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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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: 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0580894-1
PII: S 0002-9947(1980)0580894-1
Keywords: Vietoris-Smale theorem, $ U{V^k}$-map, strong shape theory, Hilbert cube manifold, finiteness obstruction
Article copyright: © Copyright 1980 American Mathematical Society