Mappings from manifolds onto manifolds
Author:
Alden Wright
Journal:
Trans. Amer. Math. Soc. 167 (1972), 479495
MSC:
Primary 57A10
MathSciNet review:
0339186
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Abstract: Let f be a compact, boundary preserving mapping from the 3manifold onto the 3manifold . Let denote the integers mod a prime p, or, if , the integers. (1) If each point inverse of f is connected and strongly 1acyclic 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 3manifold with 2sphere boundary, and replacing it by a homology 3cell. (3) If the singular set of f is contained in a 0dimensional set, then all but a locally finite collection of point inverses of f are cellular.
 [1]
Steve
Armentrout, Concerning cellular decompositions of
3manifolds with boundary, Trans. Amer. Math.
Soc. 137 (1969),
231–236. MR 0236931
(38 #5224), http://dx.doi.org/10.1090/S00029947196902369312
 [2]
R.
H. Bing, Locally tame sets are tame, Ann. of Math. (2)
59 (1954), 145–158. MR 0061377
(15,816d)
 [3]
Morton
Brown, A proof of the generalized Schoenflies
theorem, Bull. Amer. Math. Soc. 66 (1960), 74–76. MR 0117695
(22 #8470b), http://dx.doi.org/10.1090/S000299041960104004
 [4]
P.
T. Church, Differentiable monotone maps on
manifolds. II, Trans. Amer. Math. Soc. 158 (1971), 493–501.
MR
0278320 (43 #4050), http://dx.doi.org/10.1090/S0002994719710278320X
 [5]
Wolfgang
Haken, Some results on surfaces in 3manifolds, Studies in
Modern Topology, Math. Assoc. Amer. (distributed by PrenticeHall,
Englewood Cliffs, N.J.), 1968, pp. 39–98. MR 0224071
(36 #7118)
 [6]
H. Kneser, Geschlossen Flachen in dreidimensionalen Mannigfaltigkeiten, Jber. Deutsch. Math.Verein. 38 (1929), 248260.
 [7]
R.
C. Lacher, Cellularity criteria for maps, Michigan Math. J.
17 (1970), 385–396. MR 0279818
(43 #5539)
 [8]
Chris
Lacher and Alden
Wright, Mapping cylinders and 4manifolds, Topology of
Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham,
Chicago, Ill., 1970, pp. 424–427. MR 0271951
(42 #6832)
 [9]
H.
W. Lambert, Replacing certain maps of 3manifolds
by homeomorphisms, Proc. Amer. Math. Soc.
23 (1969),
676–678. MR 0256402
(41 #1058), http://dx.doi.org/10.1090/S00029939196902564022
 [10]
Wilhelm
Magnus, Abraham
Karrass, and Donald
Solitar, Combinatorial group theory: Presentations of groups in
terms of generators and relations, Interscience Publishers [John Wiley
& Sons, Inc.], New YorkLondonSydney, 1966. MR 0207802
(34 #7617)
 [11]
D.
R. McMillan Jr., Strong homotopy equivalence of
3manifolds, Bull. Amer. Math. Soc. 73 (1967), 718–722. MR 0229243
(37 #4817), http://dx.doi.org/10.1090/S000299041967118433
 [12]
D.
R. McMillan Jr., Compact, acyclic subsets of threemanifolds,
Michigan Math. J. 16 (1969), 129–136. MR 0243501
(39 #4822)
 [13]
D.
R. McMillan Jr., Acyclicity in
threemanifolds, Bull. Amer. Math. Soc. 76 (1970), 942–964.
MR
0270380 (42 #5269), http://dx.doi.org/10.1090/S000299041970125101
 [14]
J.
Milnor, A unique decomposition theorem for 3manifolds, Amer.
J. Math. 84 (1962), 1–7. MR 0142125
(25 #5518)
 [15]
T.
M. Price, A necessary condition that a cellular
upper semicontinuous decomposition of 𝐸ⁿ yield
𝐸ⁿ, Trans. Amer. Math. Soc.
122 (1966),
427–435. MR 0193627
(33 #1843), http://dx.doi.org/10.1090/S00029947196601936270
 [16]
A. H. Wright, Monotone mappings of compact 3manifolds, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1969.
 [1]
 S. Armentrout, Concerning cellular decompositions of 3manifolds with boundary, Trans. Amer. Math. Soc. 137 (1969), 231236. MR 38 #5224. MR 0236931 (38:5224)
 [2]
 R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145158. MR 15, 816. MR 0061377 (15:816d)
 [3]
 M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 7476. MR 22 #8470b. MR 0117695 (22:8470b)
 [4]
 P. T. Church, Differentiable and monotone maps on manifolds. II, Trans. Amer. Math. Soc. 158 (1971), 493503. MR 0278320 (43:4050)
 [5]
 W. Haken, Some results on surfaces in 3manifolds, Studies in Modern Topology, Math. Assoc. Amer., distributed by PrenticeHall, Englewood Cliffs, N. J., 1968, pp. 3998. MR 36 #7118. MR 0224071 (36:7118)
 [6]
 H. Kneser, Geschlossen Flachen in dreidimensionalen Mannigfaltigkeiten, Jber. Deutsch. Math.Verein. 38 (1929), 248260.
 [7]
 R. C. Lacher, Cellularity criteria for maps, Michigan Math. J. 17 (1970), 385396. MR 0279818 (43:5539)
 [8]
 R. C. Lacher and A. H. Wright, Mapping cylinders and 4manifolds, Topology of Manifolds, Markham, Chicago, Ill., 1970. MR 0271951 (42:6832)
 [9]
 H. W. Lambert, Replacing certain maps of 3manifolds by homeomorphisms, Proc. Amer. Math. Soc. 23 (1969), 676678. 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 3manifolds, Bull. Amer. Math. Soc. 73 (1967), 718722. MR 37 #4817. MR 0229243 (37:4817)
 [12]
 , Compact acyclic subsets of threemanifolds, Michigan Math. J. 16 (1969), 129136. MR 39 #4822. MR 0243501 (39:4822)
 [13]
 , Acyclicity in threemanifolds, Bull. Amer. Math. Soc. 76 (1970), 942964. MR 0270380 (42:5269)
 [14]
 J. W. Milnor, A unique decomposition theorem for 3manifolds, Amer. J. Math. 84 (1962), 17. 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), 427435. MR 33 #1843. MR 0193627 (33:1843)
 [16]
 A. H. Wright, Monotone mappings of compact 3manifolds, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1969.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203391863
PII:
S 00029947(1972)03391863
Keywords:
Monotone mapping,
3manifold,
acyclic mapping,
decomposition space,
strongly acyclic
Article copyright:
© Copyright 1972
American Mathematical Society
