A Vanishing Theorem for Donaldson Invariants

In this thesis, we investigate the conjecture that rational homology 3-spheres that admit taut foliations must also admit irreducible SU(2)-representations. Rational homology 3-spheres with taut foliations embed into 4-manifolds with non-vanishing Donaldson invariants as separating hypersurface. The main result of this thesis shows that if a 4-manifold is split by a rational homology 3-sphere whose perturbed SU(2)-representation variety (the critical set of the perturbed Chern-Simons functional) has only reducible points, then that 4-manifold's Donaldson invariants must vanish. This establishes the conjecture under an additional assumption about holonomy perturbations. The proof is based on a compactness argument, a dimension count and the invariance of Donaldson's invariants under certain perturbations. The assumption on reducibility of critical points is reflected in a positive defect in a dimension sum formula, which forces the dimension of certain moduli spaces to be negative.