Mathematische Zeitschrift

, Volume 280, Issue 1–2, pp 355–366 | Cite as

Group localization and two problems of Levine

  • Roman Mikhailov
  • Kent E. Orr


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 its 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 a union of invisible subgroups.


Spectral Sequence Group Localization Algebraic Closure Natural Homomorphism Homology Sphere 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Chebyshev LaboratorySt. Petersburg State UniversitySaint PetersburgRussia
  2. 2.St. Petersburg Department of Steklov Mathematical InstituteSaint PetersburgRussia
  3. 3.Department of MathematicsIndiana UniversityBloomingtonUSA

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