3-Manifolds with nilpotent embeddings in $S^4$

Abstract

We consider embeddings of 3-manifolds $M$ in $S^4$ such that the two complementary regions $X$ and $Y$ each have nilpotent fundamental group. If $\beta=\beta_1(M)$ is odd then these groups are abelian and $\beta\leq3$. In general, $\pi_1(X)$ and $\pi_1(Y)$ have 3-generator presentations, and $\beta\leq6$. We determine all such nilpotent groups which are torsion-free and have Hirsch length $\leq5$.