Torsion in fundamental groups of compliments of codimension 2-submanifolds of spheres

Question

Let $M^{n-2} \subset S^n$ be a smooth codimension 2 submanifold. Can there be any torsion elements in $\pi_1(S^n \setminus M)$? I know that this can not occur in the classical case of knot theory where $n = 3$. If it is possible for there to be torsion - what about in the case where $M = S^{n-2}$?

Answer 1

Yes, there are such groups. For instance, start with a simple finite superperfect group $M$, i.e. $H_i(M, {\mathbb Z})=0, i=1, 2$. For instance, the Mathieu group $M_{23}$, or the fundamental group of the 3-dimensional Poincare homology sphere. Then take the semidirect product $G=M\rtimes_{\alpha}{\mathbb Z}$, where $\alpha$ is a nontrivial automorphism of $M$, e.g. a nontrivial inner automorphism. Then one verifies that :

• The abelianization of $G$ is isomorphic to ${\mathbb Z}$.

• $H_2(G, {\mathbb Z})=0$ (use Mayer-Vietoris sequence or a spectral sequence for this calculation).

• $G$ has weight one, i.e. the normal closure of a certain element of $G$ equals $G$. For instance, you can take a generator of the ${\mathbb Z}$ factor of $G$.

• $G$ is finitely presented.

Then one can realize $G$ as the fundamental group of the complement of the complement of some $3$-dimensional sphere in $S^5$, using this paper by Kervaire.