9401032 Fintushel The main concern of this project is to study smooth simply connected 4-manifolds by using invariants obtained from gauge theory. This has three parts. The first is to understand the structure theory of the invariants themselves. For manifolds of simple type, this has been accomplished by Kronheimer and Mrowka and, using completely different techniques, by the investigator and R. Stern. The investigator and R. Stern have also proved a blowup formula for the Donaldson invariant of a general 4-manifold, and one expects that this formula, together with the techniques used in the simple-type case, will lead to a general structure theory. The second part of this project is the calculation of Donaldson invariants. The investigator intends to use his technique of 'rational blowdowns' to provide calculations. This involves giving a general formula for the Donaldson invariant of a rational blowdown, which has essentially been done by the investigator and R. Stern (some details remain to be checked) and which leads, e.g., to the calculation of complete Donaldson invariants for all elliptic surfaces and many other interesting examples. The final part of the project is to produce new examples to which these invariants can be applied in order to find some order in the classification theory. Four-dimensional manifolds are spaces which locally have the structure of 4-dimensional Euclidean space (e.g. 3 'spatial' and one 'time' dimension). These manifolds play a central role in topology, for in 4 dimensions both the powerful techniques of higher dimensional surgery and the more special and intricate techniques of lower dimensions break down. This is, of course, also the most physically relevant situation, and there is a vast literature in theoretical physics related to 4-dimensional topology. For many years 4-manifolds were studied by applying techniques from both higher and lower dimensions, and the results were mostly unsatisfactory. In 1 984, Simon Donaldson introduced a powerful technique stemming from the interplay between 4-dimensional geometry and theoretical physics via the Yang-Mills equation. Donaldson's techniques have had startling consequences, and a school of topologists and geometers has formed in the hope of using these techniques to understand 4-dimensional topology completely. (That this is not too far removed from the physics which helped spawn the theory is evidenced by the many contributions by physicists such as Witten.) In the last year, especially, progress has been very rapid. The investigator hopes to make further contributions both to the understanding of the Donaldson invariant and to the classification of 4-dimensional manifolds. ***
