The bounded and thin Whitehead theorems

Anderson, Douglas R.; Munkholm, Hans Jørgen

doi:10.1090/S0002-9939-1993-1111431-5

The bounded and thin Whitehead theorems
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by Douglas R. Anderson and Hans Jørgen Munkholm PDF

Proc. Amer. Math. Soc. 117 (1993), 551-560 Request permission

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".

References

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