Loveland has established
that if W is the set of wild points of a cellular arc that lies on a 2-sphere in E8, then
either W is empty, W is degenerate, or W contains an arc. This note considers
2-complexes rather than 2-spheres. Making strong use of Loveland’s results and
others, it is proved that a cellular arc in a 2-complex in E8 either contains an arc of
wild points or has at most one wild point that has a neighborhood in the 2-complex
homeomorphic to an open 2-cell. In the case of noncellular arcs in E3, one can
investigate “minimal cellular sets” containing the arc. A cellular hull of a subset A of
Es is a cellular set containing A such that no proper cellular set also contains A. A
characterization is given of those arcs in E3 that have cellular hulls that lie in tame
2-complexes in E3.
