4-manifolds and controlled topology

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

Abstract Award: DMS-0103976 Principal Investigator: Frank Quinn This project involves two distinct areas: 4-dimensional manifolds, and high-dimensional controlled topology. Recent successes with topological 4-manifolds suggest the time is ripe to try to formulate surgery obstructions for arbitrary fundamental groups. Improved understanding of handlebody structures and a new construction of topological field theories in dimension 4 offer hopes for combinatorially-defined invariants of smooth 4-manifolds. In the controlled area recent work with Ranicki has provided a proof of an elusive stability theorem for controlled surgery obstructions. This may point the way to a full description of the groups as generalized homology, analogous to earlier descriptions of the end and controlled h-cobordism obstructions. Such a description would have applications ranging from local structure in stratified sets to sharpening 2-torsion conclusions in some cases of the Novikov conjecture. This project explores the boundary between the discrete and continuous worlds. Geometry and analysis (continuous points of view) have revealed strange behavior in dimension 4. In terms of the mathematics used in physics, other dimensions are "classical" while dimension 4 seems to be "quantum." Topologically 4-dimensional objects are described in terms of 3-dimensional building blocks (a discrete point of view), particularly knots and links in 3-space. One objective is to bridge the gap between these two views, and in particular understand the quantum behavior from the topological point of view. In higher dimensions algebraic and qualitative topology (e.g. surgery theory) are discrete points of view. Twenty years ago the PI bridged the gap between this and local continuous topology in one case, "pseudoisotopy." This has had numerous applications. Many more applications await a widening of this bridge to include other cases, for instance algebraic K-theory and surgery, but the methods are complex and intensely technical and have so far resisted extension. The other objective of the project is to carry through these generalizations.