The Topology of Manifolds of Dimensions 3 and 4

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AbstractAward: DMS 0229035Principal Investigator: Cameron GordonThe project is a conference in 3- and 4-dimensional topology.Satisfactory accounts of manifolds of dimension greater than or equalto 5 were achieved by the late 1960's. Since then, attention hasnaturally focused on dimensions 3 and 4. Despite dramatic advances(Thurston, Freedman, Donaldson, Jones, Witten...), many major problemsin these low dimensions remain unsolved, for example the classical3-dimensional Poincare Conjecture and its smooth 4-dimensional analog.The topological classification of 1-connected 4-manifolds was achievedby Freedman in the early 1980's, but in the smooth case there is noteven a conjectural picture. By contrast, Thurston's GeometrizationConjecture provides a beautiful and coherent description of all3-manifolds, but is far from established. In the mid 1980's, Jones'discovery of his polynomial link invariant, together with work ofWitten, led to the introduction into 3-dimensional topology of methodsfrom quantum physics. The current situation in low-dimensionaltopology is that there are many different directions and methods, butlittle understanding of the relations between them. Thus in dimension3 we have hyperbolic geometry, foliations and laminations, normalsurfaces, combinatorial geometric methods, quantum invariants, finitetype invariants, Floer homology,..., while in dimension 4 there arethe Donaldson and Seiberg-Witten theories, symplectic structures,....The aim of the conference is to address these topics, and theconnections between them, and to provide a forum for interaction andexchange of ideas between experts in the different areas.Geometric topology aims to understand n-dimensional manifolds, whichare objects that locally look like ordinary n-dimensional Euclideanspace (whose points are described by n co-ordinates), but whose globalstructure might be quite complicated. The cases of dimensions 3 and 4are particularly interesting, being the dimensions of our spatial andspatial-temporal universes, and it is a striking fact that it isprecisely these dimensions that are mathematically anomalous. Therehave been many developments and considerable progress in the fields of3- and 4-dimensional topology over the last twenty-five years, but, inboth dimensions, a complete picture is still lacking. One of the aimsof the conference is to encourage interaction and collaborationbetween experts in the several different aspects and techniques thatare currently being pursued in low-dimensional topology, and inparticular between those people working in dimension 3 and those indimension 4, where the methods in the two areas tend to be quitedifferent, but where there are several hints of connections betweenthem. The current disparate state of the subject makes it difficultfor people beginning research in low-dimensional topology to get agood overview of the area, and so by encouraging the participation ofgraduate students and postdoctoral researchers, we intend that theconference should also provide an opportunity for young researchers toget a broad perspective of the present state of knowledge, directionsbegin pursued, and the main open problems. The invited speakersinclude some of the world's leading 3- and 4-dimensional topologists,whose expertise together covers all the major aspects of the twofields.