Controlled Topology and Topological Field Theory

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9705168 Quinn This project has two distinct components, in controlled topology and topological field theory. The controlled topology objective is to understand better the failure of approximate fibrations to form a bundle theory because the formation of pullbacks is obstructed. Hughes and others have made precise the sense in which approximate fibrations are the normal structure in homotopically stratified sets. This makes the pullback problem central in the study of these sets. Much is already known through the work of Hughes, Williams, Taylor, Weinberger and others, but a more satisfactory version may be possible. The other topic concerns machine computation in topological field theories defined on 2-complexes, 3-manifolds, and smooth 4-manifolds. The 2-complex case is completely implemented, and explorations are well along for field theories coming from mod p representations of Lie algebras, for p up to 13. The representation theory of quantum groups needed for 3-manifolds has been implemented, but exploration of the field theories waits on a more complete understanding of the 2-complex case. Both 2-complexes and 3-manifolds are considered test cases for the primary goal of the project, field theories on 4-manifolds. This, however, is still some years off. Some work is also being done on topological questions on which field theories might shed light: the "Andrews-Curtis" conjecture for 2-complexes, handlebody structures on 4-manifolds, and topological isotopy of 4-manifolds. The context for the first topic is the study of stratified sets. Most of the geometric objects occurring in nature are stratified: built up of layers that are manifolds. The biggest challenge is to understand how the strata fit together. For most types of stratified sets (smooth, piecewise-linear, analytic) this fitting-together is described using a bundle theory. It is complex and hard to use because the objects are complicated, but is effective and well-understood. Top ological stratified objects took longer to describe, because it turns out the fitting-together cannot be given in terms of a bundle theory. This has been extensively studied, but we do not yet have a fully satisfactory understanding. The field theories in the second part of the project are also referred to as "topological quantum field theories." They are rather idiosyncratic theories specialized to low dimensions. The ones under study are algebraic (as opposed to analytic) and are constructed using representations of groups or Lie algebras, or their deformations. These, particularly the 3-manifold versions, became popular about a decade ago. At the time, we seemed not to know enough either about deformations of Lie algebras or about 3-manifolds to get significant new information from one about the other. A particular obstacle was the inability to compute "random" invariants. This project, now in its eighth year, has numerical computation as a first goal. This is going well, but the raw output is not very revealing. The principal activity at present is finding ways to use numerical data to obtain a global qualitative understanding of a theory. ***