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

Author:
Steve Ferry

Journal:
Trans. Amer. Math. Soc. **261** (1980), 369-386

MSC:
Primary 55R65; Secondary 54C56, 55P55, 57N20, 57Q05, 57Q10

MathSciNet review:
580894

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Abstract | References | Similar Articles | Additional Information

Abstract: Our main result says that if is a map between finite polyhedra which has *k*-connected homotopy fiber, then there is an *n* such that 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 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0580894-1

Keywords:
Vietoris-Smale theorem,
-map,
strong shape theory,
Hilbert cube manifold,
finiteness obstruction

Article copyright:
© Copyright 1980
American Mathematical Society