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 . 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]**Douglas R. Anderson and Hans J. Munkholm,*Boundedly controlled topology*, Lecture Notes in Mathematics, vol. 1323, Springer-Verlag, Berlin, 1988. Foundations of algebraic topology and simple homotopy theory. MR**953961****[AM2]**Douglas R. Anderson and Hans Jørgen Munkholm,*The boundedly controlled Whitehead theorem*, Proc. Amer. Math. Soc.**117**(1993), no. 2, 561–568. MR**1111432**, 10.1090/S0002-9939-1993-1111432-7**[Ch]**T. A. Chapman,*Controlled boundary and ℎ-cobordism theorems*, Trans. Amer. Math. Soc.**280**(1983), no. 1, 73–95. MR**712250**, 10.1090/S0002-9947-1983-0712250-X**[Pe]**Erik Kjaer Pedersen,*On the bounded and thin ℎ-cobordism theorem parameterized by 𝑅^{𝑘}*, Transformation groups, Poznań 1985, Lecture Notes in Math., vol. 1217, Springer, Berlin, 1986, pp. 306–320. MR**874186**, 10.1007/BFb0072831**[S]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

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

Article copyright:
© Copyright 1993
American Mathematical Society