On links whose complements have the Lusternik-Schnirelman category one (Q1894976)

It is shown that if the Lyusternik-Shnirel'man category cat\((S^{n+2}-L)\) of a locally flat \(m\)-component link \(L\) in \(S^{n+2}\) is equal to 1, then the link exterior has the homotopy type of \((\bigvee_mS^1)\vee\left(\bigvee_{m-1}S^{n+1}\right)\). Suppose further that \(n\neq2\): then the manifold obtained by surgery along \(L\) is homeomorphic to the connected sum of \(m\) copies of \(S^1\times S^{n+1}\). A corresponding result holds in the smooth or locally flat PL case.