A controlled plus construction for crumpled laminations

Abstract:

Given a closed n-manifold M $(n > 4)$ and a finitely generated perfect subgroup P of ${\pi _1}(M)$, we previously developed a controlled version of Quillen’s plus construction, namely a cobordism (W, M, N) with the inclusion $j:N \mapsto W$ a homotopy equivalence and kernel of ${i_\# }:{\pi _1}(M) \mapsto {\pi _1}(W)$ equalling the smallest normal subgroup of ${\pi _1}(M)$ containing P together with a closed map $p:W \mapsto [0,1]$ such that ${p^{ - 1}}(t)$ is a closed n-manifold for every $t \in [0,1]$ and, in particular, $M = {p^{ - 1}}(0)$ and $N = {p^{ - 1}}(1)$. We accomplished this by constructing an acyclic map of manifolds $f:M \mapsto N$ having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when ${\pi _1}(M)$ contains no finitely generated perfect subgroups. By-products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.