Problems in 4-dimensional Topology

The early 1980’s saw enormous progress in understanding 4-manifolds: the topological Poincaré and annulus conjectures were proved, many cases of surgery and the s-cobordism theorem were settled, and Donaldson’s work showed that smooth structures are stranger than anyone had imagined. Big gaps remained: topological surgery and s-cobordisms with arbitrary fundamental group, and general classification results for smooth structures. Since then the topological work has been refined and applied, but the big problems are still unsettled. Gauge theory has flowered, but has had more to say about geometric structures (esp. complex or symplectic) than basic smooth structures. So on the foundational questions not much has happened in the last fifteen years. We might hope that this has been a period of consolidation, providing foundations for the next generation of breakthroughs. Kirby has recently completed a massive review of low-dimensional problems [Kirby]. Here the focus is on a shorter list of “tool” questions, whose solution could unify and clarify the situation. These are mostly well-known, and are repeated here mainly to give a context for comments and status reports. We warn that these formulations are implicitly biased toward positive solutions. In other dimensions when tool questions turn out to be false they still frequently lead to satisfactory solutions of the original problems in terms of obstructions (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. Some direct information about disks is now available, eg. [Kr], but 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. Section 4 concerns invariants. Finally section 5 poses a triangulation problem for certain low-dimensional stratified spaces. I would like to expand on the dedication of this paper to C. T. C. Wall. When I joined the mathematical community in the late 1960s the development of higherdimensional topology was in full swing. Surgery was hot: “everybody” seemed to be studying Wall’s monograph [W1], the solution of the Hauptvermutung was just around the corner, and the new methods were revolutionizing the study of transformation groups. However little or none of it applied to low dimensions. Few people seemed to be bothered by excluding dimensions below 5, 6 or 7, and