Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The bounded and thin Whitehead theorems

Authors: Douglas R. Anderson and Hans Jørgen Munkholm
Journal: Proc. Amer. Math. Soc. 117 (1993), 551-560
MSC: Primary 19J10; Secondary 55P10
MathSciNet review: 1111431
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Abstract: This paper deals with finite-dimensional CW complexes equipped with reference maps to a fixed metric space and maps between such complexes that respect the reference maps up to a bounded distortion. We prove two Whitehead Theorems for such maps $ f$. The Bounded Whitehead Theorem allows one to decide whether $ f$ is a bounded homotopy equivalence. The Thin Whitehead Theorem allows one to decide when a map of bound zero admits homotopy inverses of arbitrarily small bound (also on the homotopies). Both theorems come in two versions: One that deals with homotopy in all dimensions; one where homotopy in dimensions at least two is replaced by homology of "universal covers".

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Article copyright: © Copyright 1993 American Mathematical Society