A Hurewicz-type theorem for approximate fibrations

Abstract:

This paper concerns conditions on point inverses which insure that a mapping between locally compact, separable, metric ANR’s is an approximate fibration. Roughly a mapping is said to be ${\pi _i}$-movable [respectively, ${H_i}$-movable] provided that nearby fibers include isomorphically into mutual neighborhoods on ${\pi _i}$ [resp. ${H_i}$]. An earlier result along this line is that ${\pi _i}$-movability for all i implies that a mapping is an approximate fibration. The main result here is that for a $U{V^1}$ mapping, ${\pi _i}$-movability for $i \leqslant k - 1$ plus ${H_k}$- and ${H_{k + 1}}$-movability imply ${\pi _k}$-movability of the mapping. Hence a $U{V^1}$ mapping which is ${H_i}$-movable for all i is an approximate fibration. Also, if a $U{V^1}$ mapping is ${\pi _i}$-movable for $i \leqslant k$ and k is at least as large as the fundamental dimension of any point inverse, then it is an approximate fibration. Finally, a $U{V^1}$ mapping $f:{M^m} \to {N^n}$ between manifolds is an approximate fibration provided that f is ${\pi _i}$-movable for all $i \leqslant \max \{ m - n,\tfrac {1}{2}(m - 1)\}$.