Topology of Four-Manifolds

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Proposal: DMS-0072775PI: Peter TeichnerABSTRACTThis project studies the topology of 4-dimensional manifolds. Themost prominent problems are the characterization of goodfundamental groups for the 4-dimensional s-cobordism theorem, theformulation of a 4-dimensional (homology) surgery theory, and theunderstanding of the topological knot concordance group. Theexisting techniques from nilpotent/solvable group theory and higherorder intersection numbers of 2-spheres and Whitney towers in4-manifolds are to be combined with methods from L2-homology andvon Neumann algebras.One of the most exciting mathematical discoveries in the lastdecades is the fact that Euclidean space behaves very differentlydepending on whether the dimension equals or doesn't equal four(which is the dimension we live in, allowing time as the fourthdimension). More precisely, there is a unique way to do calculus inEuclidean 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 mayhave nontrivial fundamental group, so they are possibly moresimilar to our universe than Euclidean 4-space. In this case thefundamental questions are still wide open and there seems to be adeep relation to the "continuous dimension" introduced by vonNeumann in his work on quantum mechanics.