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Group localization and two problems of Levine

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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 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.

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Notes

  1. A partial list of finite superperfect groups with non-trivial third homology group can be found at http://hamilton.nuigalway.ie/Hap/www/SideLinks/About/aboutSuperperfect.html.

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Authors and Affiliations

  1. Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178, Russia

    Roman Mikhailov

  2. St. Petersburg Department of Steklov Mathematical Institute, Saint Petersburg, Russia

    Roman Mikhailov

  3. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Kent E. Orr

Authors
  1. Roman Mikhailov
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  2. Kent E. Orr
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Correspondence to Roman Mikhailov.

Additional information

This research of the first author is supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government Grant 11.G34.31.0026 and RF Presidential grant MD-381.2014.1.

The Simons Foundation generously supports the second author through Grant 209082. These results were obtained while the second author enjoyed the city of St. Petersburg and the support of the Chebyshev Laboratory, St. Petersburg State University. This author appreciates their support and the excellent research atmosphere provided.

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Mikhailov, R., Orr, K.E. Group localization and two problems of Levine. Math. Z. 280, 355–366 (2015). https://doi.org/10.1007/s00209-015-1428-5

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