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

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Abstract: We point out the sharpness of earlier results of McMillan by exhibiting a map of the *n*-sphere , 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 .

**[A]**L. Antoine,*Sur l'homeomophie de deux figures et de leur voisinages*, J. Math. Pures Appl.**86**(1921), 221-325.**[Bl]**William A. Blankinship,*Generalization of a construction of Antoine*, Ann. of Math. (2)**53**(1951), 276–297. MR**0040659****[Br]**Glen E. Bredon,*Sheaf theory*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR**0221500****[BL]**J. L. Bryant and R. C. Lacher,*Resolving zero-dimensional singularities in generalized manifolds*, Math. Proc. Cambridge Philos. Soc.**83**(1978), no. 3, 403–413. MR**0482777****[C]**J. W. Cannon,*Shrinking cell-like decompositions of manifolds. Codimension three*, Ann. of Math. (2)**110**(1979), no. 1, 83–112. MR**541330**, 10.2307/1971245**[CBL]**J. W. Cannon, J. L. Bryant, and R. C. Lacher,*The structure of generalized manifolds having nonmanifold set of trivial dimension*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 261–300. MR**537735****[E]**R. D. Edwards,*Approximating certain cell-like maps by homeomorphisms*(to appear).**[H]**Sze-tsen Hu,*Theory of retracts*, Wayne State University Press, Detroit, 1965. MR**0181977****[HW]**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493****[L]**R. C. Lacher,*Cell-like mappings. I*, Pacific J. Math.**30**(1969), 717–731. MR**0251714****[M]**D. R. McMillan Jr.,*Acyclicity in three-manifolds*, Bull. Amer. Math. Soc.**76**(1970), 942–964. MR**0270380**, 10.1090/S0002-9904-1970-12510-1**[N]**M. H. A. Newman,*The engulfing theorem for topological manifolds*, Ann. of Math. (2)**84**(1966), 555–571. MR**0203708****[S]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1979-0529237-7

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