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Bousfield-Kan completion and very large groups

Jaime Benabent Guerrero, Ramón Flores

TL;DR

The paper proves a general criterion for $R$-badness of spaces in the Bousfield–Kan $R$-completion context: if a connected space $X$ has countable $H_2(X;\mathbb{Z})$ and its $\pi_1$ surjects onto a non-commutative free group $F_2$, then $X$ is $R$-bad for $R = \mathbb{Z}/p\mathbb{Z}$ or any subring of $\mathbb{Q}$. This criterion is then applied to closed surfaces, completely resolving Murillo’s question for most genera, and extended to broad families of very large groups (e.g., Artin groups, Bestvina–Brady groups, graph braid groups), showing many classifying spaces are $R$-bad under mild finiteness hypotheses. The results illuminate the non-idempotence of $R$-completion in a wide new class of spaces and provide a practical framework for constructing non-idempotent completions. Collectively, the work connects $R$-local homotopy theory with the geometry of surfaces and the algebra of large groups, enhancing tools for studying localizations across arithmetic and homological senses.

Abstract

We establish new criteria for the $R$-badness of a space and apply it to the case of closed surfaces.

Bousfield-Kan completion and very large groups

TL;DR

The paper proves a general criterion for -badness of spaces in the Bousfield–Kan -completion context: if a connected space has countable and its surjects onto a non-commutative free group , then is -bad for or any subring of . This criterion is then applied to closed surfaces, completely resolving Murillo’s question for most genera, and extended to broad families of very large groups (e.g., Artin groups, Bestvina–Brady groups, graph braid groups), showing many classifying spaces are -bad under mild finiteness hypotheses. The results illuminate the non-idempotence of -completion in a wide new class of spaces and provide a practical framework for constructing non-idempotent completions. Collectively, the work connects -local homotopy theory with the geometry of surfaces and the algebra of large groups, enhancing tools for studying localizations across arithmetic and homological senses.

Abstract

We establish new criteria for the -badness of a space and apply it to the case of closed surfaces.

Paper Structure

This paper contains 15 sections, 14 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Bousfield-Kan $R$-Completion
  3. $R$-completion
  4. The choice of the ring
  5. The $R$-completion of a group
  6. Criteria for $R$-goodness and $R$-badness
  7. Examples of $R$-good and $R$-bad spaces
  8. The main result
  9. Verifying the hypotheses
  10. Connected Closed Surfaces
  11. Very Large Groups
  12. Artin Groups
  13. Bestvina-Brady Groups
  14. Graph Braid Groups
  15. Large Groups

Key Result

Proposition 2.2

A morphism of spaces, $f: X \to Y$, is a $R$-homology equivalence if and only if the morphism given by the $R$-completion, $R_{\infty}f : R_{\infty}X \rightarrow R_{\infty}Y$, is a homotopy equivalence.

Theorems & Definitions (32)

  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.7
  • Definition 2.8
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Lemma 3.1: BK72, Ch. IV, 5.3
  • Lemma 3.2
  • ...and 22 more