Integral Models for Spaces via the higher Frobenius

We will describe a fully faithful integral model for spaces in terms of their E-infinity algebras of cochains which assembles Mandell's p-adic homotopy theory with Sullivan's rational homotopy theory.  The key input is the development of a homotopy coherent Frobenius action on a certain subcategory of p-complete E-infinity-rings for each prime p. Using this action, we will see that the data of a space X is the data of its E-infinity-algebra of spherical cochains together with a trivialization of the Frobenius action after completion at each prime.  

We will then outline the construction of this Frobenius action. This involves constructions in equivariant homotopy theory, which produce an action of Quillen's Q-construction (on the category of abelian groups) on certain E-infinity-rings with ``genuine equivariant multiplication."  The second main idea is a ``pre-group-completed" variant of algebraic $K$-theory; we will state some basic results about this ``partial K-theory" and briefly discuss the partial K-theory of F_p.