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
Fulltext PDF Free Access
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]
Douglas
R. Anderson and W.
C. Hsiang, The functors 𝐾₋ᵢ and
pseudoisotopies of polyhedra, Ann. of Math. (2) 105
(1977), no. 2, 201–223. MR 0440573
(55 #13447)
 [2]
Hyman
Bass, Algebraic 𝐾theory, W. A. Benjamin, Inc., New
YorkAmsterdam, 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), no. 1,
13–46. MR
534693 (81f:57011)
 [5]
, Whitehead torsion and the Wall finiteness obstruction, Canad. J. Math. (to appear).
 [6]
T.
A. Chapman, Approximation results in Hilbert cube
manifolds, Trans. Amer. Math. Soc.
262 (1980), no. 2,
303–334. MR
586720 (82g:57006), http://dx.doi.org/10.1090/S00029947198005867209
 [7]
, Approximation results in finitedimensional manifolds (preprint).
 [8]
T.
A. Chapman, Lectures on Hilbert cube manifolds, American
Mathematical Society, Providence, R. I., 1976. Expository lectures from the
CBMS Regional Conference held at Guilford College, October 1115, 1975;
Regional Conference Series in Mathematics, No. 28. MR 0423357
(54 #11336)
 [9]
Marshall
M. Cohen, A course in simplehomotopy theory, SpringerVerlag,
New YorkBerlin, 1973. Graduate Texts in Mathematics, Vol. 10. MR 0362320
(50 #14762)
 [10]
D.
S. Coram and P.
F. Duvall Jr., Approximate fibrations, Rocky Mountain J. Math.
7 (1977), no. 2, 275–288. MR 0442921
(56 #1296)
 [11]
F.
T. Farrell and W.C.
Hsiang, A formula for
𝐾₁𝑅_{𝛼}[𝑇], Applications of
Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968)
Amer. Math. Soc., Providence, R.I., 1970, pp. 192–218. MR 0260836
(41 #5457)
 [12]
Steve
Ferry, Approximate fibrations with nonfinite
fibers, Proc. Amer. Math. Soc.
64 (1977), no. 2,
335–345. MR 0461505
(57 #1490), http://dx.doi.org/10.1090/S00029939197704615058
 [13]
Steve
Ferry, The homeomorphism group of a compact Hilbert cube manifold
is an 𝐴𝑁𝑅, Ann. of Math. (2)
106 (1977), no. 1, 101–119. MR 0461536
(57 #1521)
 [14]
, Approximate fibrations over (in preparation).
 [15]
Open problems in infinitedimensional topology, The Proceedings
of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979,
pp. 287–338 (1980). MR 583711
(82a:57015)
 [16]
J.
F. P. Hudson, Piecewise linear topology, University of Chicago
Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees,
W. A. Benjamin, Inc., New YorkAmsterdam, 1969. MR 0248844
(40 #2094)
 [17]
L.
S. Husch, Approximating approximate fibrations by fibrations,
Canad. J. Math. 29 (1977), no. 5, 897–913. MR 0500990
(58 #18472)
 [18]
Frank
Quinn, Ends of maps. I, Ann. of Math. (2) 110
(1979), no. 2, 275–331. MR 549490
(82k:57009), http://dx.doi.org/10.2307/1971262
 [19]
L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Thesis, Princeton University, 1965.
 [20]
L.
C. Siebenmann, A total Whitehead torsion obstruction to fibering
over the circle, Comment. Math. Helv. 45 (1970),
1–48. MR
0287564 (44 #4768)
 [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)
Similar Articles
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:
http://dx.doi.org/10.1090/S00029947198306889763
PII:
S 00029947(1983)06889763
Keywords:
Approximate fibration,
manifold,
manifold
Article copyright:
© Copyright 1983
American Mathematical Society
