A Cellular Wedge in R 3
T. B. Rushing, R. B. Sher · Proceedings of the American Mathematical Society · 1991
We construct a cellular wedge A V B c R3 such that A is not cellular. In a recent list of problems circulated by R. J. Daverman, the following question [1, El 9] is attributed to Morton Brown: If a wedge A V B c R3 is cellular, is A cellular? (Daverman was able to construct cellular wedges in R n, n > 5, having a noncellular constituent. His examples are included in an addendum to this paper.) The following example answers this question negatively. Example. Let A c R3 be the well-known Fox-Artin arc [5, Example 1.3], constructed so as to be locally polyhedral modulo its endpoints p and q. There is a piecewise linear 3-cell whose interior contains p and whose boundary is pierced by A at a single point. Guided by A, we bore out a tunnel in the 3-cell pinching down to p, and so construct a crumpled cube B such that B is not a 3-cell, Bd B is locally polyhedral modulo p, and A n B = {p}; the process of obtaining B is more precisely described in [11]. The arc A is not cellular because, as shown explicitly in [5, Example 1.3], R A is not an open 3-cell (in particular, Fox and Artin show that there is a simple closed curve in R3 -A that is not contained in the interior of a 3-cell in R3 A). On the other hand, X = A U B is cellular. The argument for this is well known, but we give a brief sketch here for completeness. To begin, note that there are arbitrarily small 3-cells C such that B n C is a disk whose interior contains p and B U C is a piecewise linear 3-cell whose boundary is pierced by A at a single point. It is easily seen that Y = X U C is cellular by constructing a 3-cell neighborhood of Y in the e-neighborhood of Y. To accomplish this, begin with a small piecewise linear sphere Y about q that is disjoint from B and is pierced by A at three points. Let these points be denoted by al, a2 , and a3, ordered moving from p to q. Let D1, D2, and D3 be pairwise disjoint small disks on X about a1, a2, and a3, respectively. Within the e-neighborhood of Y, stretch D1 in Received by the editors May 15, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 57N60.