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Homotopy versus isotopy: 2-spheres in 5-manifolds

Danica Kosanović, Rob Schneiderman and Peter Teichner

Vol. 332 (2024), No. 2, 195–218
Abstract

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 3. 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.

Keywords
2-spheres in 5-manifolds, homotopy implies isotopy, level preserving Whitney trick, metastable homotopy groups, Freedman–Quinn invariant, Dax invariant
Mathematical Subject Classification
Primary: 57K45, 57R40, 58D10
Milestones
Received: 11 April 2024
Revised: 9 July 2024
Accepted: 21 September 2024
Published: 6 December 2024
Authors
Danica Kosanović
Department of Mathematics
ETH Zürich
Zürich
Switzerland
Rob Schneiderman
Department of Mathematics
Lehman College
City University of New York
Bronx, NY
United States
Peter Teichner
Max-Planck-Institut für Mathematik
Bonn
Germany

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