Laminated decompositions involving a given submanifold (Q1059902)

An upper semicontinuous decomposition G of an \((n+1)\)-manifold M is said to be a lamination of M provided each element of G is a closed connected n-manifold. The decomposition space associated to such a lamination is always a 1-manifold (possibly with boundary). Furthermore, for each \(g\in G\), the inclusion \(g\subset M\) is a homology equivalence, though not necessarily a homotopy equivalence. This paper adresses the question of a converse to these last assertions. The main results are the following. If \(n+1\leq 3\), M admits some lamination, and N is a locally flat closed n- manifold with the inclusion \(N\subset M\) a homology equivalence, then there is a lamination G of M with \(N\in G\). For \(n+1\geq 5\), the same result holds provided the inclusion \(N\subset M\) is a homotopy equivalence. Examples are given that show that in high dimensions the weaker assumption of homology equivalence is not always sufficient. (The latter assumption is combined with a \(''\pi_ 1\)-condition'' to provide a necessary and sufficient condition.)