Homotopy versus isotopy: 2-spheres in 5-manifolds

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.