Simple type and the boundary of moduli space

Abstract

We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the “simple type” condition of Donaldson theory. Using the natural geometric representative of μ(pt) defined in [L. Sadun, Commun. Math. Phys. 178 (1996) 107–113], the boundary region of moduli space contributes 6 64 of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to the interior of the ( k+1)th ASD moduli space, intersected with two representatives of (4 times) the point class, being homologous to 58 copies of the kth moduli space. This is peculiar, since the only known embeddings of the kth moduli space into the ( k+1)th involve Taubes gluing, and the images of such embeddings lie entirely in the boundary region. When using the natural de Rham representatives of μ(pt) considered by Witten [Commun. Math. Phys. 117 (1988) 353], the boundary region contributes 1 8 of what is needed for simple type, again regardless of the topology or geometry of the underlying 4-manifold. The difference between this and the geometric representative answer is not contradictory, as the contribution of a fixed region to the Donaldson invariants is geometric, not topological.