Mathematical Sciences: The Structure of Smooth 4-Manifolds

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9626330 Stern The goal of this project is better to understand simply-connected smooth four-dimensional manifolds. In the early 1990's difficult techniques were developed and used by the principal investigator and others to prove sharp statements about the computations and structure of the Donaldson invariants (which were introduced in 1984). The October 1994 introduction of the brilliantly conceived and more easily handled Seiberg-Witten invariants has made clear the role of these Donaldson invariants in the study of smooth 4-manifolds. Now that the dust has begun to settle, it is time to review our understanding of smooth 4-manifolds. Surprisingly, most of the questions and problems related to the topology of smooth 4-manifolds present in September 1994 remain open. In particular, we still do not know how to classify simply-connected smooth 4-manifolds. The first part of this project is to determine the fundamental building blocks and to determine the operations performed on these building blocks to recover any given smooth 4-manifold. As a focal point, given a simply-connected irreducible smooth 4-manifold X, does there exist a finite collection of complex surfaces, each of which carries a pencil of curves from which X is obtained by using the following three operations: (1) fiber sum along a general fiber of the pencils; (2) local fiber sum along tori of square zero; (3) performing a topological log transform on tori of square zero? The second part of this project is to determine the effectiveness of the Seiberg-Witten and Donaldson invariants. In particular, does the deformation type of a complex surface determine its diffeomorphism type? This project will begin to focus on explicit examples (the Horikawa surfaces) that have the same Seiberg-Witten and Donaldson invariants, are known to be deformation inequivalent (i.e., not the same as complex manifolds), but are not known to be diffeomorphic. One disturbing feature of the Seiberg-Witten and Donaldson invariants is that they are defined only for manifolds for which the sum of its signature and Euler characteristic is divisible by an odd multiple of 4. The final part of this project will focus on those simply-connected 4-manifolds for which the sum of the signature and Euler characteristic is divisible by an even multiple of 4, and about which virtually nothing is known. At bottom, this project centers on the classification of objects that are locally modeled on 4-dimensional Euclidean space and upon which one can do differential calculus for real-valued functions. These objects are the so-called smooth four-dimensional manifolds. The basic technique is to extract algebraic topological data from the solution space on these manifolds of partial differential equations that arise in theoretical physics, i.e., to use gauge-theoretic techniques. It is known that one cannot expect simply to give a complete list of smooth 4-manifolds. However, one can expect to determine classifiable fundamental objects and a list of operations that one can perform on these objects in order to obtain any smooth 4-manifold. It is the goal of this project to provide these building blocks and assembly rules. Further, the behavior under these operations of known and future invariants of smooth 4-manifolds should be easily determined. Success of the project will create another strong tie between topology and theoretical physics, partly because, as noted above, the topological tools involved come from gauge theory, i.e., from quantum mechanics, and partly because any light shed on 4-manifolds bears on our understanding of the 4-dimensional space-time of relativity theory. ***