Approximation results in Hilbert cube manifolds
Author:
T. A. Chapman
Journal:
Trans. Amer. Math. Soc. 262 (1980), 303334
MSC:
Primary 57N20; Secondary 55R05
MathSciNet review:
586720
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Abstract: The purpose of this paper is to answer questions of the following type for maps between Qmanifolds: When is f close to an approximate fibration? When is f close to a fiber bundle map? When is f close to a block bundle map? For the second and third questions a series of obstructions are encountered in the lower algebraic Ktheoretic functors of Bass.
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 [1]
 D. R. Anderson and W. C. Hsiang, The functors, and pseudoisotopies of polyhedra, Ann. of Math. (2) 105 (1977), 201223. MR 0440573 (55:13447)
 [2]
 H. Bass, Algebraic Ktheory, Benjamin, New York, 1968. MR 0249491 (40:2736)
 [3]
 T. A. Chapman, Celllike mappings of Hilbert cube manifolds: Solution of a handle problem, General Topology Appl. 5 (1975), 123145. MR 0377894 (51:14063)
 [4]
 , Homotopy conditions which detect simple homotopy equivalences, Pacific J. Math. (to appear). MR 534693 (81f:57011)
 [5]
 , Approximating maps into fiber bundles by homeomorphisms, Rocky Mountain J. Math. (to appear). MR 575307 (82k:57006)
 [6]
 , Lectures on Hilbert cube manifolds, CBMS Regional Conf. Series in Math., No. 28 Amer. Math. Soc., Providence, R. I., 1976. MR 0423357 (54:11336)
 [7]
 T. A. Chapman and Steve Ferry, Fibering Hilbert cube manifolds over ANRs, Compositio Math. 36 (1978), 735. MR 515035 (80c:57010)
 [8]
 M. Cohen, A course in simplehomotopy theory, SpringerVerlag, New York, 1970. MR 0362320 (50:14762)
 [9]
 D. Coram and P. Duvall, Approximate fibrations, preprint. MR 0442921 (56:1296)
 [10]
 A. Dold, Partitions of unity in the theory of fibrations, Ann. of Math. (2) 78 (1963), 223255. MR 0155330 (27:5264)
 [11]
 R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 6388. MR 0283802 (44:1032)
 [12]
 F. T. Farrell and W. C. Hsiang, A formula for , Proc. Sympos. Pure Math., vol. 17, Amer. Math. Soc., Providence, R. I., 1970. MR 0260836 (41:5457)
 [13]
 Steve Ferry, Approximate fibrations with nonfinite fibers, Proc. Amer. Math. Soc. 64 (1977), 335345. MR 0461505 (57:1490)
 [14]
 , The homeomorphism group of a compact Hilbert cube manifold is an ANR, Ann. of Math. (2) 106 (1977), 101119. MR 0461536 (57:1521)
 [15]
 M. Mather, Counting homotopy types of manifolds, Topology 4 (1965), 9394. MR 0176470 (31:742)
 [16]
 L. C. Siebenmann, A total Whitehead torsion obstruction, Comment. Math. Helv. 45 (1970), 148. MR 0287564 (44:4768)
 [17]
 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
 [18]
 C. T. C. Wall, Finiteness conditions for CW complexes, Ann. of Math. (2) 81 (1965), 5569. MR 0171284 (30:1515)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005867209
PII:
S 00029947(1980)05867209
Keywords:
Hilbert cube manifolds,
engulfing,
block bundles,
aproximate fibration,
Whitehead torsion
Article copyright:
© Copyright 1980 American Mathematical Society
