Topology of Four-Manifolds

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Proposal: DMS-0072775 PI: Peter Teichner ABSTRACT This project studies the topology of 4-dimensional manifolds. The most prominent problems are the characterization of good fundamental groups for the 4-dimensional s-cobordism theorem, the formulation of a 4-dimensional (homology) surgery theory, and the understanding of the topological knot concordance group. The existing techniques from nilpotent/solvable group theory and higher order intersection numbers of 2-spheres and Whitney towers in 4-manifolds are to be combined with methods from L2-homology and von Neumann algebras. One of the most exciting mathematical discoveries in the last decades is the fact that Euclidean space behaves very differently depending on whether the dimension equals or doesn't equal four (which is the dimension we live in, allowing time as the fourth dimension). More precisely, there is a unique way to do calculus in Euclidean n-space except for n=4 where there are uncountably many such theories. These are the "exotic" structures on Euclidean 4-space, discovered by Donaldson, Freedman and others around 1983. This project deals with similar questions on more complicated 4-dimensional manifolds. For example, these may have nontrivial fundamental group, so they are possibly more similar to our universe than Euclidean 4-space. In this case the fundamental questions are still wide open and there seems to be a deep relation to the "continuous dimension" introduced by von Neumann in his work on quantum mechanics.