Controlled \(K\)-theory. (English) Zbl 0835.57013

Ranicki, Andrew; Yamasaki, Masayuki

Topology Appl. 61, No. 1, 1-59 (1995).

As an editor, the senior author of this paper is well known for his very commendable effort to make authors write informative introductions to their papers. It is a pleasure to report that this paper lives up to such a demand, and thus to be able to quote briefly from the introduction without feeling that as a reviewer one ought do better: “In this paper we develop the controlled algebra of projections, define the \(\widetilde {K}_0\)-groups directly, and relate the controlled \(\widetilde {K}_0\) and \(Wh\)-groups to each other by various exactness properties. The algebraic methods are used to give a self-contained treatment of the following results: (1) A homeomorphism of finite CW complexes is simple. This is the topological invariance of Whitehead torsion, originally proved by Chapman. (2) Every compact ANR has the homotopy type of a finite CW complex. This is the Borsuk conjecture, originally proved by West. (3) The results of Ferry and Chapman generalizing (1) and (2), by which an \(\varepsilon\)-domination (respectively \(\varepsilon\)-homotopy equivalence) for sufficiently small \(\varepsilon\) implies the vanishing of the ordinary Wall finiteness obstruction (respectively Whitehead torsion).” The above quote describes the contents of the paper very adequately. However, it would be unfair not to warn the reader that the algebra needed for the proofs is occasionally quite unpleasant, mainly because the authors have to keep track of a large number of awkward “epsilons” such as \((90(n+1)+250) \cdot 2700 \varepsilon\) and complicated equivalence relations such as \(\sim^{n, Y^{17\varepsilon}}_{16\varepsilon}\). Thus, this reviewer agrees with the anonymous referee who (according to the introduction) “asked if there is a categorical approach to our ‘stable isomorphisms’ and ‘stably exact’ sequences, and even went so far as to suggest an appropriate category. …Regrettably, we have not been able to provide such a categorical treatment in this paper”.

Reviewer: H.J.Munkholm (Odense)

MSC:

57Q10	Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
19J05	Finiteness and other obstructions in \(K_0\)
19J10	Whitehead (and related) torsion
55N15	Topological \(K\)-theory
54C55	Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
57Q12	Wall finiteness obstruction for CW-complexes

Keywords:

controlled \(K\)-theory; controlled algebra of projections; controlled \(\widetilde {K}_ 0\) and \(Wh\)-groups; topological invariance of Whitehead torsion; compact ANR; homotopy type of a finite CW complex; Borsuk conjecture; \(\varepsilon\)-domination; \(\varepsilon\)-homotopy equivalence; Wall finiteness obstruction

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