Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties

Author: Robert J. Daverman
Journal: Proc. Amer. Math. Soc. 75 (1979), 177-182
MSC: Primary 57N60; Secondary 57N10, 57N15
MathSciNet review: 529237
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We point out the sharpness of earlier results of McMillan by exhibiting a map of the n-sphere $ {S^n},n \geqslant 5$, onto itself having acyclic but non-cell-like polyhedra as its nondegenerate point inverses and for which the image of the set of nondegenerate point inverses is a Cantor set K. Of necessity, K is wildly embedded, and it has the unusual additional property that every self-homeomorphism of K extends to a self-homeomorphism of $ {S^n}$.

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

  • [A] L. Antoine, Sur l'homeomophie de deux figures et de leur voisinages, J. Math. Pures Appl. 86 (1921), 221-325.
  • [Bl] W. A. Blankinship, Generalization of a construction of Antoine, Ann. of Math. (2) 53 (1951), 276-297. MR 0040659 (12:730c)
  • [Br] G. E. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR 0221500 (36:4552)
  • [BL] J. L. Bryant and R. C. Lacher, Resolving zero-dimensional singularities in generalized manifolds, Math. Proc. Cambridge Philos. Soc. 83 (1978), 403-413. MR 0482777 (58:2830)
  • [C] J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three (to appear). MR 541330 (80j:57013)
  • [CBL] J. W. Cannon, J. L. Bryant and R. C. Lacher, The structure of generalized manifolds having nonmanifold set of trivial dimension (to appear). MR 537735 (80h:57026)
  • [E] R. D. Edwards, Approximating certain cell-like maps by homeomorphisms (to appear).
  • [H] S. T. Hu, Theory of retracts, Wayne State Univ. Press, Detroit, 1965. MR 0181977 (31:6202)
  • [HW] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N. J., 1948. MR 0006493 (3:312b)
  • [L] R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717-731. MR 0251714 (40:4941)
  • [M] D. R. McMillan, Jr., Acyclicity in three-manifolds, Bull. Amer. Math. Soc. 76 (1970), 942-964. MR 0270380 (42:5269)
  • [N] M. H. A. Newman, The engulfing theorem for topological manifolds, Ann. of Math. (2) 84 (1966), 555-571. MR 0203708 (34:3557)
  • [S] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N60, 57N10, 57N15

Retrieve articles in all journals with MSC: 57N60, 57N10, 57N15

Additional Information

Keywords: Cell-like, acyclic, upper semicontinuous, decomposition, disjoint disks property, generalized manifold, wild Cantor set, strongly homogeneous embedding
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society