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 $\mathbf{R}^{5}$.