## 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 $$\beta\leq3$$. 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 $$\le5$$.

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

: Primary 57N13.

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 Tuesday, December 10, 2019