Topology Atlas Document # topd-66
From TopCom, Volume 11, #1
Also available as original PDF document (6 pages).
The University of Texas at Austin, March 21-23, 2002
Partially supported by NSF Grant DMS-0129227
Most of the geometric topology represented at the conference was 3-dimensional topology, so this report will concentrate on that particular area.
Thurston's Geometrization Conjecture, formulated approximately 25 years ago, provides a coherent and simple (at least compared with the known situation in higher dimensions) picture of 3-manifolds, and has given direction to much research in the subject, especially the use of methods from hyperbolic geometry. A more recent trend in the subject has been an increased interest in algorithmic and computational questions about 3-manifolds. The main reason for this is the fact that 3-dimensional topology appears to be immune from the logical problems that put limits on our understanding of various questions about groups and about manifolds of higher dimensions: the unsolvability of the isomorphism and word problems for finitely presented groups, of the homeomorphism problem for manifolds of dimension greater than 3, etc.. Indeed it would follow from the Geometrization Conjecture that most of the corresponding problems in dimension 3 are solvable. The first result of this kind was Haken's proof in 1962 that the knot problem is solvable. This was followed by work of Waldhausen, Jaco-Shalen, Johannson, and Hemion, culminating in the proof of the solvability of the homeomorphism problem for Haken manifolds (although Matveev has recently pointed out, and closed, a gap in that program); the solvability of the word problem for fundamental groups of Haken 3-manifolds was established earlier by Waldhausen. A secondary reason for the current interest in questions about computability is simply the increased availability and use of computers. Rubinstein's solution of the recognition problem for the 3-sphere around 1992, with the consequent revival of the theory of normal surfaces, has been the impetus for much current work on algorithmic questions in dimension 3.
At the conference, the talks of Agol, Li, Manning and Schleimer all dealt with such questions: Agol showed that the Thurston norm is NP-computable; Li showed that there is an algorithm to recognize Seifert fiber spaces; Manning showed (based on an outline by Casson) that (assuming a solution to the word problem for the fundamental group) one can decide whether or not a 3-manifold is hyperbolic; Schleimer showed that there is a polynomial time algorithm to decide whether or not a given submanifold of a 3-manifold is a canonical compression body - this implies that recognizing S3 is in NP.
The talks of Hodgson and Przeworski were on questions about hyperbolic 3-manifolds. Hodgson described joint work with Kerckhoff establishing a universal bound on the number of non-hyperbolic Dehn fillings along a cusp of a hyperbolic 3-manifold, while Przeworski discussed lower bounds on the volume of a closed hyperbolic 3-manifold coming from tube packing estimates, which in turn derive from the study of Euclidean double lattice packings.
Ever since Whitehead's famous example in 1935, it has been known that there are many (indeed uncountably many) exotic non-compact 3-manifolds. However, there are situations in which the pathology which can exist in general is conjectured to be absent; perhaps the simplest example of this is the conjecture that the universal covering of an aspherical 3-manifold with infinite fundamental group is R3 . Another is the conjecture that the ends of a complete hyperbolic 3-manifold with finitely generated fundamental group are topologically tame (proved by Bonahon in the case where the fundamental group is not a free product). At the conference, Gabai talked about a criterion for a 3-manifold to have tame ends; this was related to work of Poenaru.
It has been known for some time that, for (irreducible) 3-manifolds with infinite fundamental group, the property of being Haken, i.e. containing an incompressible surface, is, one might say, quite rare. Since most of the main problems about 3-manifolds are known to be true for Haken manifolds, this is perhaps disappointing. However, it is also known that for many problems it is sometimes enough to solve the problem virtually, i.e. in some finite covering, and thus a good deal of attention has been directed towards the virtual Haken conjecture (due to Waldhausen), which asserts that every irreducible 3-manifold with infinite fundamental group has a finite-sheeted covering which is Haken. Boyer's talk at the conference dealt with closely related algebraic questions about the virtual orderability of 3-manifold groups.
A different sort of algebraic method in low-dimensional topology was discussed in the talks of Harvey and Cochran, based on looking at the homology modules (over the group ring of the covering group) of certain covering spaces. Using infinite cyclic coverings, Harvey showed that there are closed 3-manifolds with arbitrarily high first Betti number whose fundamental group do es not map onto a free group of rank greater than 1. (This question had arisen in some work on quantum invariants of 3-manifolds.) Cochran discussed the use of solvable coverings, which along with methods from von Neumann algebras, have led, in joint work of Cochran, Orr and Teichner, to dramatic new advances in the understanding of knot concordance.
A fruitful concept in knot theory, which has had several important ramifications for 3-dimensional topology, is that of thin position, intro duced by Gabai in 1987 for the purpose of proving Property R. It played a key role in Gordon and Luecke's proof that knots are determined by their complements, and in Thompson's version of the solution of the recognition problem for S3 and related developments in the theory of Heegaard splittings of 3-manifolds. One defines the width w(K) of a knot5C K in S3 to be an appropriate count of the number of intersections of K with certain horizontal planes when K is in thin position; it is a kind of refined version of the bridge number br(K). A classical theorem of Schubert asserts that br(K) - 1 is additive with respect to connected sum of knots, and a natural question is whether w(K) is also additive. This is unsolved, but Schultens in her talk showed that w(K1 # K2) is at least (w(K1)+w(K2))/2.
An active area of research in 3-dimensional topology which was not covered at the conference is the theory of laminations. The condition on a 3-manifold of having an essential lamination is weaker than being Haken, and so a good deal of effort has gone into trying, on the one hand, to prove that many non-Haken 3-manifolds contain such laminations, and on the other, to extend to the case of laminar manifolds the various theorems that are known to hold for Haken manifolds. There has been progress on both of these programs. However, recent results have put limitations on the applicability of the theory: at one time it was considered possible that all hyperbolic 3-manifolds are laminar (although many are non-Haken), but this is now known not to be the case.
Set-theoretic topology may be described as a happy melding of general topology and the modern set-theoretic methods that have been proven to be essential in the solution of many long-standing topological questions. The field has built close ties to related disciplines such as real analysis, topological groups, and functional analysis, as well as to many other areas of mathematics in which convergence plays a crucial role.
Some of the topological questions in which set-theory turns out to be important do not appear at first glance to be distinctively set-theoretic in nature. A very interesting case in point is the problem of the existence of metrizable spaces in which the standard small and large inductive dimensions disagree by more than 1. Forty years ago, Prabir Roy constructed a metrizable space where these dimensions differ by exactly one, but it was not known until approximately four years ago, by an example constructed by S.. Mrowka, that the difference could be two. At this conference, J. Kulesza spoke about his interesting extension of Mrowka's examples to obtain differences of any postive integer whatso ever. However, both Mrowka's and Kulesza's examples, unlike Roy's difference-one example, are not ZFC examples, and none are known. Indeed, their examples are destroyed by the continuum hypothesis! They are valid under a different, very strong, axiom which was shown to be consistent relative to the existence of certain large cardinals by Dougherty in 1997. Future work will focus on determining if the topological problem has a standard ZFC solution or not.
Another excellent example of set-theory and topological interaction is the solution, first announced at this conference by P. Larson, of a fascinating question asked by S. Watson about 20 years ago. In the early 1980's, a string of results by Rudin, Gruenhage, Junnila, and others suggested that it may be the case that, consistently, all perfectly normal locally compact spaces are paracompact; Watson was the one who explicitly asked if this were so. The key to the solution of this problem turned out to be the study of the new so-called P-max models of set-theory developed by H. Woodin at UC Berkeley. P. Larson, a recent student of Woodin, came to Toronto a couple of years ago to work with topologists and set-theorists there. Subsequently, he and Todorcevic showed that there is a P-max type model of set-theory which solved a 52 year old problem of M. Katetov, whether every compact Hausdorff space with completely normal square must be metrizable. At that time, it was thought that a solution of Watson's problem might also be near, but the methods needed some significant further development. Now it has happened: Larson, Todorcevic and Tall developed obtained a P-max type model in which every perfectly normal locally compact space is paracompact. There remain several other very basic questions about the structure of perfectly normal compacta which might be settled by refinements of this obviously very powerful technique.
Functional analysis provides another fertile ground for interaction with set-theoretic topology. The talks by R. Pol and D. Burke at this conference on their joint work regarding the Borel structure of function spaces C(K), where K is compact, is an excellent case in point. Their quite general results illustrate their power in a corollary which answers an old question of A.H. Stone: a separately continuous function whose domain is the product of two compact spaces must be Borel measurable. In another vein, G. Gruenhage gave a talk in which he described how a simple topological game gives a useful characterization of Eberlein compacts, the weakly compact subsets of Banach spaces, as well as a characterization of when a function space C(X) with the compact-open topology, where X is locally compact, has the Baire property.
The Spring Topology and Dynamics Conference continues to bring together researchers in continuum theory and in dynamical systems with a broad range of interests and expertise. The international flavor is strong, with a large contingent of topologists from Mexico in attendance. Below are a few highlights of the conference and their relation to ongoing threads of research in continuum theory and in dynamical systems.
Areas of progress in continuum theory well-represented at the conference include work on classification of continua by their hyperspaces (notably by the Mexican topologists), and remarkable theorems and examples in the theory of homogeneous continua. An important result presented was an example by Janusz Prajs of a homogeneous, path-connected, non-locally-connected continuum. This answers in the negative the long-standing question of whether every homogeneous path-connected continuum must be locally connected. There are still many open questions in the classification of homogeneous continua.
There is continuing interest in the study of families of inverse limits on the interval, particularly in terms of classification and properties of the inverse limit continua X = (I,f) obtained which are implied by the dynamics of the bonding map f on the interval. Of special note is the recent proof by Christopher Mouron that if X is a chainable continuum (i.e., an inverse limit of intervals) and X admits a homeomorphism of positive topological entropy, then X must contain an indecomposable subcontinuum. Mouron's proof last year that no tree-like continuum admits an expansive homeomorphism is another significant recent contribution to the overlap of continuum theory and dynamical systems.
Topological models for "natural" dynamical systems are usually proposed in an attempt to understand the dynamics of a complicated dynamical system "by analogy" to a simplified case where one can obtain rigorous results. For example the "topological Lorenz attractor," was introduced to analogize the conjectured "strange attractors," which seemed to exist for some parameter values of Lorenz's simple system of differential equations, since it was so hard to prove anything about Lorenz's "natural" system. It was shown that for an open set of parameter values in the topological system, the topological attractor was an indecomposable continuum. Recently, Warwick Tucker went beyond analogy and obtained the result that the original Lorenz differential equations support a strange attractor which persists under small perturbations of the parameters in the underlying system of differential equations. This is one of several recent examples where rigorous computer-based computational techniques have been used to obtain topological theorems.
Combinatorial group theory is the study of finite presentations (or more generally finite 2-complexes) using the techniques of combinatorics and combinatorial topology. Geometric group theory is of much more recent vintage, and its main object of study is the metric spaces that discrete groups can act on in a reasonably nice way. The object is to use the metric (that is, the geometry) of these spaces to derive results about the group structure. There is a significant overlap between the two approaches since the classes of finitely presented groups amenable to a combinatorial approach typically are negatively (or at least non-positively) curved when viewed metrically.
Throughout the 1990s two ma jor theories of curvature were developed and elaborated: large-scale negative curvature and local non-positive curvature. The groups associated to these two types of spaces are word hyperbolic groups (as known as Gromov hyperbolic groups) and CAT(0) groups. The exact relationship between these two classes of groups is still unclear, although it is conjectured that every word hyperbolic group is the fundamental group of a compact non-positively curved space (i.e. it is also a CAT(0) group). One difficulty in attacking this conjecture is the lack of examples which are known to be word hyperbolic without being known to be CAT(0), small cancellation groups and some one-relator groups being significant exceptions. [A similar difficulty arise in the study of Thurston's geometrization conjecture.] At the conference Noel Brady described recent work with Jon McCammond in which they show that metric small cancellation groups are fundamental groups of high dimensional non-positively curved cube complexes. D. Wise has independently shown a similar result. Despite this progress, the general conjecture remains wide open.
Three fundamental problems about finite presentations were laid out by Max Dehn in 1910: the word problem, conjugacy problem, and isomorphism problem. There is a fundamental tension in the field between the fact that nice algorithms exist for large classes of presentations and the fact that, based on results coming from logic and computer science, there cannot exist a general algorithm for any of Dehn's three fundamental questions which works for an arbitrary finite presentation. The main problem in combinatorial group theory is to decide which classes of presentations fall on which side of this divide. Both types of curvature initally seemed to avoid the types of undecidability results associated with Dehn's problems. At the conference, however, Martin Bridson described some of the "horrors" which can "lurk amongst the subgroups of seemingly benign groups". In particular, he showed that the isomorphism problem can be undecidable for subgroups of a fixed CAT(0) group. That this was true for the direct product of two free groups was already known, but Bridson's techniques give a broad class of examples which exhibit this phenomenon. This behavior is in contrast with the solution of the isomorphism problem for torsion-free word hyperbolic group by Zlil Sela using actions on R-trees. On the positive side, Mark Feighn described his recent work with Guo-An Diao showing that many questions about graphs of free groups (the quintessential word hyperbolic groups) were not only decidable, but also possess rather reasonable algorithmic solutions.
Despite certain undecidable aspects, the class of non-positively curved groups is still sufficiently well-behaved that it is worth while to show that various types of groups are non-positively curved. At the conference Jon McCammond discussed results with Murray Elder and John Meier showing that 3-manifold satisfying certain combinatorial restrictions have fundamental groups which are word-hyperbolic and CAT(0). The proof proceeds by giving the underlying 3-manifold a metric of non-positive curvature. Similarly, Noel Brady (with John Crisp) described an example of a group which does not act on a 2-dimensional non-positively curved space but which does act on a 3-dimensional one. Finally, Tadeusz Januzkiewicz discussed hyperbolic Coxeter groups of large dimensions. He showed that if a Gromov hyperbolic Coxeter group W is an n-dimensional virtual Poincaré duality group, then n is at most 61.
The study of an arbitrary word hyperbolic group has always been difficult because of the lack of a detailed (general) structure theory. This lack is currently being remedied by Zlil Sela in a series of papers in which he studies the set of solutions of equations over word hyperbolic groups. In addition to solving the Tarskii conjecture (about whether non-abelian free groups have the same elementary theory) Sela characterizes which word hyperbolic groups are elementary equivalent a free group. At the conference he described how every word hyperbolic group has an elementary core and two word hyperbolic groups are elementary equivalent if and only if their elementary cores are isomorphic. This is very deep result which will have enormous consequences for the field.
Finally, strong connections have recently been developed between geometric group theory and functional analysis. Functional analysts have recently used techniques in geometric group theory to show results about maps between Banach spaces (Johnson, Lindenstrauss, Preissman, and Schechtmann) and have used results about word hyperbolic groups to detail the structure of the corresponding C*-algebras (Dykema, de la Harpe). In his talk Igor Mineyev described his work with Guoliang Yu in which averaging constructions can be used to prove the Baum-Connes conjecture for all word hyperbolic groups.
Copyright © 2006 by Topology Atlas. All rights reserved. Published February 3, 2006.