Notes on One-Parameter Deformations of Cohomology Theories

Thanks to Dr. Lubin for graciously helping me derive and understand one-parameter families of formal group laws, thanks to Eric Peterson for introducing me to Morava’s Forms of K-theory, thanks to Agnes Beaudry for pointing out that there was a neater way to check for Landweber-exactness.

Today, I want to discuss the opposite procedures of deformations and contractions of complex-orientable cohomology theories.

Really, all I want is to illustrate the fact that, with the appropriate combinations of both procedures, we obtain new cohomology theories. So, we’re going to  examine the construction of one-parameter families of cohomology theories via one-parameter deformations of formal group laws — of particular interest is the case where a continuous deformation causes an increase of chromatic height.

Motivation and Story Leading To This Construction

I was playing with the concept of group contraction, and thinking about the construction of elliptic cohomology theories.

I accidentally constructed a model of Morava E-theory of height 2 at the prime 3. (We didn’t realize that it was $E_{(2)}(3)$ at first, we just thought it was some weird cohomology theory.)

I found this kind of enlightening and so I want to show you how I came across it.

 

Morava constructed a family of elliptic cohomology theories by deforming K-theory (well, by deforming the multiplicative group associated to K-theory). His construction can be viewed as a recipe:

  1. start with a point of interest (an algebraic group)
    e.g., $\mathbb{C}^\times$
    e.g., $y^3 = x^3 – x $ over $\mathbb{F}_3$
  2. deform that point (create a family of algebraic groups indexed by one parameter) 
    e.g., $\mathbb{C}^\times/q^{\mathbb{Z}}$ where $q:= e^{2\pi i}$, we vary the norm of $0 \leq |q| < 1$.
    e.g., $y^3 = x^3 + tx^2 – x$ over $\mathbb{F}_3[[t]]$
  3. look at the formal group laws associated to your family, this is still indexed by one parameter (in fact, they can be viewed as ONE formal group law, if you keep the parameter formal)
  4. either apply the Landweber exact functor theorem to the whole family stalkwise (specializing the parameter), or apply the Landweber exact functor theorem to the ONE formal group law (keeping the parameter formal).

Let me say this again, because when I explain this to people they like me to say it twice.

Morava’s deformation method is a recipe which consists of 4 steps:

  1. construct a continuous family of smooth algebraic groups indexed by q
  2. construct a continuous family of formal group laws indexed by q
  3. construct a family indexed (indexed by q) of contra-functors from Top \to AbGrp (“potential” cohomology theories — we don’t know if they are exact yet)
  4. prove that each member of this family of contravariant functors is exact (or treat the family as one functor, keeping the variable formal, and prove that this functor is exact)

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