A geometric realization of $C$-groups

It is shown that for each $C$-group $G$ and each $n\geqslant 2$ there exists an $n$-dimensional compact orientable manifold without boundary $X_n\subset S^{n+2}$ such that $\pi_1(S^{n+2}\setminus X_n)\simeq G$. Furthermore, the well-known representation of Riemann surfaces ($(n=2)$) as a union of finitely many copies of the Riemann sphere with slits glued together is generalized to the $n$-dimensional case.