Problems in Low-dimensional Topology

Four-dimensional topology is in an unsettled state: a great deal is known, but the largest-scale patterns and basic unifying themes are not yet clear. Kirby has recently completed a massive review of low-dimensional problems [Kirby], and many of the results assembled there are complicated and incomplete. In this paper the focus is on a shorter list of \tool" questions, whose solution could unify and clarify the situation. However we warn that these formulations are implicitly biased toward positive solutions. In other dimensions tool questions are often directly settled one way or the other, and even a negative solution leads to a general conclusion (eg. surgery obstructions, Whitehead torsion, characteristic classes, etc). In contrast, failures in dimension four tend to be indirect inferences, and study of the failure leads nowhere. For instance the failure of the disk embedding conjecture in the smooth category was inferred from Donaldson's nonexistence theorems for smooth manifolds. And although some direct information about disks is now available, eg. [Kr], it does not particularly illuminate the situation. Topics discussed are: in section 1, embeddings of 2-disks and 2-spheres needed for surgery and s-cobordisms of 4-manifolds. Section 2 describes uniqueness questions for these, arising from the study of isotopies. Section 3 concerns handlebody structures on 4-manifolds. Finally section 4 poses a triangulation problem for certain low-dimensional strati ed spaces. This paper was developed from a lecture given at the International Conference on Surgery and Controlled Topology, held at Josai University in September 1996. I would like to express my thanks to the organizers, particularly Masayuki Yamasaki, and to Josai University for their great hospitality.