Finiteness Theorems for Approximate Fibrations

Abstract

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 it,-movability, which says roughly that nearby point inverses include isomorphically on the ith shape group into a mutual neighborhood. Suppose/: Mm-> N" is a UVX mapping which is w.-movable for i < k -1, and n > k + 1. An earlier paper proved that / is an approximate fibration when m < 2k -1. If instead m = 2k, this paper proves that there is a locally finite set S c N such th!df\f~l(N -5) 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 c N such that f\f~\N -E) is an approximate fibration.