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

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
PDFBibTeX
XMLCite

\textit{A. Ranicki} and \textit{M. Yamasaki}, Topology Appl. 61, No. 1, 1--59 (1995; Zbl 0835.57013)

### References:

[1] | Bass, H., Algebraic \(K\)-Theory (1968), Benjamin: Benjamin New York · Zbl 0174.30302 |

[2] | Bass, H.; Heller, A.; Swan, R. G., The Whitehead group of a polynomial extension, Publ. Math. I.H.E.S., 22, 61-79 (1964) · Zbl 0248.18026 |

[3] | Chapman, T. A., The topological invariance of Whitehead torsion, Amer. J. Math., 96, 488-497 (1974) · Zbl 0358.57004 |

[4] | Chapman, T. A., Homotopy conditions which detect simple homotopy equivalences, Pacific J. Math., 80, 13-46 (1979) · Zbl 0412.57015 |

[5] | Chapman, T. A., Invariance of torsion and the Borsuk conjecture, Canad. J. Math., 32, 1333-1341 (1980) · Zbl 0539.57009 |

[6] | Chapman, T. A., Controlled Simple Homotopy Theory and Applications, (Lecture Notes in Mathematics 1009 (1983), Springer: Springer Berlin) · Zbl 0548.57001 |

[7] | Cohen, M. M., A Course in Simple-Homotopy Theory, (Graduate Texts in Mathematics, 10 (1973), Springer: Springer Berlin) · Zbl 0261.57009 |

[8] | Connell, E. H.; Hollingsworth, J., Geometric groups and Whitehead torsion, Trans. Amer. Math. Soc., 140, 161-181 (1969) · Zbl 0191.53904 |

[9] | Connolly, F.; Koźniewski, T., Rigidity and crystallographic groups I, Invent. Math., 99, 25-48 (1990) · Zbl 0692.57017 |

[10] | Ferry, S., The homeomorphism group of a compact Hilbert cube manifold is an ANR, Ann. of Math., 106, 101-119 (1977) · Zbl 0375.57014 |

[11] | Ferry, S.; Hambleton, I.; Pedersen, E., A survey of bounded surgery theory and applications, (Algebraic Topology and its Applications. Algebraic Topology and its Applications, Mathematical Sciences Research Institute Publications 27 (1994), Springer: Springer Berlin), 57-80 · Zbl 0840.57020 |

[12] | Hatcher, A. E., Higher simple homotopy theory, Ann. of Math., 102, 101-137 (1975) · Zbl 0305.57009 |

[13] | Higman, G., The units of group rings, (Proc. London Math. Soc. (2), 46 (1940)), 231-248 · JFM 66.0104.04 |

[14] | Lück, W.; Ranicki, A., Chain homotopy projections, J. Algebra, 120, 361-391 (1989) · Zbl 0671.18005 |

[15] | Milnor, J., Whitehead torsion, Bull. Amer. Math. Soc., 72, 358-426 (1966) · Zbl 0147.23104 |

[16] | Quinn, F. S., Ends of maps I, Ann. of Math., 110, 275-331 (1979) · Zbl 0394.57022 |

[17] | Quinn, F. S., Ends of maps II, Invent. Math., 68, 353-424 (1982) · Zbl 0533.57008 |

[18] | Quinn, F. S., Geometric algebra, (Proceedings 1983 Rutgers Topology Conference. Proceedings 1983 Rutgers Topology Conference, Lecture Notes in Mathematics 1126 (1985), Springer: Springer Berlin), 182-198 · Zbl 0589.57033 |

[19] | Ranicki, A., The algebraic theory of finiteness obstruction, Math. Scand., 57, 105-126 (1985) · Zbl 0589.57018 |

[20] | Ranicki, A., The algebraic theory of torsion I, (Proceedings 1983 Rutgers Topology Conference. Proceedings 1983 Rutgers Topology Conference, Lecture Notes in Mathematics 1126 (1985), Springer: Springer Berlin), 199-237 · Zbl 0567.57013 |

[21] | Ranicki, A., Algebraic and geometric splittings of the \(K\)- and \(L\)-groups of polynomial extensions, (Proceedings Symposium on Transformation Groups. Proceedings Symposium on Transformation Groups, Poznań 1985. Proceedings Symposium on Transformation Groups. Proceedings Symposium on Transformation Groups, Poznań 1985, Lecture Notes in Mathematics 1217 (1986), Springer: Springer Berlin), 321-364 · Zbl 0615.57017 |

[22] | Ranicki, A., Lower \(K\)- and \(L\)-theory, (London Mathematical Society Lecture Note Series 178 (1992), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0752.57002 |

[23] | Wall, C. T.C., Finiteness conditions for CW complexes, Ann. of Math., 81, 55-69 (1965) · Zbl 0152.21902 |

[24] | West, J. E., Mapping Hilbert cube manifolds to ANRs, Ann. of Math., 106, 1-18 (1977) · Zbl 0375.57013 |

[25] | Whitehead, J. H.C., Simple homotopy types, Amer. J. Math., 72, 1-57 (1950) · Zbl 0040.38901 |

[26] | Yamasaki, M., \(L\)-groups of crystallographic groups, Invent. Math., 88, 571-602 (1987) · Zbl 0622.57022 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.