Approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections
Author:
R. E. Goad
Journal:
Trans. Amer. Math. Soc. 258 (1980), 8797
MSC:
Primary 57N99; Secondary 55R65
MathSciNet review:
554320
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper, we prove that approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections. The principal tools are the torus trick developed by Kirby and Siebenmann, a surgery theorem concerning homotopy structures on torii due to Hsiang and Wall, a theorem on the space of homeomorphisms of the torus due to Hamstrom and a generalization of hereditary homotopy equivalence developed by the author.
 [CD 1]
D.
S. Coram and P.
F. Duvall Jr., Approximate fibrations, Rocky Mountain J. Math.
7 (1977), no. 2, 275–288. MR 0442921
(56 #1296)
 [CD 2]
Donald
Coram and Paul
Duvall, Approximate fibrations and a movability condition for
maps, Pacific J. Math. 72 (1977), no. 1,
41–56. MR
0467745 (57 #7597)
 [CF 1]
T.
A. Chapman and Steve
Ferry, Hurewicz fiber maps with ANR fibers, Topology
16 (1977), no. 2, 131–143. MR 0448356
(56 #6663)
 [CF 2]
, Hurewicz fiberings of ANR's (preprint).
 [DH]
E.
Dyer and M.E.
Hamstrom, Completely regular mappings, Fund. Math.
45 (1958), 103–118. MR 0092959
(19,1187e)
 [FTW]
F.
T. Farrell, L.
R. Taylor, and J.
B. Wagoner, The Whitehead theorem in the proper category,
Compositio Math. 27 (1973), 1–23. MR 0334226
(48 #12545)
 [Go]
R. E. Goad, Local homotopy properties of maps and approximation of fibre bundle projections, Thesis, University of Georgia, 1976.
 [Ha]
MaryElizabeth
Hamstrom, The space of homeomorphisms on a torus, Illinois J.
Math. 9 (1965), 59–65. MR 0170334
(30 #572)
 [H]
L.
S. Husch, Approximating approximate fibrations by fibrations,
Canad. J. Math. 29 (1977), no. 5, 897–913. MR 0500990
(58 #18472)
 [HW]
W.c.
Hsiang and C.
T. C. Wall, On homotopy tori. II, Bull. London Math. Soc.
1 (1969), 341–342. MR 0258044
(41 #2691)
 [Ki]
R. C. Kirby, Lectures on triangulation of manifolds, Notes, UCLA, 1969.
 [KS]
R. C. Kirby and L. C. Siebenmann, Foundations of topology, Notices Amer. Math. Soc. 16 (1969), 848.
 [Ko]
George Kozlowski, Variants of homotopy equivalence, contributed lecture, CBMS/NSF Regional Conference on the theory of infinite dimensional manifolds and its applications to topology, October 1115, 1975, Guilford College, Greensboro, N. C.
 [Mi]
John
Milnor, On spaces having the homotopy type of
a 𝐶𝑊complex, Trans. Amer.
Math. Soc. 90
(1959), 272–280. MR 0100267
(20 #6700), http://dx.doi.org/10.1090/S00029947195901002674
 [Si 1]
L.
C. Siebenmann, Approximating cellular maps by homeomorphisms,
Topology 11 (1972), 271–294. MR 0295365
(45 #4431)
 [Si 2]
, The obstruction to finding a boundary for an open manifold of dimension greater than five, Thesis, Princeton University, 1965.
 [St]
Norman
Steenrod, The Topology of Fibre Bundles, Princeton
Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J.,
1951. MR
0039258 (12,522b)
 [Su]
D. P. Sullivan, Triangulating homotopy equivalences, Thesis, Princeton University, 1966.
 [Wa]
C.
T. C. Wall, Surgery on compact manifolds, Academic Press,
LondonNew York, 1970. London Mathematical Society Monographs, No. 1. MR 0431216
(55 #4217)
 [CD 1]
 D. S. Coram and P. F. Duvall, Approximate fibrations, Rocky Mountain J. Math. 7 (1977), 275288. MR 0442921 (56:1296)
 [CD 2]
 , Approximate fibrations and a movability condition for maps, Pacific J. Math. 72 (1977), 4156. MR 0467745 (57:7597)
 [CF 1]
 T. A. Chapman and S. Ferry, Hurewicz fiber maps with ANR fibers, Topology 6 (1977), 121143. MR 0448356 (56:6663)
 [CF 2]
 , Hurewicz fiberings of ANR's (preprint).
 [DH]
 E. Dyer and M.E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1957), 103118. MR 0092959 (19:1187e)
 [FTW]
 F. T. Farrell, L. R. Taylor and J. B. Wagoner, The Whitehead theorem in the proper category, Compositio Math. 27 (1973), 123. MR 0334226 (48:12545)
 [Go]
 R. E. Goad, Local homotopy properties of maps and approximation of fibre bundle projections, Thesis, University of Georgia, 1976.
 [Ha]
 M.E. Hamstrom, The space of homeomorphisms on a torus, Illinois J. Math. 9 (1965), 5965. MR 0170334 (30:572)
 [H]
 L. S. Husch, Approximating approximate fibrations by fibrations, Canad. J. Math. 29 (1977), 897913. MR 0500990 (58:18472)
 [HW]
 W.C. Hsiang and C. T. C. Wall, On homotopy tori. II, Bull. London Math. Soc. 1 (1969), 341342. MR 0258044 (41:2691)
 [Ki]
 R. C. Kirby, Lectures on triangulation of manifolds, Notes, UCLA, 1969.
 [KS]
 R. C. Kirby and L. C. Siebenmann, Foundations of topology, Notices Amer. Math. Soc. 16 (1969), 848.
 [Ko]
 George Kozlowski, Variants of homotopy equivalence, contributed lecture, CBMS/NSF Regional Conference on the theory of infinite dimensional manifolds and its applications to topology, October 1115, 1975, Guilford College, Greensboro, N. C.
 [Mi]
 John Milnor, On spaces having the homotopy type of a CWcomplex, Trans. Amer. Math. Soc. 90 (1959), 272280. MR 0100267 (20:6700)
 [Si 1]
 L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271294. MR 0295365 (45:4431)
 [Si 2]
 , The obstruction to finding a boundary for an open manifold of dimension greater than five, Thesis, Princeton University, 1965.
 [St]
 Norman Steenrod, The topology of fibre bundles, Princeton Univ. Press, Princeton, N. J., 1951. MR 0039258 (12:522b)
 [Su]
 D. P. Sullivan, Triangulating homotopy equivalences, Thesis, Princeton University, 1966.
 [Wa]
 C. T. C. Wall, Surgery on compact manifolds, Academic Press, New York, 1970. MR 0431216 (55:4217)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
57N99,
55R65
Retrieve articles in all journals
with MSC:
57N99,
55R65
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005543202
PII:
S 00029947(1980)05543202
Article copyright:
© Copyright 1980
American Mathematical Society
