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), 87-97

MSC:
Primary 57N99; Secondary 55R65

DOI:
https://doi.org/10.1090/S0002-9947-1980-0554320-2

MathSciNet review:
554320

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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.

**[C-D 1]**D. S. Coram and P. F. Duvall,*Approximate fibrations*, Rocky Mountain J. Math.**7**(1977), 275-288. MR**0442921 (56:1296)****[C-D 2]**-,*Approximate fibrations and a movability condition for maps*, Pacific J. Math.**72**(1977), 41-56. MR**0467745 (57:7597)****[C-F 1]**T. A. Chapman and S. Ferry,*Hurewicz fiber maps with ANR fibers*, Topology**6**(1977), 121-143. MR**0448356 (56:6663)****[C-F 2]**-,*Hurewicz fiberings of ANR*'s (preprint).**[D-H]**E. Dyer and M.-E. Hamstrom,*Completely regular mappings*, Fund. Math.**45**(1957), 103-118. MR**0092959 (19:1187e)****[F-T-W]**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]**M.-E. 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), 897-913. MR**0500990 (58:18472)****[H-W]**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.**[K-S]**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 11-15, 1975, Guilford College, Greensboro, N. C.**[Mi]**John Milnor,*On spaces having the homotopy type of a CW-complex*, Trans. Amer. Math. Soc.**90**(1959), 272-280. MR**0100267 (20:6700)****[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 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)**

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0554320-2

Article copyright:
© Copyright 1980
American Mathematical Society