  For every $n \ge 3$, we construct 2-component links in $S^{n+1}$ that are a
split by an integer homology $n$-sphere, but not by $S^n$. In the special case
$n=3$, i.e. that of 2-links in $S^4$, we produce an infinite family of links
$L_\ell$ and of integer homology spheres $Y_\ell$ such that the link $L_\ell$
is (topologically or smoothly) split by $Y_\ell$ and by no other integer
homology sphere in the family.
