| This dissertation provides several contributions to stable homotopy theory. It is divided into the following three chapters:;Generalized Witt schemes in algebraic topology. We analyze the even-periodic cohomology of the space BU and some of its relatives using the language of formal schemes as developed by Strickland. In particular, we connect E0(BU) to the theory of Witt vectors and lambda-rings. We use these connections to study the effect of the coproduct arising from the tensor product on generalized Chern classes. We then exploit this connection to simultaneously construct Husemoller's splitting of HZ*p (BU) and Quillen's splitting of MU (p).;Hinfinity orientations on BP. In joint work with Niles Johnson, we show, at the primes 2 and 3, that no map from MU to BP defining a universal p-typical formal group law on BP is Hinfinity . In particular, no such map is Einfinity.;This builds on McClure's work on determining if Quillen's orientation on BP is an H2infinity map. By direct computation, we show that the necessary condition he derives for Quillen's orientation to be H2infinity fails at the primes 2 and 3. We go on to show that this implies the more general result above.;We also provide a reinterpretation of McClure's conditions in the language of formal group laws.;Hinfinity ≠ Einfinity. We give an example of a spectrum with an Hinfinity structure which does not rigidify to an E3 structure. It follows that not every Hinfinity ring spectrum comes from an underlying Einfinity ring spectrum. After comparing definitions, we see the counterexample to the transfer conjecture constructed by Kraines and Lada can be used to construct this spectrum. |
