No abstract

No abstract

No abstract

Then, since J has properties (P and Q, sets G, Ti, and Di may be constructed so that the sets G u u Ti u u Di and ( u u Zi u u Ti are homeomorphic.The extension of this homeomorphism to a self-homeomorphism of R3 which maps J onto a is then made. In obtaining this extension strong use is made of the result due to Alexander7 that one of the complementary domains of a polyhedral torus in spherical 3-space is homeomorphic to a complementary domain of a standard torus.It is known8 that if Si, S1' and S2, S2' are two pairs of polyhedral spheres in RI, with Si contained in the interior of Si' (Int Si'), then there is a homeomorphism of the closure of the set (Int Si') n (Ext Si) onto the closure of (Int S2') n (Ext S2).It is evident that quite a diverse collection of types of regions are bounded by pairs of disjoint polyhedral tori, but an extension of the above-mentioned result to suitably restricted pairs of "concentric" polyhedral tori is made. This provides a chief tool for the proof of the above theorem.III. If the 1-manifold J has properties (P and Q at each point, then J is locally tame and conversely.* This work was begun jointly during the tenure of a Naitional Science Foundation Fellowship by Griffith. I L. Antoine, "Sur la possibility d'entendre l'hom6omorphisme de deux figures a leur voisinage,"

No abstract