Laminations, finitely generated perfect groups, and acyclic maps (Q1821391)

Suppose M is a compact, connected \((n+1)\)-manifold whose boundary consists of two components \(N_ 1\) and \(N_ 2\). A lamination of \((M,N_ 1,N_ 2)\) is an upper semicontinuous decomposition G of M into closed n-manifolds with \(N_ k\in G\), \(k=1,2\). If \(i_ 1: N_ 1\to M\) and \(i_ 2: N_ 2\to M\) are homotopy equivalences and \(n\neq 3\), then classical theory supplies a lamination of \((M,N_ 1,N_ 2)\) since \(M- N_ 2\) is then homeomorphic to \(N_ 1\times [0,1)\). The authors consider the case where \(i_ 2\) is a homotopy equivalence while \(i_ 1\) is not. They verify under that assumption on \(i_ 2\) that \(H_*(M,N_ 1)=0\) and that the kernel of \(i_{\#}: \pi_ 1(N_ 1)\to \pi_ 1(M)\) is perfect. They prove as their main result that if, in addition, \(\ker nel(i_{\#})\) is the normal closure of a finitely generated perfect group, then a lamination may be imposed on \((M,N_ 1,N_ 2)\). They ask whether this extra hypothesis on \(\ker nel(i_{\#})\) is necessary.