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.

AMS Subject Classification: Primary 57N13.