The goal of this project is to prove some conjectured relations among gauge theoretic invariants of smooth four-manifolds and to investigate whether it is possible to define a new family of invariants for four-manifolds with b^+ even. The first step is to prove some analytic results on the Taubesian gluing maps which model the ends of the moduli space of SO(3) monopoles. These results will complete the proof of the SO(3) monopole cobordism formula, giving a relation between the Seiberg-Witten and the Donaldson invariants and a relation between the Seiberg-Witten and the spin invariants. These relations resemble Witten's conjecture relating the Seiberg-Witten and the Donaldson invariants except that they contain some unknown coefficients depending only on the topological type of the underlying four-manifold. The second step is an extension of Taubes' work on the gluing maps for the moduli space of anti-self-dual connections to prove that the bubbletree compactification of the moduli space of anti-self-dual connections is a manifold with boundary. This result will provide a framework to prove a relation between the Donaldson and spin invariants. It is hoped that combining the three relations will yield new topological constraints on these gauge theoretic invariants. Finally, the analytic work on the SO(3) monopole invariants should answer the question of whether it is possible to define gauge-theoretic invariants for four-manifolds with b^+ even by using the moduli space of SO(3) monopoles.
Manifolds are shapes which locally resemble the Euclidean space of our everyday experience. The solution set of n equations in (n+d) variables will usually be a d-dimensional manifold: thus manifolds appear throughout mathematics and its applications, from the knot theory used to describe DNA to the manifolds appearing in string theory. Two manifolds are diffeomorphic if one can be stretched, without wrinkling, into the other. Deciding when four-dimensional manifolds are diffeomorphic has proven to be a particularly difficult problem. An invariant is a rule assigning an algebraic object such as a number or polynomial to a topological space in such a manner that the algebraic object does not change when the space is stretched. Hence if I is an invariant and X and Y are topological spaces with I(X) not equal to I(Y) then X cannot be stretched into Y so X and Y are not diffeomorphic. The goal of this project is to look for relations between different invariants and to find new invariants.
