UBC Mathematics Department
Until 1994, the only known invariants of smooth four-manifolds were the Donaldson polynomials. The definition of these invariants relied upon a construction called the Uhlenbeck compactification and it was thought by some that the invariant's sensitivity to the differential structure of the manifold depended upon this compactification. However, work of Kronheimer-Mrowka on the structure of Donaldson polynomials and the new Seiberg-Witten invariant have shown that the Donaldson polynomial is really only sensitive to certain cohomology classes, similar to the canonical class of an algebraic surface. The moduli space of PU(2) monopoles gives a method to prove a relation between the Donaldson and Seiberg-Witten invariant as well as some understanding of the role the Uhlenbeck compactification plays in Donaldson theory.