Group localization and two problems of Levine

Roman Mikhailov; Kent E. Orr

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arXiv · 1306.6065

Group localization and two problems of Levine

Roman Mikhailov·Kent E. Orr

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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.

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Roman Mikhailov, Kent E. Orr. 2013-06-25. Group localization and two problems of Levine. https://arxiv.org/abs/1306.6065

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