$\mathbb{Q}_p$-Homotopy Types and Applications to Topology and Algebraic Geometry

Abstract

We develop a $\mathbb{Q}_p$-homotopy theory for $p$-complete spaces. To a $p$-complete space $X$, we associate a commutative differential graded algebra over $\mathbb{Q}_p$ by rectifying the $E_\infty$-algebra $S^*(X;\widehat{\mathbb{Z}}_p)\otimes_{\widehat{\mathbb{Z}}_p} \mathbb{Q}_p$ of singular cochains. For nilpotent $p$-complete finite type spaces, we prove that the minimal model of this algebra recovers the $\mathbb{Q}_p$-homotopy groups and Whitehead products, in direct analogy with Sullivan's rational homotopy theory. We also prove that, for a non-simply-connected $p$-complete space, the Lie algebra dual to its $1$-minimal model is the Lie algebra of the continuous Mal'cev $\mathbb{Q}_p$-completion of the fundamental group. We apply the $\mathbb{Q}_p$-homotopy theory to several questions in topology and algebraic geometry, including finite realization problems for $p$-complete spaces, finiteness properties of \'etale homotopy types, formality of smooth proper varieties, Galois representations on \'etale homotopy groups, and constraints on \'etale fundamental groups.