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 $\beta \le 3$. 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 $\le 5$.

arXiv commentv2: new paragraph in \S2. v3: Lemma 4 and Theorem 5 have been excised, to become part of a group-theoretic paper "Nilpotent groups with balanced presentation"

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• Nilpotent groups with balanced presentations