Controlled \(K\)-theory
type: Item
instance of: scholarly article
title: Controlled \(K\)-theory
zbMATH Open document ID: 0835.57013
DOI: 10.1016/0166-8641(94)00017-W
author: Andrew Ranicki
author: Masayuki Yamasaki
published in: Topology and its Applications
publication date: 1996-04-16T00:00:00Z
full work available at URL: https://eudml.org/doc/224881
review text: 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. \dots Regrettably, we have not been able to provide such a categorical treatment in this paper''.
reviewed by: Hans Jørgen Munkholm
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 19J05
Mathematics Subject Classification ID: 19J10
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 57Q12
zbMATH DE Number: 719497
zbMATH Keywords: controlled \(K\)-theory
zbMATH Keywords: controlled algebra of projections
zbMATH Keywords: controlled \(\widetilde {K}_ 0\) and \(Wh\)-groups
zbMATH Keywords: topological invariance of Whitehead torsion
zbMATH Keywords: compact ANR
zbMATH Keywords: homotopy type of a finite CW complex
zbMATH Keywords: Borsuk conjecture
zbMATH Keywords: \(\varepsilon\)-domination
zbMATH Keywords: \(\varepsilon\)-homotopy equivalence
zbMATH Keywords: Wall finiteness obstruction
MaRDI profile type: Publication
cites work: Q5562621
cites work: The Whitehead group of a polynomial extension
cites work: Topological Invariance of Whitehead Torsion
cites work: Homotopy conditions which detect simple homotopy equivalences
cites work: Invariance of Torsion and the Borsuk Conjecture
cites work: Controlled simple homotopy theory and applications
cites work: Q5676841
cites work: Geometric Groups and Whitehead Torsion
cites work: Rigidity and crystallographic groups. I
cites work: The homeomorphism group of a compact Hilbert cube manifold is an ANR
cites work: Q4280930
cites work: Higher simple homotopy theory
cites work: Chain homotopy projections
cites work: Whitehead torsion
cites work: Ends of maps. I
cites work: Ends of Maps. II
cites work: Q3717865
cites work: The algebraic theory of finiteness obstruction.
cites work: Q3683205
cites work: Q4722861
cites work: Q4003910
cites work: Finiteness conditions for CW-complexes. I, II.
cites work: Mapping Hilbert cube manifolds to ANR's: A solution of a conjecture of Borsuk
cites work: Simple Homotopy Types
cites work: L-groups of crystallographic groups
description: scientific article; zbMATH DE number 719497