In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. My question: how do you prove this?
I'm not very familiar with PL theory, but I have a sketch in the smooth case:
(Proof sketch): Let $f: S^1 \to S^n$ be a loop that misses $L^k$. Since $S^n$ is simply connected, the map $f$ can be extended to a map $ F: D^2 \to S^n$. Assume that $ F$ is transverse to $L^k$. If $ F(D^2) \cap L^k$ is nonempty, then $ F^{-1}(L^k)$ is a codimension $3$ submanifold of $D^2$, a contradiction. It follows that $ F(D^2) \cap L^k$ is empty, so $ F$ can be viewed as a map into $S^n \setminus L^k$ extending $f$. Thus $f$ is nullhomotopic in $S^n\setminus L^k$.
If correct, can this proof be adapted to the PL setting?
