Semicellularity, decompositions and mappings in manifolds
Author:
Donald Coram
Journal:
Trans. Amer. Math. Soc. 191 (1974), 227244
MSC:
Primary 57A60
MathSciNet review:
0356068
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Abstract: If X is an arbitrary compact set in a manifold, we give algebraic criteria on X and on its embedding to determine that X has an arbitrarily small, closed neighborhood each component of which is a pconnected, piecewise linear manifold which collapses to a qdimensional subpolyhedron from some p and q. This property generalizes cellularity. The criteria are in terms of UV properties and AlexanderSpanier cohomology. These criteria are then applied to decide when the components of a given compact set in a manifold are elements of a decomposition such that the quotient space is the nsphere. Conversely, algebraic criteria are given for the point inverses of a map between manifolds to have arbitrarily small neighborhoods of the type mentioned above; these criteria are considerably weaker than for an arbitrary compact set.
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 S. Armentrout, UV properties of compact sets, Trans. Amer. Math. Soc. 143 (1969), 487498. MR 42 #8451. MR 0273573 (42:8451)
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 R. J. Bean, Repairing embeddings and decompositions in , Duke Math. J. 36 (1969), 379385. MR 39 #4820. MR 0243499 (39:4820)
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 D. Coram, Semicellularity of compact subsets of manifolds, Ph. D. thesis, University of Wisconsin, Madison, 1970.
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 P. F. Duvall, Jr., Neighborhoods of 1connected ANR's in high dimensional piecewise linear manifolds, Ph. D. thesis, University of Georgia, Athens, Ga., 1967.
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 , Weakly flat spheres, Michigan Math. J. 16 (1969), 117124. MR 39 #7604. MR 0246300 (39:7604)
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 J. G. Hocking and G. S. Young, Topology, AddisonWesley, Reading, Mass., 1961. MR 23 #A2857. MR 0125557 (23:A2857)
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 John Hollingsworth and R. B. Sher, Triangulating neighborhoods in topological manifolds, General Topology and Appl. 1 (1971), 345348. MR 45 #6011. MR 0296952 (45:6011)
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 J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, Benjamin, New York, 1969. MR 40 #2094. MR 0248844 (40:2094)
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 J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
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 R. C. Lacher, Celllike mappings. I, Pacific J. Math. 30 (1969), 717731. MR 40 #4941. MR 0251714 (40:4941)
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 , Cellularity criterion for maps, Michigan Math. J. 17 (1970), 385396. MR 43 #5539. MR 0279818 (43:5539)
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 R. C. Lacher and D. R. McMillan, Jr., Partially acyclic mappings between manifolds, Amer. J. Math. 94 (1972), 246266. MR 0301743 (46:898)
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 W. B. E. Lickorish, Polyhedral handlebody theory, Mimeographed lecture notes, University of Wisconsin, Madison, 1969.
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 D. R. McMillan, Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327337. MR 28 #4528. MR 0161320 (28:4528)
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 , Acyclicity in threemanifolds, Bull. Amer. Math. Soc. 76 (1970), 942964. MR 42 #5269. MR 0270380 (42:5269)
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 R. T. Miller, Approximating codimension 3 embeddings, Proceedings of the Georgia Topology Conference, 1970.
 [22]
 R. B. Sher, Defining subsets of manifolds by cells, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 363365. MR 41 #1059. MR 0256403 (41:1059)
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 J. Stallings, The piecewiselinear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481488. MR 26 #6945. MR 0149457 (26:6945)
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 , On topologically unknotted spheres, Ann. of Math. (2) 77 (1963), 490503. MR 26 #6946. MR 0149458 (26:6946)
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 , Homology and central series of groups, J. Algebra 2 (1965), 170181. MR 31 #232. MR 0175956 (31:232)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403560683
PII:
S 00029947(1974)03560683
Keywords:
Neighborhoods of compacta,
UV properties,
extending decompositions,
mappings on manifolds
Article copyright:
© Copyright 1974 American Mathematical Society
