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Transactions of the American Mathematical Society

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Cellular maps between polyhedra


Author: James P. Henderson
Journal: Trans. Amer. Math. Soc. 272 (1982), 527-537
MSC: Primary 57Q55; Secondary 54E60, 57N60, 57Q37
DOI: https://doi.org/10.1090/S0002-9947-1982-0662050-3
MathSciNet review: 662050
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Abstract: A compact subset $ X$ of a polyhedron $ P$ is cellular in $ P$ if there is a pseudoisotopy of $ P$ shrinking precisely $ X$ to a point. A proper surjection $ f:P\rightarrow Q$ is cellular if each point inverse of $ f$ is cellular in $ P$. We give certain conditions under which cellular maps between polyhedra are approximable by homeomorphisms. An example of a cellular map which is not approximable by homeomorphisms is also given.


References [Enhancements On Off] (What's this?)

  • [1] Ethan Aiken, Manifold phenomena in the theory of polyhedra, Trans. Amer. Math. Soc. 143 (1969), 413-473. MR 0253329 (40:6544)
  • [2] F. D. Ancel, Engulfing the tracks of a proper homotopy, notes.
  • [3] S. Armentrout, Cellular decompositions of $ 3$-manifolds that yield $ 3$-manifolds, Mem. Amer. Math. Soc. No. 107 (1971). MR 0413104 (54:1225)
  • [4] J. W. Cannon, Shrinking cell-like decompositions of manifolds: codimension three, Ann. of Math. (2) 110 (1979), 83-112. MR 541330 (80j:57013)
  • [5] R. D. Edwards, Approximating certain cellular maps by homeomorphisms, manuscript.
  • [6] R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63-88. MR 0283802 (44:1032)
  • [7] R. Geoghegan, Open problems in infinite-dimensional topology, Topology Proc. 4 (1979), 287-338. MR 583711 (82a:57015)
  • [8] Michael Handel, Approximating stratum preserving CE maps between CS sets of stratum preserving homeomorphisms, Geometric Topology (L. C. Glaser and T. B. Rushing, Eds.), Lecture Notes in Math., vol. 438, Springer-Verlag, New York, 1975, pp. 245-250. MR 0407854 (53:11624)
  • [9] J. Henderson, Cellularity in polyhedra, Topology Appl. 12 (1981), 267-282. MR 623735 (83b:57014)
  • [10] -, Approximating cellular maps between low dimensional polyhedra, Pacific J. Math. (to appear). MR 675404 (84b:57012)
  • [11] -, A discussion of results and problems related to cellularity in polyhedra, Proc. 1980 Summer Topology Conference (Austin, Texas) (to appear). MR 711986 (84i:57016)
  • [12] R. C. Lacher, Locally flat strings and half strings, Proc. Amer. Math. Soc. 18 (1967), 299-304. MR 0212805 (35:3670)
  • [13] D. R. McMillan, Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327-337. MR 0161320 (28:4528)
  • [14] L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271-294. MR 0295365 (45:4431)
  • [15] -, Deformations of homeomorphisms on stratified sets, Comment. Math. Helv. 74 (1972), 123-163.
  • [16] J. R. Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481-488. MR 0149457 (26:6945)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0662050-3
Keywords: Cellular map, polyhedron, approximation, stratification
Article copyright: © Copyright 1982 American Mathematical Society

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