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

Ferry, Steve

doi:10.1090/S0002-9947-1980-0580894-1

A stable converse to the Vietoris-Smale theorem with applications to shape theory
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Trans. Amer. Math. Soc. 261 (1980), 369-386 Request permission

Abstract:

Our main result says that if $f: X \to Y$ is a map between finite polyhedra which has k-connected homotopy fiber, then there is an n such that $f \times {\text {id:}} X \times {I^n} \to Y$ 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 $U{V^1}$ 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.

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