Cobordisme des surfaces plongées dans S⁴

We show that a closed connected surface embedded in S⁴ = ∂B5 bounds a handlebody of dimension 3 embedded in B⁵ 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 S⁵. We also give a new proof of Rohlin's theorem on embeddings of 3-manifolds into R⁵.