4-Manifold topology and related topics

University Of Texas At Austin, Austin TX

Investigators

Abstract

A major thrust of the project concerns Cappell-Shaneson homotopy 4-spheres. These have long been considered the most likely counterexamples to the smooth 4-dimensional Poincare Conjecture. Recent work by Akbulut, and a simpler, more general approach by the PI, have shown that infinitely many of these are standard. The PI plans to extend his method further, trying to show that all CS-spheres are standard. He will also study other homotopy 4-spheres containing fibered 2-knots, accumulating evidence suggesting that the conjecture is true. The project includes various other avenues of research related to the PI's expertise. He will search for compact domains of holomorphy in complex surfaces, using one of his recent theorems. It now suffices to locate smoothly embedded compact 4-manifolds admitting suitable handle decompositions. He hopes to find (or rule out) examples such as pseudoconvex embeddings of preassigned homology spheres, and a compact domain of holomorphy in C^2 homotopy equivalent to the 2-sphere (violating a conjecture of Forstneric). A related theorem of the PI allows one to construct topological embeddings of 3-manifolds with a generalized pseudoconvexity property; he will investigate these in more detail. Another project is to study what manifolds of dimension 4 and higher can be realized as orbit spaces of vector fields in Euclidean space. Preliminary research shows that such manifolds must be simply connected, but many 4-manifolds with nontrivial 2-homology can arise. Various other investigations, concerning compact 4-manifolds, symplectic structures and Lefschetz pencils will also be pursued. The Poincare Conjecture for 3-manifolds has now been proved by Perelman, a full century after it was first proposed. Its generalization to higher dimensions was proved in the 1960's except in dimension 4. That last version was proved around 1981 by Freedman. However, the original conjecture was made for topological manifolds, so that one is allowed to crinkle the manifold in complicated ways. When manifolds are used in practice, in geometry, analysis, physics, economics and so on, one normally wants to be able to apply calculus, so one must disallow crinkling and work entirely with smooth manifolds. The smooth analog of the Poincare Conjecture has been understood in dimensions >4 since the 1960's, and is equivalent to the topological version (hence solved) in dimensions <4. However, the smooth 4-dimensional Poincare Conjecture is still mysterious, and is the last fundamental open question remaining from the initial heyday of manifold topology a half-century ago. There have been many potential counterexamples constructed, homotopy 4-spheres that might not be the standard 4-sphere, but none has been shown to actually be nonstandard. It has also been quite difficult to show that any of these examples are standard, but the PI has been in the forefront of research in this direction. His new methods have dispensed with a large family of potential counterexamples that were constructed in the 1970's. He intends to further investigate this problem, adding evidence that the conjecture may be true after all, in spite of the prevailing belief to the contrary in recent decades. He will also study other problems involving 4-manifolds and other classical mathematical objects.