Finiteness theorems for approximate fibrations

Abstract:

This paper concerns conditions on the point inverses of a mapping between manifolds which insure that it is an approximate fibration almost everywhere. The primary condition is ${\pi _i}$-movability, which says roughly that nearby point inverses include isomorphically on the $i$th shape group into a mutual neighborhood. Suppose $f:{M^m} \to {N^n}$ is a $U{V^1}$ mapping which is ${\pi _i}$-movable for $i \leqslant k - 1$, and $n \geqslant k + 1$. An earlier paper proved that $f$ is an approximate fibration when $m \leqslant 2k - 1$. If instead $m = 2k$, this paper proves that there is a locally finite set $S \subset N$ such that $f|{f^{ - 1}}(N - S)$ is an approximate fibration. Also if $m = 2k + 1$ and all of the point inverses are FANR’s with the same shape, then there is a locally finite set $E \subset N$ such that $f|{f^{ - 1}}(N - E)$ is an approximate fibration.