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

TL;DR

This paper investigates embeddings of 3-manifolds into 4-spheres where the complementary regions have nilpotent fundamental groups, classifying possible groups based on Betti number and torsion-free conditions.

Contribution

It characterizes the nilpotent fundamental groups of the complementary regions in such embeddings, providing bounds and classifications for torsion-free cases with low Hirsch length.

Findings

If Betti number is odd, the groups are abelian with Betti ≤ 3.

In general, the groups have 3-generator presentations with Betti ≤ 6.

All torsion-free nilpotent groups with Hirsch length ≤ 5 are classified.

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 $\leq5$.