Splitting manifold approximate fibrations

J. L. Bryant, P. Kirby

Suppose $M$ is a topological $m$-manifold, $X$ is a generalized $n$-manifold satisfying the disjoint disks property (\ddp), $m> n\ge 5$, $f\colon M\to X$ is an approximate fibration, with fiber the shape of a closed topological manifold $F$, and $Y$ is a closed, 1-\lcc, codimension three subset of $X$. We examine conditions under which $f$ is controlled homeomorphic to an approximate fibration $g\colon M\to X$ such that $g\vert g^{-1}(Y)\colon g^{-1}(Y)\to Y$ is, in some sense, an improvement of $f\vert f^{-1}(Y)$. One of the main results is that if $Y$ is a generalized manifold, and if $f\vert f^{-1}(Y)\colon f^{-1}(Y)\to Y$ is fiberwise shape equivalent to a manifold approximate fibration $p\colon E\to Y$, and $Wh(\pi_1(F)\times\mathbb{Z}^k) = 0$, $k = 0, 1, \ldots$, then $f$ is controlled homeomorphic to a manifold approximation $g\colon M\to X$ such that $g\vert g^{-1}(Y)\colon g^{-1}(Y)\to Y$ controlled homeomorphic to $p\colon E\to Y$.