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

We show that subpresentations of aspherical prounipotent presentations over fields of characteristic zero and subpresentations of aspherical pro-$p$-presentations are aspherical. The results are regarded as affirmative answers to the rational and $p$-adic analogues of J. H. C. Whitehead conjecture. Methods of affine group schemes make it possible to unify the presentation for pro-$p$-groups and pro-unipotent groups in characteristic zero. This approach, in particular, allows us to put a rigour mathematical reason under the philosophy proposed by J.-P. Serre; applications of our results to the classical Whitehead's problem are discussed as well.