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

We show that subpresentations of aspherical prounipotent presentations over fields of zero characteristics and subpresentations of aspherical pro-p-presentations are aspherical, the results are regarded as affirmative answers to rational and p-adic analogs 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 zero characteristic. This approach, in particular, allows us to put a strict mathematical reason under the philosophy proposed by J.-P. Serre; applications of our results to the classical Whitehead‘s problem are discussed as well.