4-manifolds and controlled topology

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AbstractAward: DMS-0103976Principal Investigator: Frank QuinnThis project involves two distinct areas: 4-dimensionalmanifolds, and high-dimensional controlled topology. Recentsuccesses with topological 4-manifolds suggest the time is ripeto try to formulate surgery obstructions for arbitraryfundamental groups. Improved understanding of handlebodystructures and a new construction of topological field theoriesin dimension 4 offer hopes for combinatorially-defined invariantsof smooth 4-manifolds. In the controlled area recent work withRanicki has provided a proof of an elusive stability theorem forcontrolled surgery obstructions. This may point the way to a fulldescription of the groups as generalized homology, analogous toearlier descriptions of the end and controlled h-cobordismobstructions. Such a description would have applications rangingfrom local structure in stratified sets to sharpening 2-torsionconclusions in some cases of the Novikov conjecture.This project explores the boundary between the discrete andcontinuous worlds. Geometry and analysis (continuous points ofview) have revealed strange behavior in dimension 4. In terms ofthe mathematics used in physics, other dimensions are "classical"while dimension 4 seems to be "quantum." Topologically4-dimensional objects are described in terms of 3-dimensionalbuilding blocks (a discrete point of view), particularly knotsand links in 3-space. One objective is to bridge the gap betweenthese two views, and in particular understand the quantumbehavior from the topological point of view. In higher dimensionsalgebraic and qualitative topology (e.g. surgery theory) arediscrete points of view. Twenty years ago the PI bridged the gapbetween this and local continuous topology in one case,"pseudoisotopy." This has had numerous applications. Many moreapplications await a widening of this bridge to include othercases, for instance algebraic K-theory and surgery, but themethods are complex and intensely technical and have so farresisted extension. The other objective of the project is tocarry through these generalizations.