Rational and $p$-adic analogs of J.H.C. Whitehead's conjecture

Abstract

We show that subpresentations of aspherical prounipotent presentations over fields of zero characteristics and subpresentations of aspherical pro-ppresentations are aspherical. Since simplicial prounipotent groups are a model of finite type objects in Quillen's rational homotopy category and as simplicial pro-pgroups should be considered, following Bousfield-Kan, as p-adic homotopy theory, the results are regarded as affirmative answers to rational and p-adic analogs of J.H.C. Whitehead's conjecture.