Unstable Cohomology Operations: Computational Aspects of Plethories

Generalised cohomology theories are a broad class of powerful invariants in algebraic topology. Unstable cohomology operations are a useful piece of structure associated to a such a theory and as a result the collection of these operations is of interest. Traditionally, these operations have been studied through the medium of Hopf rings. However, a Hopf ring does not readily admit algebraic structure corresponding to composition of operations. Stacey and Whitehouse showed that the unstable cohomology operations naturally admit the structure of an esoteric algebraic gadget termed a plethory. This plethory contains all the information of the Hopf ring together with additional structure corresponding to the composition of operations. In this thesis, I shall introduce the algebraic theory of plethories and extend with results which will aid computations. I will then illustrate, in a direct
fashion, how the unstable cohomology operations admit the structure of a plethory and discuss the implications in this context. Finally, I shall perform
some computations of the plethory of unstable cohomology operations for some familiar cohomology theories.