Mappings from -manifolds onto -manifolds

Author:
Alden Wright

Journal:
Trans. Amer. Math. Soc. **167** (1972), 479-495

MSC:
Primary 57A10

DOI:
https://doi.org/10.1090/S0002-9947-1972-0339186-3

MathSciNet review:
0339186

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Abstract: Let *f* be a compact, boundary preserving mapping from the 3-manifold onto the 3-manifold . Let denote the integers mod a prime *p*, or, if , the integers. (1) If each point inverse of *f* is connected and strongly 1-acyclic over , and if is orientable for , then all but a locally finite collection of point inverses of *f* are cellular. (2) If the image of the singular set of *f* is contained in a compact set each component of which is strongly acyclic over , and if is orientable for , then can be obtained from by cutting out of a compact 3-manifold with 2-sphere boundary, and replacing it by a -homology 3-cell. (3) If the singular set of *f* is contained in a 0-dimensional set, then all but a locally finite collection of point inverses of *f* are cellular.

**[1]**S. Armentrout,*Concerning cellular decompositions of*3-*manifolds with boundary*, Trans. Amer. Math. Soc.**137**(1969), 231-236. MR**38**#5224. MR**0236931 (38:5224)****[2]**R. H. Bing,*Locally tame sets are tame*, Ann. of Math. (2)**59**(1954), 145-158. MR**15**, 816. MR**0061377 (15:816d)****[3]**M. Brown,*A proof of the generalized Schoenflies theorem*, Bull. Amer. Math. Soc.**66**(1960), 74-76. MR**22**#8470b. MR**0117695 (22:8470b)****[4]**P. T. Church,*Differentiable and monotone maps on manifolds*. II, Trans. Amer. Math. Soc.**158**(1971), 493-503. MR**0278320 (43:4050)****[5]**W. Haken,*Some results on surfaces in*3-*manifolds*, Studies in Modern Topology, Math. Assoc. Amer., distributed by Prentice-Hall, Englewood Cliffs, N. J., 1968, pp. 39-98. MR**36**#7118. MR**0224071 (36:7118)****[6]**H. Kneser,*Geschlossen Flachen in dreidimensionalen Mannigfaltigkeiten*, Jber. Deutsch. Math.-Verein.**38**(1929), 248-260.**[7]**R. C. Lacher,*Cellularity criteria for maps*, Michigan Math. J.**17**(1970), 385-396. MR**0279818 (43:5539)****[8]**R. C. Lacher and A. H. Wright,*Mapping cylinders and*4-*manifolds*, Topology of Manifolds, Markham, Chicago, Ill., 1970. MR**0271951 (42:6832)****[9]**H. W. Lambert,*Replacing certain maps of*3-*manifolds by homeomorphisms*, Proc. Amer. Math. Soc.**23**(1969), 676-678. MR**41**#1058. MR**0256402 (41:1058)****[10]**W. Magnus, A. Karrass and D. Solitar,*Combinatorial group theory*:*Presentations of groups in terms of generators and relations*, Pure and Appl. Math., vol. 13, Interscience, New York, 1966. MR**34**#7617. MR**0207802 (34:7617)****[11]**D. R. McMillan, Jr.,*Strong homotopy equivalence of*3-*manifolds*, Bull. Amer. Math. Soc.**73**(1967), 718-722. MR**37**#4817. MR**0229243 (37:4817)****[12]**-,*Compact acyclic subsets of three-manifolds*, Michigan Math. J.**16**(1969), 129-136. MR**39**#4822. MR**0243501 (39:4822)****[13]**-,*Acyclicity in three-manifolds*, Bull. Amer. Math. Soc.**76**(1970), 942-964. MR**0270380 (42:5269)****[14]**J. W. Milnor,*A unique decomposition theorem for*3-*manifolds*, Amer. J. Math.**84**(1962), 1-7. MR**25**#5518. MR**0142125 (25:5518)****[15]**T. M. Price,*A necessary condition that a cellular upper semicontinuous decomposition of**yield*, Trans. Amer. Math. Soc.**122**(1966), 427-435. MR**33**#1843. MR**0193627 (33:1843)****[16]**A. H. Wright,*Monotone mappings of compact*3-*manifolds*, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1969.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0339186-3

Keywords:
Monotone mapping,
3-manifold,
acyclic mapping,
decomposition space,
strongly acyclic

Article copyright:
© Copyright 1972
American Mathematical Society