Simple Type is Not a Boundary Phenomenon

Abstract

This is an expository article, explaining recent work by D. Groisser and myself [GS] on the extent to which the boundary region of moduli space contributes to the ``simple type'' condition of Donaldson theory. The presentation is intended to complement [GS], presenting the essential ideas rather than the analytical details. It is shown that 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 a statement about the interior of moduli space, namely that the interior of the (k+1)st ASD moduli space, intersected with two representatives of (4 times) the point class, be homologous to 58 copies of the (k)-th moduli space. This is peculiar, since the only known embeddings of the (k)-th moduli space into the (k+1)st involve Taubes patching, and the image of such an embedding lies entirely in the boundary region.