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Invariance of torsion and the Borsuk conjecture. (English) Zbl 0539.57009
The author generalizes the notions of homotopy equivalence and finite domination to allow for \(\epsilon\)-control in a parameter space. This leads to refinements of the well-known results of: (a) J. H. C. Whitehead [Am. J. Math. 72, 1-57 (1950; Zbl 0040.389)] that a homotopy equivalence f: \(X\to Y\) of compact CW complexes determines a torsion element \(\tau(f)\in Wh \pi_ 1(Y)\) which vanishes if and only if f is a simple homotopy equivalence; and (b) C. T. C. Wall [Ann. Math., II. Ser. 81, 55-69 (1965; Zbl 0152.219)] that if the space X is finitely dominated, then there is an obstruction \(\sigma(X)\in K_ 0\pi_ 1(X)\) which vanishes if and only if X has the homotopy type of a compact polyhedron. Respectively following from these improved theorems as almost immediate corollaries are the results of: (a) the author [Am. J. Math. 96, 488-497 (1974; Zbl 0358.57004)] that any topological homeomorphism of compact CW complexes is a simple homotopy equivalence; and (b) J. E. West [Ann. Math., II. Ser. 106, 1-18 (1977; Zbl 0375.57013)] that any compact metric ANR has the homotopy type of a compact polyhedron.
Reviewer: R.Sher

MSC:
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
55P10 Homotopy equivalences in algebraic topology
55M15 Absolute neighborhood retracts
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
57Q05 General topology of complexes
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