Group localization and two problems of Levine
Abstract
A. K. Bousfield's H 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 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 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.
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