Vol. 27, No. 2, 1968

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ISSN: 0030-8730
Uncountably many almost polyhedral wild (k 2)-cells in Ek for k 4

Leslie C. Glaser

Vol. 27 (1968), No. 2, 267–273
Abstract

In [1] infinitely many almost polyhedral wild arcs were constructed in E3 so as to have an end point as the “bad” point. In [5] uncountably many almost polyhedral wild arcs were constructed in E3 with an interior point as the “bad” point. In [4] Doyle and Hocking constructed an almost polyhedral wild disk in E4 with the property that the proof of the nontameness is perhaps the most elementary possible. They state that essentially the same construction yields a wild (n 2)-disk in En for n 4. Here, making use of the construction given in [4], we prove that for each k 4, there exist uncountably many almost polyhedral wild (k 2)-cells in Ek. To obtain the above result we also prove that for each k 3, there exist countably many polyhedral locally flat (k 2)-spheres in Ek so that the fundamental groups of the complements of these spheres are all distinct and given any two of these groups, one is not the surjective image of the other.

Mathematical Subject Classification
Primary: 55.20
Secondary: 54.00
Milestones
Received: 30 January 1968
Published: 1 November 1968
Authors
Leslie C. Glaser