${\mathbb{Q}}_p$-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 $S^{\ast }(X;{\hat{\mathbb{Z}}}_p){\otimes }_{{\hat{\mathbb{Z}}}_p}{\mathbb{Q}}_p$ 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 étale homotopy types, formality of smooth proper varieties, Galois representations on étale homotopy groups, and constraints on étale fundamental groups.