Cobordism of surfaces embedded into \(S^4\)
The authors show that a closed connected surface embedded in \(S^4=\partial B^5\) bounds a handlebody of dimension 3 embedded in \(B^5\) if and only if the Euler number of its normal bundle vanishes. (The proof is inspirated from the proof given by \textit{M. A. Kervaire} to the fact that an \(n\)-sphere always bounds an \((n+1)\)-disc embedded in the disc \(B^{n+3}\), see [Bull. Soc. Math. Fr. 93, 225--271 (1965; Zbl 0141.21201)]). This characterization is used to prove that two closed connected surfaces embedded in \(S^4\) are cobordant if and only if they are abstractly diffeomorphic to each other and the Euler numbers of their normal bundles coincide. As an application it is shown that a given Heegaard decomposition of a 3-manifold can be realized in \(S^5\). The authors also give a new proof of \textit{V. A. Rohlin}'s theorem of embeddings of 3-manifolds into \(\mathbb{R}^5\) [Sov. Math. Dokl. 6, 153--156 (1965); translated from Dokl. Akad. Nauk SSSR 160, 549--551 (1965; Zbl 0132.20101)].
scientific article; zbMATH DE number 5015896