Conditions implying that a 2-sphere is almost tame
L. D. Loveland · Transactions of the American Mathematical Society · 1968
Introduction.Recently, Hempel [15] proved that a 2-sphere S in S3 is tame if S is free and satisfies an additional Condition (A).It is not known whether S is tame if it is free ; however, each complementary domain of a free 2-sphere must be an open 3-cell [18].We show that 5 has at most two wild points if S satisfies (A) and each component of S3 -S is an open 3-cell.Thus it appears that Hempel needed the full force of freeness only to rid 51 of two wild points.Burgess [11] proved that a 2-sphere S in 53 is tame modulo two points if each component of S3 -S is an open 3-cell and S can be locally peripherally collared.In the next section we define a surface to be locally annular if it satisfies certain conditions more general than those in Burgess' definition of locally peripherally collared.We show in §5 that a surface is locally annular if it satisfies Hempel's Condition (A).Furthermore Burgess' result remains valid with "locally annular" replacing "locally peripherally collared" (see §4).No examples were given in [15] of surfaces which fail to satisfy Condition (A).It follows from our results in §5 that the examples described in [1] and [13] each fail to satisfy (A).These examples also fail to be locally annular.We show that a closed connected 2-manifold in S3 is tame if it is free and locally annular (see §6).All the main results of [15] follow as corollaries, and in addition we remove the condition in the hypothesis of Theorem 4 of [15] that the 2-manifold M be a 2-sphere.In §3 we develop a characterization of tame surfaces in S3 which turns out to be useful in both § §4 and 6.