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
π
i
{\pi _i}
-movable [respectively,
H
i
{H_i}
-movable] provided that nearby fibers include isomorphically into mutual neighborhoods on
π
i
{\pi _i}
[resp.
H
i
{H_i}
]. An earlier result along this line is that
π
i
{\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
U{V^1}
mapping,
π
i
{\pi _i}
-movability for
i
⩽
k
−
1
i \leqslant k - 1
plus
H
k
{H_k}
- and
H
k
+
1
{H_{k + 1}}
-movability imply
π
k
{\pi _k}
-movability of the mapping. Hence a
U
V
1
U{V^1}
mapping which is
H
i
{H_i}
-movable for all i is an approximate fibration. Also, if a
U
V
1
U{V^1}
mapping is
π
i
{\pi _i}
-movable for
i
⩽
k
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
U{V^1}
mapping
f
:
M
m
→
N
n
f:{M^m} \to {N^n}
between manifolds is an approximate fibration provided that f is
π
i
{\pi _i}
-movable for all
i
⩽
max
{
m
−
n
,
1
2
(
m
−
1
)
}
i \leqslant \max \{ m - n,\tfrac {1}{2}(m - 1)\}
.