Frank Quinn

Ends of maps, II

Versions of the finiteness obstruction and simple homotopy theory “within ε overX” are developed. This provides a setting for obstructions to the map analogs of the end ands-cobordism theorems for manifolds. These are applied to study equivariant mapping cylinder neighborhoods in topological group actions, triangulations of locally triangulable spaces, and block bundle structures on approximate fibrations.

Subexponential groups in 4{manifold topology

We present a new, more elementary proof of the Freedman{Teichner result that the geometric classication techniques (surgery, s{cobordism, and pseudoisotopy) hold for topological 4{manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reformulate the growth estimates in terms of coarse geometry.

The stable topology of 4-manifolds

Abstract The stable theory (which allows connected sums with S 2 × S 2 ) is unified and extended using current 4-manifold techniques. Principal new results are a stable 5-dimensional s -cobordism theorem, and the fact that 1-connected smooth 4-manifold pairs stably have handle decompositions with no 1-handles.

A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today

T he physical sciences all went through “revolutions”: wrenching transitions in which methods changed radically and became much more powerful. It is not widely realized, but there was a similar transition in mathematics between about 1890 and 1930. The first section briefly describes the changes that took place and why they qualify as a “revolution”, and the second describes turmoil and resistance to the changes at the time. The mathematical event was different from those in science, however. In science, most of the older material was wrong and discarded, while old mathematics needed precision upgrades but was mostly correct. The sciences were completely transformed while mathematics split, with the core changing profoundly but many applied areas, and mathematical science outside the core, relatively unchanged. The strangest difference is that the scientific revolutions were highly visible, while the significance of the mathematical event is essentially unrecognized. The section “Obscurity” explores factors contributing to this situation and suggests historical turning points that might have changed it. The main point of this article is not that a revolution occurred, but that there are penalties for not being aware of it. First, precollege mathematics education is still based on nineteenth-century methodology, and it seems to me that we will not get satisfactory outcomes until this changes [9]. Second, the mathematical community is adapted to the social and intellectual environment of the midand late twentieth century, and this environment is changing in ways likely to marginalize core mathematics. But core mathematics provides the skeleton that supports the muscles and sinews of science and technology; marginalization will lead to a scientific analogue of osteoporosis. Deliberate management [2] might avoid this, but only if the disease is recognized.

Roadkill on the electronic highway? The threat to the mathematical literature

This article begins with an analysis of reliability and usability in the mathematical literature. Mathematical practice is seen to be adapted to very high standards in these respects, and quite sensitive to even modest declines in quality. Unfortunately most scenarios for the transition to electronic publication suggest serious loss of quality. This is a problem special to mathematics: other sciences are adapted to lower standards, and should be much less sensitive to changes. Some ways to prevent this decline are suggested.

Contemporary Proofs for Mathematics Education

In contemporary mathematical practice, the primary importance of proof is the advantage it provides to users: proofs enable very high levels of reliability. This essay explores use of this sort of proof, and methods mathematicians use to implement them, in pre-college mathematics. Examples include methods for multiplying integers (including large ones), multiplication of polynomials, solving equations, and standardizing quadratic functions. The point of view also reveals drawbacks of real-world applications (word problems).

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

The Triangulation of Manifolds: Topology, Gauge Theory, and History

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the questions.