Abstract
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We give a complete obstruction for two homotopic embeddings of a 2-sphere into a
5-manifold to be isotopic. The results are new even though the methods are classical, the
main tool being the elimination of double points via a level preserving Whitney move in
codimension .
Moreover, we discuss how this recovers a particular case of a result of Dax on
metastable homotopy groups of embedding spaces. It follows that “homotopy implies
isotopy” for 2-spheres in simply connected 5-manifolds and for 2-spheres admitting
algebraic dual 3-spheres.
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Keywords
2-spheres in 5-manifolds, homotopy implies isotopy, level
preserving Whitney trick, metastable homotopy groups,
Freedman–Quinn invariant, Dax invariant
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Mathematical Subject Classification
Primary: 57K45, 57R40, 58D10
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Milestones
Received: 11 April 2024
Revised: 9 July 2024
Accepted: 21 September 2024
Published: 6 December 2024
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© 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences Publishers).
Distributed under the Creative Commons
Attribution License 4.0 (CC BY). |
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