Formal Group Laws

It has been some time since we studied at the correlation between formal group laws, which were certain power series that looked like Taylor expansion of multiplication on a Lie group, and complex oriented cohomology theories. In particular, we learned that these two completely separate notions had a common universal object. The universal formal group law over the Lazard ring was the same as the formal group law determined by the universal complex oriented cohomology theory — complex cobordism cohomology. Ever since that time we have not encountered formal group laws in any interesting manner, but, today is the day where we do so. The continuation of studying formal group laws — and later, formal groups — will be very important in understanding the field of chromatic homotopy theory, as they are highly linked. In some sense, the algebraic geometry of formal groups corresponds to the stable homotopy theory of complex oriented cohomology theories. One very important feature of this correspondence is the concept of height. The algebraic geometry of formal groups can be filtered by a variable called height, and this — through the correspondence — gives a filtration on spectra. In this blog post we will define this concept of height, and produce some new spectra in light of this new technology. ...

Recently we have covered a lot of heavy topology and abstract mathematics, so today I thought we would cover something else — something maybe a bit easier to grasp. We will introduce the concept of formal group laws, and a bit on why they are interesting. Introduction and definition To not just spew out the definition straight away, we look at a situation where formal group laws arise very naturally. Let $G$ be a one-dimensional commutative Lie group (Think here of the real numbers $\mathbb{R}$ or the circle group $S^1$). This group has a continuous product, $m:G\times G\longrightarrow G$, which locally can be described by a real-valued function in two variables. This function has a Taylor expansion around the origin, which is a power series in two variables. Denote this power series by $F$. This power series satisfies some axioms because of the group structure we have on $G$, these axioms are identity, commutativity and associativity. In more mathematical terms this means that the following statements hold ...