EMBEDDING SURFACES IN 3-MANIFOLDS

EMBEDDING SURFACES IN 3-MANIFOLDS ByR. H . B I N G

Is a 2-sphere in E3 tame if it can be pierced at each point by a straight line interval?(x) Is it tame if it can be pierced along each arc in it by a tame disk? Is a 2-sphere in E3 tame if it is homogeneous under a space homeomorphism? Is a 2-sphere in E3 tame if each Sierpinski curve in it can be pushed to either side? Is it tame if it can be approximated from either side by a singular 2-sphere? Does one obtain E3 if the interior of a wild 2-sphere is replaced by the interior of a tame one? What restrictions imply that a surface is tamely embedded in a 3-manifold? These are samples of some of the interesting unsolved problems about the embedding of surfaces in 3-manifolds. In addition to raising questions about the embedding of surfaces in 3manifolds, recent discoveries were discussed. A surface is known to be tame if its complement is 1-ULC. It is tame if it can be pushed to either side with a map. It is tame if it can be homeomorphically approximated from either side. The fact that surfaces can be approximated by polyheral surfaces has been useful in proving some of these results and may be the means for learning more about the embedding of surfaces in 3-manifolds. One advantage of working with polyhedral surfaces instead of arbitrary ones is that they intersect in decent sets. However, it has been shown recently that if in E3, Lis a, straight line, S is a 2-sphere, and e >0, then there is a homeomorphism h of E3 onto itself such that h does not move any point more than e, h is fixed outside an e-neighborhood of L, and S'h(L) does not have infinitely many points. A simple proof of this result would improve the proof of the approximation theorems. Suppose SX,S2 are two arbitrary surfaces in E3. One might wonder if it is possible to move Sx slightly with a space homeomorphism so that the resulting 2-sphere intersects S2 "nicely". Hand in hand with proofs go counterexamples. Examples have recently been given of wild spheres all of whose arcs are tame. It has been shown however that all wild surfaces contain tame arcs and can be pierced by tame arcs. The set of points that belong to tame arcs in the surfaces is precisely the set of points of the surface at which it can be pierced by tame arcs. Does a simple closed curve bound a disk in a 3-manifold if the curve can be shrunk to a point in its own complement? This is an unsolved version of Dehn's lemma. If a singular plane P in E3 links two points in E3, does each e-neighbor(*) Following the lecture, M. K. Fort, Jr. answered the first question raised in this lecture in the negative by giving an example of a wild 2-sphere (which might be called a "wild porcupine") which could be pierced at each point by a straight line interval. The example is described in a paper submitted to the Proc. Amer. Math. Soc.

458

B. H. BING

hood of P contain a non-singular plane that links the two points? This version of the sphere theorem remains unanswered partly because in Papakyriakopoklos' proof of Dehn's lemma, the splitting and resewing along double lines introduced long feelers. Suppose D is a disk on a polyhedral 2-sphere S and e > 0. Is there a positive number à > 0 such that if g is a piecewise linear map of S into E3 that is fixed on S — D and moves no point more than ô, then each neighborhood of g(S) contains a 2-sphere whose homeomorphic distance from S is less than ei A contemplation of E3 and a study of recent discoveries about 3-manifolds reveals that although we have learned much about 3-space in recent years, there is much yet to learn.