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Problems in Low-Dimensional Topology

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Problems in Low-Dimensional Topology Edited by Rob Kirby Berkeley - 22 Dec 95 Contents 1 Knot Theory 7 2 Surfaces 85 3 3-Manifolds 97 4 4-Manifolds 179 5 Miscellany 259 Index of Conjectures 282 Index 284 Old Problem Lists 294 Bibliography 301 1 2 CONTENTS Introduction In April, 1977 when my first problem list [38,Kirby,1978] was finished, a good topologist could reasonably hope to understand the main topics in all of low dimensional topology. But at that time Bill Thurston was already starting to greatly influence the study of 2- and 3-manifolds through the introduction of geometry, especially hyperbolic. Four years later in September, 1981, Mike Freedman turned a subject, topological 4-manifolds, in which we expected no progress for years, into a subject in which it seemed we knew everything. A few months later in spring 1982, Simon Donaldson brought gauge theory to 4-manifolds with the first of a remarkable string of theorems showing that smooth 4-manifolds which might not exist or might not be diffeomorphic, in fact, didn’t and weren’t. Exotic R4’s, the strangest of smooth manifolds, followed. And then in late spring 1984, Vaughan Jones brought us the Jones polynomial and later Witten a host of other topological quantum field theories (TQFT’s). Physics has had for at least two decades a remarkable record for guiding mathematicians to remarkable mathematics (Seiberg–Witten gauge theory, new in October, 1994, is the latest example). Lest one think that progress was only made using non-topological techniques, note that Freedman’s work, and other results like knot complements determining knots (Gordon- Luecke) or the Seifert fibered space conjecture (Mess, Scott, Gabai, Casson & Jungreis) were all or mostly classical topology. So editing a problem list in 1994 is a very different task than in 1977. It would not have been possible for this editor without an enormous amount of help from others. For no particular reason, I did not keep track at first of who provided help with the Updates of the lists from 1977 and 1982 [39,Kirby,1984], so there are no names attached to the Updates. However Geoff Mess alone must have provided me with half the Updates in dimensions 2 and 3, as well as others. I also received much help from Joel Hass, Cameron Gordon, Dieter Kotschick, Walter Neumann, Peter Teichner, Larry Taylor, Jonathan Hillman, Peter Kronheimer, Marty Scharlemann, Ron Stern, Andrew Casson, Francis Bonahon, Paulo Ney de Souza, Hyam Rubinstein, Lee Rudolph, Robert Myers, Bob Gompf, Selman Akbulut, Chuck Livingston, Tom Mrowka, Mike Freedman, Pat Gilmer, Michel Boileau, Peter Scott, 3 4 CONTENTS Abby Thompson, Steve Bleiler, Curt McMullen, Raymond Lickorish, Tsuyoshi Kobayashi, Yasha Eliashberg, Yukio Matsumoto, Danny Ruberman, Zarkoˇ Biˇzaca, Wolfgang Metzler, Jim Milgram, Dave Gabai, Darryl McCullough, and many others who contributed to Updates involving their own work. One can imagine an editor who reads and understands hundreds of papers so as to personally write and vouch for each Update; that is not the case here. Rather I am indebted to many erudite and hardworking friends. The new Problems often have a name or names attached; occasionally the name is the originator of a conjecture or question, but most of the time is the person who helped me write the problem. I’d like to thank also a handful of people who reviewed parts of the penultimate draft: Mess, Gordon, Neumann, Kotschick, Akbulut and Gompf. One might expect, with a problem list of this size, that the list is all inclusive. Wrong. Of course I have made attempts to cover obvious areas, but I never wished to take on the task of covering everything. For example, laminations are already beautifully covered by Dave Gabai in another problem list in these Proceedings. In the 1977 list, I particularly tried to get problems involving related subjects, but this time, that task was too daunting and no great effort was made. There are not as many problems involving contact structures, graph theory, dynamics, for example, as there could have been. Will Kazez suggested this task in June 1992, no doubt hoping that I would be done shortly after the August, 1993, Georgia conference. But not much was accomplished before the Georgia conference at which many of the problems were proposed. More were added at conferences in Warwick (August, 1993), Oberwolfach (September 1993), the Park City gauge theory conference (July, 1994), Huia, New Zealand (December, 1994), Princeton (January, 1995), Max Planck Institute, Bonn (May, 1995), and Gokova, Turkey (May 1995); and many problems turned up by e-mail or through personal contacts. Another half year would have elapsed if I had not had the help of Paulo Ney de Souza for Updates, computer drawn figures, wise advise on editing, and (with Faye Yeager) high tech typing and editing. He is primarily responsible for the huge task of organizing the bibliography and making it complete and accurate. Significant further help in LATEXing was provided by Larry Taylor, and in producing figures by Silvio Levy, Leo Tenenblat, and Jonas Gomes. The format of the 1977 list has, in a Procrustean way, been continued. There are still five chapters, one each for the four dimensions plus miscellaneous. The old problems keep their numbers, except that the 1982 list of 4-manifold problems with a few 2-in-4 knot problems, have had the N dropped, e.g. Problem N4.45 became Problem 4.45. Then the new Problems continue the numbering, so, for example, in Chapter 1, Problems 1.1 through 1.51 are from 1977, Problems 1.52 through 1.57 are from 1982, and Problems 1.58 through 1.105 are new. This means that, for example, problems on knot groups can appear in three different sections CONTENTS 5 rather than together. But there is liberal cross referencing, as in the table of contents for each chapter. The old Problems are stated without changes, except for completed references and cor- rections of errors. The texts of the Problems contain abbreviated references, typically just the author, journal, and year, which should usually be enough for the reader to guess the paper rather than pause to consult the bibliography where the full citation occurs. There are indices of conjectures, and an index of mathematical terms including symbols, knots and manifolds. Finally, there is a list of old problem lists. The manuscript was prepared in LATEX2based on a specially enlarged version of TEX running on a SUN Sparcstation 20 at UC Berkeley over the last year, merging several pre- existing documents. Each problem was kept as a single file that could be formatted indi- vidually and sent out by e-mail. These files are then called by a master file that formats the whole document using several standard LATEX packages as well as in-house developed ones and a package developed by Larry Taylor enabling each bibliographic item to list the problems in which it is cited. All graphics have been produced in PostScript; most of them were drawn by the Math- ematica program NiceKnots by Silvio Levy, with some of them further manipulated by CorelDraw and labels introduced using the geompsfi package from the Geometry Center, Minn. Care was taken to prepare a source document for later translation in HTML, PDF, and other electronic formats, which is now work in progress. Berkeley April 25, 1996 6 CONTENTS Chapter 1 Knot Theory • Problems 1.1–1.51 (1977), 1.52–1.57 (1982), 1.58–1.105 (new). • S1 ,→ S3, 1.1–1.47, 1.52, 1.53, 1.58–1.101. • S2 ,→ S4, 1.48–1.51, 1.54–1.57, 1.103–1.105. • Braids, 1.7, 1.8, 1.84, 1.100. • Knot groups, 1.9–1.14, 1.57, 1.85, 1.86. • Properties P, R, 1.15–1.18, 1.82. • Branched covers, 1.21–1.29, 1.74. • Concordance and slice knots, 1.19, 1.30–1.47, 1.52, 1.53, 1.93–1.97. • Various genera, 1.1, 1.20, 1.40–1.42, 1.83. • Crossing, unknotting and tunnel numbers, 1.63–1.73. • Hyperbolic knots, 1.75–1.77. • Dehn surgery, 1.15–1.18, 1.77–1.82. • Jones polynomial, etc., 1.87–1.92. • Contact, complex structures, 1.98–1.102. 7 8 CHAPTER 1. KNOT THEORY Introduction Definitions and notation: most definitions are given in the problems, but here are some that are widely used. A diagram for a knot K in S3 is, intuitively, how we draw a knot on the blackboard; that is, it is a generic projection onto a plane (but preserving over and undercrossings) of a representative in the isotopy class of submanifolds in S3 which is K. Then the crossing number c(K)ofaknotKin S3 is the minimal number of crossings in a diagram for K, the minimum being taken over all possible diagrams of K. The unknotting number u(K)ofaknotKin S3 is the minimum, taken over all diagrams of K, of the number of crossings which must be changed to obtain a diagram of the unknot. The tunnel number t(K)ofaknotKin S3 is the minimal number of arcs which must be added to the knot, forming a graph with three edges at a vertex, so that the complement in S3 (of an open regular neighborhood of the graph) is a handlebody. The boundary of this handlebody is then a minimal Heegaard splitting of the knot complement. This graph is the simplest graph, formed by adding arcs to the knot, which (allowing edges to slide over edges) can be moved into a plane, yet contains the knot.
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