On the complementary regions of a closed hyper surface

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

Let $X$ and $Y$ be the complementary regions of a closed hyper surface $M$ in
$S^4$, labeled so that $\chi(X)\leq\chi(Y)$. Suppose that $\pi_1(X)$ is
nilpotent and has Hirsch length $h$, and let $F$ be its torsion subgroup. We
show that if $h\leq1$ then $H_2(F;\mathbb{Z})=0$, and if $h=2$ then
$\pi_1(X)\cong\mathbb{Z}^2$.