch Spectral Sequence.

    So what is Chromatic Homotopy? Well one way to find out is to read the lecture notes as they get written. Or you can look at Jacob’s own lecture notes available on the course website.

    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

    E^q: Top \to Ab

    from topological spaces to abelian groups which satisfies all of the usual axioms of cohomology except the dimension axiom: E^q(pt) 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 E^q(X) for reasonable spaces X, like X =\mathbb{CP}^n. 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 E^*(X) into a graded ring.

    Definition: A complex oriented cohomology theory is a multiplicative cohomology theory E equipped with an isomorphism of graded rings

    E^*( \mathbb{CP}^\infty) \cong E^*(pt) [[t]].

    What does this mean? Well as we know \mathbb{CP}^\infty 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

    c_1(L) \in E^*(X)

    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

    E^*(\mathbb{CP}^\infty \times \mathbb{CP}^\infty) \cong E^*(pt) [[ u,v]].

    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

    f(u,v) \in E^*(pt) [[u,v]].

    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

    E*(\mathbb{CP}^\infty) \cong E^*(pt)[[t]]

    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

    E*(\mathbb{CP}^\infty) \cong E^*(pt)[[t]] ?

    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

    E_2^{p,q} = H^p(X; E_q(pt)) \Rightarrow E^{p+q}(X).

    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 \mathbb{CP}^\infty, we know the ordinary cohomology very well. For example this gives the following corollary:

    Corollary: If E is even, i.e. E^q(pt) = 0 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

    A \stackrel{f}{\to} A \stackrel{g}{\to} C \stackrel{h}{\to} A

    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:

    \emptyset \subset X_0 \subset X_1 \subset \cdots \subset X

    Meanwhile, if E is a cohomology theory, then for (CW) pairs (X,A) we get relative cohomology groups E^q(X,A) = \tilde E^q(X/A). Moreover, these fit into long exact sequences, just as for ordinary cohomology.

    Now we can build our exact couple as follows:

    A = \sum_p E^*(X_p)

    C = \sum_p E^*(X_p, X_{p-1})

    and we have

    f = i^*: A = \sum_p E^*(X_p)  \to \sum_p E^*(X_{p-1}) = A

    g= \partial: A = \sum_p E^*(X_{p-1})  \to \sum_p E^*(X_p, X_{p-1}) = C

    h = j^*: C = \sum_p E^*(X_p, X_{p-1})  \to \sum_p E^*(X_{p}) = A .

    Notice that

    C = \sum_p E^{p+q}(X_p, X_{p-1}) = \tilde E^{p+q}( \vee S^p) = C^p_\text{CW}(X; E_q(pt))

    consists of the ordinary CW-cochains of X with coefficients in E_q(pt). This is the E_1-term. With more work one can identify the E_2-term of the spectral sequence with the ordinary cohomology of this chain complex. That’s it!

    7 thoughts on “Chromatic Stable Homotopy Theory and the AHSS

    1. Chris, in regards to your Example 2.7 (with the stars), we only need to show that complex K-theory gives that formal group law. If we define the first Chern class of a line bundle L to be [L] – 1 (so that the trivial bundle has class 0), then the formula holds.

    2. Thanks for your support everyone!

      @Steven: I see that the choice [L] -1 (basically) gives that formal group law. The thing that was confusing me though is the following logic:

      (1) This gives a map L –> Z classifying that FGL (it has a_11 =1, and all others zero).

      (2) If we try to define a cohomology theory by

      E(X) = MU(X) \otimes_L Z

      we get something which I don’t think exists as a cohomology theory. In particular we would see that

      E(pt) = L \otimes_L Z = Z

      so we must have ordinary cohomology. In particular this is NOT K-theory since K(pt) = Z[b, b^{-1}]. I’m pretty convinced that ordinary cohomology can’t support the FGL we are imposing on it.

      I think the out is that R=Z is not flat over L, so we don’t get a cohomology theory by process (2). Instead I think we need to take R= Z[b, b^{-1}]. It seems to work out better if I put the Chern class in degree 2, like I’m supposed to. I’ll probably add this to the next rendition of the notes.

    3. Perhaps the right way to see what’s going on with K-theory is that the FGL x+y+xy does not have the requisite grading–a_11=1 is in grading 0, not 2. So to have any hope of making it compatible with topology, you have to introduce some element b in degree 2 and change the FGL to x+y+bxy. Furthermore, for this new FGL to be equivalent to the original one when you forget the grading, b should be invertible.

    4. @ Eric.

      Yeah. That’s basically what I’m thinking too. If you want

      E(X) = MU(X) \otimes_L R

      to be a cohomology theory (i.e. to have a good Z grading), then the map from L to R better respect the grading. In the K-theory case, the different choices of Chern class give you theories which are isomorphic, but which don’t always have a Z grading, just a Z/2 grading. This is not so important since K-theory is 2-periodic, but it still seems like a confusion-trap.

      This brings up another point, which is the choice of gradings. Jacob is grading the Lazard ring so that a_{i,j} has degree 2(i + j -1). This is how Adams grades it in his book (which is one of the few standard references):

      “Stable Homotopy and Gen. Homology”

      This seems like a fine grading if (like Adams) you are looking at homology theories since MU_*(pt) = \pi_*(MU) = L with this grading. But so far in this class we have been looking at COHOMOLOGY theories. It seems like we should use the convention that L = MU^*(pt), which is the same ring but with the opposite grading, namely

      |a_{i,j}| = -2(i + j -1)

      This seems to make much more sense to me vis-a-vis Chern classes having degree +2.

    5. this is a great post. It’s a shame I cannot find a sequel to this post, covering in an informal way some of the contents of your notes on Lurie’s chormatic course (which I find daunting).

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