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

Andrey M. Mikhovich

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.

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--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 -adic analogs of J.H.C. Whitehead's conjecture.

Paper Structure

This paper contains 23 sections, 35 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Homotopy theory of discrete presentations
  3. Presentations, crossed modules and second homotopy groups
  4. Simplicial presentation as a model of $\Omega K(X|R)$
  5. Representations and cohomology of affine group schemes
  6. Affine group schemes and their representations
  7. Cohomology of affine group schemes
  8. The Lyndon-Hochschild-Serre spectral sequence
  9. Topological modules as a dual category
  10. Prounipotent groups and their presentations
  11. Prounipotent groups
  12. Fixed points and coinvariants
  13. Presentations of prounipotent groups
  14. Simplicial presentations of prounipotent groups
  15. Subpresentations
  16. ...and 8 more sections

Key Result

theorem 1

Let $(X|R_0)$ be a subpresentation of an aspherical prounipotent presentation $(X|R)$ (if the base field $k$ has a positive characteristics, we assume that $(X|R)$ is a pro-$p$-presentation), then $(X|R_0)$ is also aspherical.

Theorems & Definitions (91)

  • definition 1
  • theorem 1
  • definition 2
  • definition 3
  • definition 4
  • lemma 1
  • proof
  • proposition 1
  • proposition 2
  • proposition 3
  • ...and 81 more