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ENGEL RELATIONS IN 4-MANIFOLD TOPOLOGY

Part of: Special aspects of infinite or finite groups Topological manifolds

Published online by Cambridge University Press:  16 August 2016

MICHAEL FREEDMAN  and
VYACHESLAV KRUSHKAL
MICHAEL FREEDMAN
Affiliation:
Microsoft Station Q, University of California, Santa Barbara, CA 93106-6105, USA; michaelf@microsoft.com
VYACHESLAV KRUSHKAL
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA; krushkal@virginia.edu

Abstract

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We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional (4D) topological surgery conjecture. The A–B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surgery problems, defined using ramifications of homotopically trivial links. More generally we show how $n$ -Engel relations arise from higher-order double points of surfaces in 4-space.

MSC classification

Primary: 57N13: Topology of $E^4$, $4$-manifolds
Secondary: 20F45: Engel conditions
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e22
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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