Embedding seifert manifolds in S⁴

Using an obstruction based on Donaldson’s theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces smoothly embed in S⁴. We also find constraints on the Seifert invariants of Seifert 3-manifolds which embed in S⁴when either the base orbifold is non-orientable or the first Betti number is odd. In addition, we construct some new embeddings and use these, along with the d and μ invariants, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds.