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

J. A. Hillman

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 $\beta \le 3$. In general, ${\pi }_1(X)$ and ${\pi }_1(Y)$ have 3-generator presentations, and $\beta \le 6$. We determine all such nilpotent groups which are torsion-free and have Hirsch length $\le 5$.

Keywords: embedding, homologically balanced, nilpotent, 3-manifold, restrained, surgery.

AMS Subject Classification: Primary 57N13.