Derived mapping spaces as models for localizations

This work focuses on a generalization of the models for rational homotopy theory developed by D. Sullivan and D. Quillen and p-adic homotopy developed by M. Mandell to K(1)-local homotopy theory. The work is divided into two parts. The first part is a reflection on M. Mandell's model for p-adic homotopy theory. Reformulating M. Mandell's result in terms of an adjunction between p-complete, nilpotent spaces of finite type and a subcategory of commutative HIF,-algebras, the main theorem shows that the unit of this adjunction induces an isomorphism between the unstable HF, Adams spectral sequence and the HIF, Goerss-Hopkins spectral sequence. The second part generalizes M. Mandell's model for p-adic homotopy theory to give a model for K(1)-localization. The main theorem gives a model for the K(1)localization of an infinite loop space as a certain derived mapping space of K(1)local ring spectra. This result is proven by analyzing a more general functor from finite spectra to a mapping space of K^-algebras using homotopy calculus, and then taking the continuous homotopy fixed points with respect to the prime to p Adams operations. Thesis Supervisor: Mark Behrens Title: Associate Professor of Mathematics