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.
