Constructing approximate fibrations
Authors:
T. A. Chapman and Steve Ferry
Journal:
Trans. Amer. Math. Soc. 276 (1983), 757774
MSC:
Primary 55R65; Secondary 57N15, 57N30
MathSciNet review:
688976
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Abstract: In this paper two results concerning the construction of approximate fibrations are established. The first shows that there are approximate fibrations which are homotopic to bundle maps but which cannot be approximated by bundle maps. Here can be a compact manifold or some topological manifold, . The second shows how to construct approximate fibrations whose fibers do not have finite homotopy type, for any of Euler characteristic zero. Here can be a compact manifold and only has to be an ANR, or can be an manifold, , and must then also be a topological manifold.
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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 theory, Benjamin, New York, 1968. MR 0249491 (40:2736)
 [3]
 D. Carter, private communication, (an example), January 1980.
 [4]
 T. A. Chapman, Homotopy conditions which detect simple homotopy equivalences, Pacific J. Math. 80 (1979), 1346. MR 534693 (81f:57011)
 [5]
 , Whitehead torsion and the Wall finiteness obstruction, Canad. J. Math. (to appear).
 [6]
 , Approximation results in Hilbert cube manifolds, Trans. Amer. Math. Soc. 262 (1980), 303334. MR 586720 (82g:57006)
 [7]
 , Approximation results in finitedimensional manifolds (preprint).
 [8]
 , Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., no. 28, Amer. Math. Soc., Providence, R. I., 1976. MR 0423357 (54:11336)
 [9]
 M. Cohen, A course in simplehomotopy theory, SpringerVerlag, Berlin and New York, 1970. MR 0362320 (50:14762)
 [10]
 D. Coram and P. Duvall, Approximate fibrations, Rocky Mountain J. Math. 7 (1977), 275288. MR 0442921 (56:1296)
 [11]
 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)
 [12]
 S. Ferry, Approximate fibrations with nonfinite fibers, Proc. Amer. Math. Soc. 64 (1977), 335345. MR 0461505 (57:1490)
 [13]
 , The homeomorphism group of a compact Hilbert cube manifold is an ANR, Ann. of Math. (2) 106 (1977), 101119. MR 0461536 (57:1521)
 [14]
 , Approximate fibrations over (in preparation).
 [15]
 R. Geoghegan, Open problems in infinitedimensional topology (preprint). MR 583711 (82a:57015)
 [16]
 J. F. P. Hudson, Piecewise linear topology, Benjamin, New York, 1969. MR 0248844 (40:2094)
 [17]
 L. S. Husch, Approximating approximate fibrations by fibrations. II (preprint). MR 0500990 (58:18472)
 [18]
 F. Quinn, Ends of maps. I, Ann. of Math (2) 110 (1979), 275331. MR 549490 (82k:57009)
 [19]
 L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Thesis, Princeton University, 1965.
 [20]
 , A total Whitehead torsion obstruction, Comment. Math. Helv. 45 (1970), 148. MR 0287564 (44:4768)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198306889763
PII:
S 00029947(1983)06889763
Keywords:
Approximate fibration,
manifold,
manifold
Article copyright:
© Copyright 1983 American Mathematical Society
