IMBEDDING DECOMPOSITIONS OF E3 IN El

Our first observation is that if the decomposition contains only two nondegenerate elements, then such an imbedding is always possible. However, if the nondegenerate elements of the decomposition consist of nine tamely imbedded (actually planar) circles, appropriately linked, then such an imbedding is not possible. We conjectured that the same was true in case the nondegenerate elements were three planar circles each pair of which was linked, but a neat proof by Goblirsch [4] shows our intuition was faulty. THEOREM 1. If a monotone decomposition of E3 contains only two nondegenerate elements, then the decomposition space is imbeddable in E4. PROOF. Let Ci and C2 be the nondegenerate elements and consider E3 to be the Xi, X2, x3 plane in E4. Let P1 and P2 be points in E4 with X4 coordinates 1 and -1 respectively. Let Bi be the cone over C from Pi, i= 1, 2. Let U1 and U2 be disjoint neighborhoods of C1 and C2 respectively, and let fi be a function mapping E3 onto the unit interval such that the complement of Us is sent to 0 and precisely all