Hopf Rings in Algebraic Topology

These are colloquium style lecture notes about Hopf rings in algebraic topology. They were designed for use by non-topologists and graduate students but have been found helpful for those who want to start learning about Hopf rings. They are not “up to date,” nor are they intended to be, but instead they are intended to be introductory in nature. Although these are “old” notes, Hopf rings are thriving and these notes give a relatively painless introduction which should prepare the reader to approach the current literature. This is a brief survey about Hopf rings: what they are, how they arise, examples, and how to compute them. There are very few proofs. The bulk of the technical details can be found in either [RW77] or [Wil82], but a “soft” introduction to the material is difficult to find. Historically, Hopf algebras go back to the early days of our subject matter, homotopy theory and algebraic topology. They arise naturally from the homology of spaces with multiplications on them, i.e. H-spaces, or “Hopf” spaces. In our language, this homology is a group object in the category of coalgebras. Hopf algebras have become objects of study in their own right, e.g. [MM65] and [Swe69]. Hopf algebras were also able to give great insight into complicated structures such as with Milnor’s work on the Steenrod algebra [Mil58]. However, when spaces have more structure than just a multiplication, their homology produces even richer algebraic stuctures. In particular, with the development of generalized cohomology theories, we have seen that the spaces which classify them have a structure mimicking that of a graded ring. The homology of all these spaces reflects that fact with a rich algebraic structure: a Hopf ring, or, a ring in the category of coalgebras. This then is the natural tool to use when studying generalized homology theories with the aim of developing them to the point of being useful to the average working algebraic topologist. Hopf rings lead to elegant descriptions of the answers and new techniques for computing them. The first known reference to Hopf rings was in Milgram’s paper [Mil70]. Lately, the theoretical background for Hopf rings has been greatly expanded, [Goe99] and [HT98], i.e. Hopf rings have begun to be studied as objects in their own right, and more and more applications are being found for them. An incomplete list of references is: [Mil70] [RW77] [RW80] [TW80] [Wil82] [Wil84] [Har91] [Str92] [Tur93] [Kas94] [BJW95] [GRT95] [HH95] [HR95] [Kas95] [KST96] [RW96] [ETW97] [Tur97] [HT98] [Goe99] [BWb] [BWa] [Kas]. Although these notes have been around for awhile, the demand for them has been increasing and so it seemed it might be appropriate to publish them as a survey. They are intended to be readable to all in the way that colloquium lectures should be understandable to graduate students and those in other fields entirely. However, their main use seems to be to serve as an introduction to the material for