Constructing approximate fibrations

Authors:
T. A. Chapman and Steve Ferry

Journal:
Trans. Amer. Math. Soc. **276** (1983), 757-774

MSC:
Primary 55R65; Secondary 57N15, 57N30

DOI:
https://doi.org/10.1090/S0002-9947-1983-0688976-3

MathSciNet review:
688976

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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*-*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), 13-46. 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), 303-334. MR**586720 (82g:57006)****[7]**-,*Approximation results in finite-dimensional 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 simple-homotopy theory*, Springer-Verlag, Berlin and New York, 1970. MR**0362320 (50:14762)****[10]**D. Coram and P. Duvall,*Approximate fibrations*, Rocky Mountain J. Math.**7**(1977), 275-288. 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), 335-345. MR**0461505 (57:1490)****[13]**-,*The homeomorphism group of a compact Hilbert cube manifold is an ANR*, Ann. of Math. (2)**106**(1977), 101-119. MR**0461536 (57:1521)****[14]**-,*Approximate fibrations over*(in preparation).**[15]**R. Geoghegan,*Open problems in infinite-dimensional 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), 275-331. 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), 1-48. MR**0287564 (44:4768)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55R65,
57N15,
57N30

Retrieve articles in all journals with MSC: 55R65, 57N15, 57N30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0688976-3

Keywords:
Approximate fibration,
-manifold,
-manifold

Article copyright:
© Copyright 1983
American Mathematical Society