Approximation results in Hilbert cube manifolds

Author:
T. A. Chapman

Journal:
Trans. Amer. Math. Soc. **262** (1980), 303-334

MSC:
Primary 57N20; Secondary 55R05

DOI:
https://doi.org/10.1090/S0002-9947-1980-0586720-9

MathSciNet review:
586720

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to answer questions of the following type for maps between *Q*-manifolds: 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 *K*-theoretic functors of Bass.

**[1]**D. R. Anderson and W. C. Hsiang,*The functors*,*and pseudo-isotopies of polyhedra*, Ann. of Math. (2)**105**(1977), 201-223. MR**0440573 (55:13447)****[2]**H. Bass,*Algebraic K-theory*, Benjamin, New York, 1968. MR**0249491 (40:2736)****[3]**T. A. Chapman,*Cell-like mappings of Hilbert cube manifolds: Solution of a handle problem*, General Topology Appl.**5**(1975), 123-145. 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), 7-35. MR**515035 (80c:57010)****[8]**M. Cohen,*A course in simple-homotopy theory*, Springer-Verlag, 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), 223-255. MR**0155330 (27:5264)****[11]**R. D. Edwards and R. C. Kirby,*Deformations of spaces of imbeddings*, Ann. of Math. (2)**93**(1971), 63-88. 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), 335-345. MR**0461505 (57:1490)****[14]**-,*The homeomorphism group of a compact Hilbert cube manifold is an ANR*, Ann. of Math. (2)**106**(1977), 101-119. MR**0461536 (57:1521)****[15]**M. Mather,*Counting homotopy types of manifolds*, Topology**4**(1965), 93-94. MR**0176470 (31:742)****[16]**L. C. Siebenmann,*A total Whitehead torsion obstruction*, Comment. Math. Helv.**45**(1970), 1-48. MR**0287564 (44:4768)****[17]**E. H. Spanier,*Algebraic topology*, McGraw-Hill, New York, 1966. MR**0210112 (35:1007)****[18]**C. T. C. Wall,*Finiteness conditions for CW complexes*, Ann. of Math. (2)**81**(1965), 55-69. MR**0171284 (30:1515)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57N20,
55R05

Retrieve articles in all journals with MSC: 57N20, 55R05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0586720-9

Keywords:
Hilbert cube manifolds,
engulfing,
block bundles,
aproximate fibration,
Whitehead torsion

Article copyright:
© Copyright 1980
American Mathematical Society