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

DOI:
https://doi.org/10.1090/S0002-9939-1993-1111431-5

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 . The Bounded Whitehead Theorem allows one to decide whether 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".

**[AM1]**D. R. Anderson and H. J. Munkholm,*Boundedly controlled topology*, Lecture Notes in Math., vol. 1323, Springer-Verlag, New York and Heidelberg, 1988. MR**953961 (89h:57029)****[AM2]**-,*The boundedly controlled Whitehead Theorem*, Proc. Amer. Math. Soc.**117**(1992), 561-568. MR**1111432 (93d:19008)****[Ch]**T. A. Chapman,*Controlled boundary and*-*cobordism theorems*, Trans. Amer. Math. Soc.**280**(1983), pp. 73-95. MR**712250 (85e:57041)****[Pe]**E. K. Pedersen,*On the bounded and thin*-*cobordism theorem parametrized over*, Transformation Groups (Poznan 1985), Lecture Notes in Math., vol. 1217, Springer-Verlag, New York and Heidelberg, 1986, pp. 306-319. MR**874186 (88g:57036)****[S]**E. H. Spanier,*Algebraic topology*, McGraw-Hill, New York, 1966. MR**0210112 (35:1007)**

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1111431-5

Article copyright:
© Copyright 1993
American Mathematical Society