Cobordisme des surfaces plongées dans $S^4$

Abstract

We 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. Using this characterization, we show 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, we show that a given Heegaard decomposition of a 3-manifold can be realized in $S^{5}$. We also give a new proof of Rohlin's theorem on embeddings of 3-manifolds into $\mathbf{R}^{5}$.