Ebook: Combinatorial Models for Certain Structures in Formal Group Theory and Algebraic Topology

Reference:

C. Lenart, Combinatorial Models for Certain Structures in Formal Group Theory and Algebraic Topology, Ph.D. thesis, University of Manchester, May 1996.
Abstract:

This work in algebraic combinatorics is concerned with a new, combinatorial approach to the study of certain structures in algebraic topology and formal group theory. Our approach is based on several invariants of combinatorial structures which are associated with a formal group law, and which generalise classical invariants. There are three areas covered by our research, as explained below.
Our first objective is to use the theory of incidence Hopf algebras developed by G.-C. Rota and his school in order to construct and study several Hopf algebras of set systems equipped with a group of automorphisms. These algebras are mapped onto certain algebras arising in algebraic topology and formal group theory, such as binomial and divided power Hopf algebras, covariant bialgebras of formal group laws, as well as the Hopf algebroid of cooperations in complex cobordism. We identify the projection maps as certain invariants of set systems, such as the umbral chromatic polynomial, which is studied in its own right. Computational applications to formal group theory and algebraic topology are also given.

Secondly, we generalise the necklace algebra defined by N. Metropolis and G.-C. Rota, by associating an algebra of this type with every formal group law over a torsion free ring; this algebra is a combinatorial model for the group of Witt vectors associated with the formal group law. The cyclotomic identity is also generalised. We present combinatorial interpretations for certain generalisations of the necklace polynomials, as well as for the actions of the Frobenius operator and of the p-typification idempotent. For an important class of formal group laws over the integers, we prove that the associated necklace algebra is also defined over the integers; this implies the existence of a ring structure on the corresponding group of Witt vectors.

Thirdly, we study certain connections between formal group laws and symmetric functions, such as those concerning an important map from the Hopf algebra of symmetric functions over a torsion free ring to the covariant bialgebra of a formal group law over the same ring. Applications in this area include: generating function identities for symmetric functions which generalise classical ones, generators for the Lazard ring, and a simplified proof of a classical result concerning Witt vectors.