Finiteness theorems for approximate fibrations

Authors:
D. S. Coram and P. F. Duvall

Journal:
Trans. Amer. Math. Soc. **269** (1982), 383-394

MSC:
Primary 55R65; Secondary 55P55, 57N55

MathSciNet review:
637696

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns conditions on the point inverses of a mapping between manifolds which insure that it is an approximate fibration almost everywhere. The primary condition is -movability, which says roughly that nearby point inverses include isomorphically on the th shape group into a mutual neighborhood. Suppose is a mapping which is -movable for , and . An earlier paper proved that is an approximate fibration when . If instead , this paper proves that there is a locally finite set such that is an approximate fibration. Also if and all of the point inverses are FANR's with the same shape, then there is a locally finite set such that is an approximate fibration.

**[B]**Karol Borsuk,*A note on the theory of shape of compacta*, Fund. Math.**67**(1970), 265–278. MR**0266169****[Bi]**R. H. Bing,*Decompositions of 𝐸³*, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall. Englewood Cliffs, N.J., 1962, pp. 5–21. MR**0141088****[Bo]**N. Bourbaki,*Elements of mathematics: General topology*, Addison-Wesley, Reading, Mass., 1966.**[C]**Donald Coram,*Semicellularity, decompositions and mappings in manifolds*, Trans. Amer. Math. Soc.**191**(1974), 227–244. MR**0356068**, 10.1090/S0002-9947-1974-0356068-3**[C-D]**D. S. Coram and P. F. Duvall Jr.,*Approximate fibrations*, Rocky Mountain J. Math.**7**(1977), no. 2, 275–288. MR**0442921****[C-D]**Donald Coram and Paul Duvall,*Approximate fibrations and a movability condition for maps*, Pacific J. Math.**72**(1977), no. 1, 41–56. MR**0467745****[C-D]**D. S. Coram and P. F. Duvall Jr.,*Mappings from 𝑆³ to 𝑆² whose point inverses have the shape of a circle*, General Topology Appl.**10**(1979), no. 3, 239–246. MR**546098****[C-D]**D. Coram and P. Duvall,*Nondegenerate 𝑘-sphere mappings between spheres*, The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979, pp. 67–82 (1980). MR**583689****[C-D]**D. S. Coram and P. F. Duvall Jr.,*A Hurewicz-type theorem for approximate fibrations*, Proc. Amer. Math. Soc.**78**(1980), no. 3, 443–448. MR**553392**, 10.1090/S0002-9939-1980-0553392-4**[D-H]**P. F. Duvall and L. S. Husch,*Fundamental dimension of fibers of approximate fibrations*, Proceedings of the 1978 Topology Conference (Univ. Oklahoma, Norman, Okla., 1978), I, 1978, pp. 53–57 (1979). MR**540477****[D-S]**Jerzy Dydak and Jack Segal,*Shape theory*, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978. An introduction. MR**520227****[F-W]**Ronald Fintushel and John J. Walsh,*Singularly fibered homotopy spheres*, Indiana Univ. Math. J.**30**(1981), no. 2, 179–192. MR**604278**, 10.1512/iumj.1981.30.30015**[H]**L. S. Husch,*Fibres of Hurewicz and approximate fibrations*, Math. Scand.**43**(1978/79), no. 1, 44–48. MR**523823****[L]**R. C. Lacher,*Cellularity criteria for maps*, Michigan Math. J.**17**(1970), 385–396. MR**0279818****[L]**R. C. Lacher,*Finiteness theorems in the study of mappings between manifolds*, Proceedings of the University of Oklahoma Topology Conference Dedicated to Robert Lee Moore (Norman, Okla., 1972) Univ. Oklahoma, Norman, Okla., 1972, pp. 79–96. MR**0370593****[L]**R. C. Lacher,*Some mapping theorems*, Trans. Amer. Math. Soc.**195**(1974), 291–303. MR**0350743**, 10.1090/S0002-9947-1974-0350743-2**[L]**L. C. Glaser and T. B. Rushing (eds.),*Geometric topology*, Lecture Notes in Mathematics, Vol. 438, Springer-Verlag, Berlin-New York, 1975. MR**0362309****[L]**R. C. Lacher,*Cell-like mappings and their generalizations*, Bull. Amer. Math. Soc.**83**(1977), no. 4, 495–552. MR**0645403**, 10.1090/S0002-9904-1977-14321-8**[L-M]**R. C. Lacher and D. R. McMillan Jr.,*Partially acyclic mappings between manifolds*, Amer. J. Math.**94**(1972), 246–266. MR**0301743****[M]**Sibe Mardešić,*Strongly movable compacta and shape retracts*, Proceedings of the International Symposium on Topology and its Applications (Budva, 1972) Savez Društava Mat. Fiz. i Astronom., Belgrade, 1973, pp. 163–166. MR**0334143****[O]**R. H. Overton,*Čech homology for movable compacta*, Fund. Math.**77**(1973), no. 3, 241–251. MR**0322797****[S]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[Sp]**S. Spież,*On a characterization of shapes of several compacta*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**24**(1976), no. 4, 257–263 (English, with Russian summary). MR**0410650****[W]**David C. Wilson,*On constructing monotone and 𝑈𝑉¹ mappings of arbitrary degree*, Duke Math. J.**41**(1974), 103–109. MR**0339250**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55R65,
55P55,
57N55

Retrieve articles in all journals with MSC: 55R65, 55P55, 57N55

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0637696-9

Keywords:
Approximate fibration,
movability,
mapping between manifolds

Article copyright:
© Copyright 1982
American Mathematical Society