On the higher Frobenius

Abstract

Given a homotopy invariant of a space, one can ask how much of the space can be recovered from that invariant. This question was first addressed in work of Quillen and Sullivan on rational homotopy theory in the 1960's and in work of Dwyer-Hopkins and Mandell on p-adic homotopy theory in the 1990's. In this thesis, we describe a way to unify these ideas and recover a space in its entirety, rather than up to an approximation. The approach is centered around the study of the higher Frobenius map. First defined by Nikolaus and Scholze, the higher Frobenius map generalizes to E[subscript infinity]-ring spectra the classical Frobenius endomorphism for rings in characteristic p. Our main result is that there is an action of the circle group on (a certain subcategory of) p-complete [subscript infinity]-rings whose monodromy is the higher Frobenius. Using this circle action, we give a fully faithful model for a simply connected finite complex X in terms of Frobenius-fixed [subscript infinity]-rings.