Symmetric Functions, Formal Group Laws, and Lazard's Theorem

Abstract

Lazard's theorem is a central result in formal group theory; it states that the ring over which the universal formal group law is defined (known as the Lazard ring) is a polynomial algebra over the integers with infinitely many generators. This ring also shows up in algebraic topology as the complex cobordism ring. The main aim of this paper is to show that the polynomial structure of the Lazard ring follows from the polynomial structure of a certain subalgebra of symmetric functions with integer coefficients. The connection between symmetric functions and the Lazard ring is provided by a certain Hopf algebra map from symmetric functions to the covariant bialgebra of a formal group law. We study this map by deriving formulas for the images of certain symmetric functions; in passing, we use this map to prove some symmetric function and Catalan number identities. Based on the above results, we prove Lazard's theorem, and present an application to the construction of certainp-typical formal group laws over the integers. Combinatorial methods play a major role throughout this paper.