Operations in Complex - Oriented Cohomology Theories Related to Subgroups of Formal Groups

Using the character theory of Hopkins-Kuhn-Ravenel and the total power operation in complex cobordism of tom Dieck, we develop a theory of power operations in Landweber-exact cohomology theories. We give a description of the total power operation in terms of the theory of subgoups of formal group laws developed by Lubin. We apply this machinery in two cases. For the cohomology theory Eh, we obtain a formal-group theoretic condition on the oientation which is an obstruction to the compatibility of H, structures in MU and Eh. We show that there is a unique choice of orientation for which this obstruction vanishes, allowing us to build a large family of unstable cohomology operations based on Eh. We show that the the multiplicative formal group law of K-theory satisfies our condition. In elliptic cohomology, our machinery is naturally related to quotients of elliptic curves by finite subgroups. We adapt our machinery to elliptic cohomology, and produce both the Adams operation and versions of the Hecke operators of Baker as power operations. Thesis Supervisor: Haynes Miller Title: Professor of Mathematics