Some Applications of Incidence Hopf Algebras to Formal Group Theory and Algebraic Topology

1. Introduction Hopf algebras are achieving prominence in combinatorics through the innuence of G.-C. Rota and his school, who developed the theory of incidence Hopf algebras (see 7], 15], 16]). The aim of this paper is to show that incidence Hopf algebras of partition lattices provide an eecient combinatorial framework for formal group theory and algebraic topology. We start by showing that the universal Hurwitz group law (respectively the universal formal group law) are generating functions for certain leaf-labelled trees (repectively plane trees with colored leaves); the proof uses the combinatorial expression for Lagrange inversion in terms of leaf-labelled trees, which is due to M. Haiman and W. Schmitt 5]. A formal group law identity with a combinatorial proof (similar to the one in 8]) is also presented. The relevance of Hopf algebras and formal group theory to algebraic topology (and in particular to K-theory and bordism theory) is well-known. The relevance of the Roman-Rota umbral calculus, as an elegant and illuminating framework for computations, became clear through the work of N. Ray, F. Clarke, A. Baker et al. (see 12], 11], 2], 3]). Applications of incidence Hopf algebra techniques to algebraic topology were given recently by N. Ray and W. Schmitt. In this work, we discuss some of their applications from a diierent point of view, and present other examples of computations (both classical and new) which can be expressed in a very concise form using the incidence Hopf algebra framework. Such applications include: coactions of MU (MU) and K (K), expressing the images of the coeecients of the universal formal group law under the Hurewicz homomorphism