Our main result says that if
f
:
X
→
Y
f:\,X\, \to \,Y
is a map between finite polyhedra which has k-connected homotopy fiber, then there is an n such that
f
×
id:
X
×
I
n
→
Y
f\, \times \,{\text {id:}}\,X\, \times \,{I^n} \to Y
is homotopic to a map with k-connected point-inverses. This result is applied to give an algebraic characterization of compacta shape equivalent to locally n-connected compacta. We also show that a
U
V
1
U{V^1}
compactum can be “improved” within its shape class until its homotopy theory and strong shape theory are the same with respect to finite dimensional polyhedra.