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 S4. 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 S4, 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 S4. 1991 Mathematics Subject Classification: 57N13.
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