Mathematical Sciences: Topology of 4-Manifolds

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9703996 Teichner This project deals with topological problems related to 4-dimensional manifolds. The main question to be answered is for which fundamental groups the higher dimensional techniques of surgery and s-cobordism theorems work. The existing techniques include sophisticated constructions of Casson-handles that answer the question in the affirmative for a large class of groups. In addition, there is a proposed nilpotent group theory method for locating obstructions for free fundamental groups. Both methods will also be applied to the link homotopy classification of disjoint 2-spheres in 4-space as well as to questions concerning concordance classes of knots and links in 3-space. One of the most exciting mathematical discoveries in the last decade is the fact that Euclidean n-space behaves very differently depending on whether the dimension n equals or doesn't equal 4 (which is the dimension we live in, allowing time as the fourth dimension). More precisely, we have all learned calculus in one real variable, and some of us went on to learn calculus in n variables, i.e., the theory of smooth functions on Euclidean n-space. But very few of us were told that for n=4 there are (uncountably!) many such theories, whereas for all other dimensions there is a unique one, namely the one we were taught. In mathematical terms, I am talking about 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 (technically, those which have nontrivial fundamental group). One concrete difference from Euclidean 4-space is that in a general 4-manifold, not every vector-field corresponds to a potential energy function. ***