We determine
strong constraints on the generalized Euler invariants of Seifert bundles over non-orientable base orbifolds
which may embed as topologically locally flat submanifolds of $S^4$. In particular, a circle bundle over a
non-orientable surface $F$ embeds if and only if it embeds as the boundary of a regular neighbourhood of an
embedding of $F$ in $S^4$, and we show that precisely thirteen geometric 3-manifolds with elementary amenable
fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter
class may be obtained by 0-framed surgery on a link which is the union of two slice links, and so embeds
smoothly in $S^4$.
1991 Mathematics Subject Classification: 57N13.
