In this post, I will talk about chromatic homotopy theory, which you can call “chromotopy”. This post assumes familiarity with spectra and the notions of Chern classes of line bundles; the reader who is not familiar with this notion is referred to (the very brief) http://www.mit.edu/~sanathd/papers/htpy.pdf.
Recall that there is a universal line bundle over such that every line bundle over a space can be obtained by pulling back over maps . The first Chern class is of interest in this situation. This satisfies where is a line bundle over and . Since acts as a moduli space for complex line bundles, this means that the Chern class of any line bundle is determined by , which lies in . Recall that a property of the Chern class is that . How can we specify the Chern class purely in terms of cohomology?
We claim that where , i.e., is of degree . To see this, use the Serre spectral sequence (we refer the reader to http://www.mit.edu/~sanathd/papers/spectral%20sequences.pdf for very brief notes). Consider the fibration . The Serre spectral sequence then reads , where we are implicitly assuming coefficients in . Now, the group is if , and is zero otherwise. Hence, the page looks like:
Where the only nontrivial differentials are shown above. Since is if , and is zero otherwise, and lies on the line , we see that . From this it follows that for all . Clearly via each even cohomology group of must be isomorphic, so it suffices to determine . But this is simply , finishing our computation of the cohomology groups of . The ring structure follows from the multiplicative structure on the spectral sequence, so ; taking the limit finishes the proof.
Using this computation, one can compute the -cohomology of for more general cohomology theories . Let be a multiplicative cohomology theory (so that has the structure of a graded ring for every ; this is equivalent to asking that (the spectrum associated via Brown representability to) is a homotopy ring spectrum). Say that is a complex-orientable cohomology theory if the Atiyah-Hirezebruch spectral sequence degenerates on the second page. Then clearly where . A complex-orientable cohomology theory with a choice of is called complex-oriented. With a similar computation, we see that , where and are the pullbacks of along the projections .
In ordinary cohomology, we can define to be the first Chern class . Then the Chern class of any line bundle is the pullback in cohomology of . Is this consistent with the Chern class preserving tensor products? By Yoneda’s lemma, the tensor product of line bundles can be thought of as a multiplication inducing . But this is simply a map . Consider ; this is just . On the other hand, . The formula gives the formula .
What about the more general case of a complex-oriented cohomology theory? In this case, denote by the cohomology class where . We see that
for some function . Consequently, . This function clearly satisfies conditions reflecting the geometry of line bundles, namely:
- Let denote the trivial line bundle; then , so .
- , so .
- , so .
If you have studied elliptic curves (or number theory, or algebraic geometry, etc.), you’ll notice these relations are exactly those of a formal group law. What this shows is the following:
Proposition: A complex-oriented cohomology theory gives a formal group law over .
What are some examples of this? As we saw above with ordinary cohomology, the formal group law that arises is the additive formal group law . Suppose , complex K-theory. This is complex-oriented. Without proof, we claim that the Chern class of a line bundle is . Then . This is:
In other words, the formal group law for is , which can easily be checked to satisfy the conditions for being a formal group law. Consider a formal group law . Then a formal group law over a ring can be considered to be a map where is the ideal generated by the relations for being a formal group law (in terms of ), namely:
- if , and otherwise, and similarly for .
- for all .
- Some complicated relations, not reproduced here, corresponding to associativity.
Hence there is a univeral formal group law over such that every formal group law over a ring can be obtained by pushing the universal formal group law to . Consequently, a complex-oriented cohomology theory has a map . Lazard proved the following important result.
Proposition: The Lazard ring is a polynomial algebra such that there is a grading on with of degree .
Shifting over to topology, we see that there is a universal example of a complex-oriented spectrum. Define to be the Thom spectrum of , defined as the homotopy colimit of the sequence where is the spectrum . For example, is the sphere spectrum and . The universal bundle of rank on is classified by a map . In particular, the map is given by the complex orientation of (because ). These maps are compatible in the sense that the following diagram commutes for :
Consequently we get a map , which is a map of ring spectra. In other words, we have:
Proposition: Consider the inclusion , giving a complex orientation of . Then the map taking to the set of complex orientations of given by taking to is a bijection.
Therefore is the universal complex oriented spectrum. Milnor and Novikov proved:
with . Now, is isomorphic to with the grading reversed – but this looks suspiciously like the Lazard ring. Quillen bridged this topological side with algebra in the following result.
Theorem: The map classifying the formal group law of is an isomorphism.
Now, given an -module , we can attempt to define a new cohomology theory via , but this does not necessarily have excision or Mayer-Vietoris. Therefore there needs to be some conditions on the map making a cohomology theory. To do this, we must use the language of stacks. We’ll pick up from here in the next post (but if you’re interested, the next part of the story is written down here: http://www.mit.edu/~sanathd/papers/chromatic%20homotopy%20theory.pdf).