On punctured 3-manifolds in 5-sphere
Let \(M\) be a closed connected orientable smooth 3-manifold and \(H_1(M,\mathbb{Z})\) without torsion. Let \(M^0\) denote the punctured manifold \(M\smallsetminus \text{Int }D^3\). Then \(M\) can be embedded in the 5-sphere \(S^5\) [\textit{M. Hirsch}, Ann. Math., II. Ser. 74, 494-497 (1961; Zbl 0102.38604)]. The present paper contains the classification of embeddings of \(M^0\) in \(S^5\) up to isotopy. The classification is by the Euler class of the normal bundle of the embedding and a certain kind of Seifert linking form associated to the embedding. Similar results are proved in higher dimensions. In this case \(M\) is assumed to be a smooth closed \((n - 2)\)-connected manifold of dimension \(2n-1\), \((n\geq 3)\), which is stably parallelizable and has \(n\)-th homology group \(H_n(M,\mathbb{Z})\) without torsion. The embeddings of the punctured manifolds \(M^0\) in \(S^{2n+1}\) are classified up to almost isotopy.
scientific article; zbMATH DE number 1347345