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

TL;DR

This paper proves that certain aspherical presentations over fields of characteristic zero and pro-$p$ groups are aspherical, providing affirmative answers to rational and $p$-adic versions of Whitehead's conjecture.

Contribution

It establishes asphericity of subpresentations in rational and $p$-adic contexts, extending Whitehead's conjecture to these settings.

Findings

Subpresentations of aspherical prounipotent presentations are aspherical.

Subpresentations of aspherical pro-$p$-presentations are aspherical.

Results support rational and $p$-adic analogs of Whitehead's conjecture.

Abstract

We show that subpresentations of aspherical prounipotent presentations over fields of zero characteristics and subpresentations of aspherical pro-$p$-presentations are aspherical, an application to subpresentations of aspherical discrete presentations is also included. Following Bousfield-Kan, Quillen and Sullivan the results are regarded as affirmative answers to rational and $p$-adic analogs of J.H.C. Whitehead's conjecture.