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Our main result says that if f: X -Y is a map between finite polyhedra which has k-connected homotopy fiber, then there is an n such that f x id: X x I' 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 UV' 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.

A stable converse to the Vietoris-Smale theorem with applications to shape theory