Group localization and two problems of Levine

TL;DR

This paper addresses longstanding questions in group localization and algebraic closure, demonstrating that certain kernels are not always perfect or invisible, thus clarifying the structure of these localizations.

Contribution

It resolves an old problem in Bousfield $H\mathbb Z$-localization and answers two questions of Levine about algebraic closure, showing kernels are not always perfect or invisible.

Findings

The kernel of the Bousfield $H\mathbb Z$-localization map is not always a $G$-perfect subgroup.

The kernel of the algebraic closure map is not always an invisible subgroup.

Provides new insights into the structure of group localizations and closures.

Abstract

A. K. Bousfield's $H\mathbb Z$-localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine's algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield $H\mathbb Z$-localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group $G$ to it's Bousfield $H\mathbb Z$-localization is not always a $G$-perfect subgroup. In the case of algebraic closure of groups, we prove the analogous result that this kernel is not always an invisible subgroup.