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
Fulltext 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 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.
 [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]
T.
A. Chapman, Celllike mappings of Hilbert cube manifolds: solution
of a handle problem, General Topol. Appl. 5 (1975),
123–145. MR 0377894
(51 #14063)
 [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]
T.
A. Chapman, Approximating maps into fiber bundles by
homeomorphisms, Rocky Mountain J. Math. 10 (1980),
no. 2, 333–350. MR 575307
(82k:57006), http://dx.doi.org/10.1216/RMJ1980102333
 [6]
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)
 [7]
T.
A. Chapman and Steve
Ferry, Fibering Hilbert cube manifolds over ANRs, Compositio
Math. 36 (1978), no. 1, 7–35. MR 515035
(80c:57010)
 [8]
Marshall
M. Cohen, A course in simplehomotopy theory, SpringerVerlag,
New YorkBerlin, 1973. Graduate Texts in Mathematics, Vol. 10. MR 0362320
(50 #14762)
 [9]
D.
S. Coram and P.
F. Duvall Jr., Approximate fibrations, Rocky Mountain J. Math.
7 (1977), no. 2, 275–288. MR 0442921
(56 #1296)
 [10]
Albrecht
Dold, Partitions of unity in the theory of fibrations, Ann. of
Math. (2) 78 (1963), 223–255. MR 0155330
(27 #5264)
 [11]
Robert
D. Edwards and Robion
C. Kirby, Deformations of spaces of imbeddings, Ann. Math. (2)
93 (1971), 63–88. MR 0283802
(44 #1032)
 [12]
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)
 [13]
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
 [14]
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)
 [15]
Micheal
Mather, Counting homotopy types of manifolds, Topology
3 (1965), 93–94. MR 0176470
(31 #742)
 [16]
L.
C. Siebenmann, A total Whitehead torsion obstruction to fibering
over the circle, Comment. Math. Helv. 45 (1970),
1–48. MR
0287564 (44 #4768)
 [17]
Edwin
H. Spanier, Algebraic topology, McGrawHill Book Co., New
YorkToronto, Ont.London, 1966. MR 0210112
(35 #1007)
 [18]
C.
T. C. Wall, Finiteness conditions for
𝐶𝑊complexes, Ann. of Math. (2) 81
(1965), 56–69. MR 0171284
(30 #1515)
 [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)
Similar Articles
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:
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
