Links with homotopically trivial complements are trivial (Q1910407)

Suppose that \(L\subset S^{n+2}\) is a (smooth, PL locally flat or locally flat) \(m\)-component homotopically trivial link (i.e., \(S^{n+2}\smallsetminus L\) has the homotopy type of one point union of \(m\) copies of \(S^1\) and \(m-1\) copies of \(S^{n+1}\)). Then the author proves that \(L\) is trivial if \(n\neq 2\). In fact, he shows that such a link is a boundary link for any \(n\). Now the proof follows from the unlinking criterion of boundary links by \textit{M. A. Gutiérrez} [Trans. Am. Math. Soc. 171, 491-499 (1972; Zbl 0222.57013)] for \(n\geq 4\), from the splitting theorem by \textit{S. E. Cappell} [Invent. Math. 33, 69-170 (1976; Zbl 0348.57017)] for \(n=3\) and from the loop theorem for \(n=1\).