Gluing maps, moduli spaces of connections and Donaldson invariants

In Chapter 1, we follow P. Feehan’s iterated conformal blow-ups method, to check that neighbourhoods of boundary points of the compactified moduli space Aik of anti-selfdual connections of charge k which lie on the diagonal of a symmetric sum of copies of the underlying 4-manifold X, are constructible by a gluing process. We then observe that a natural stratification of the associated space of gluing data with respect to the number of points with scale zero, leads to the definition of a space J Aik which is such that every weakly convergent sequence of Aik converges into JAik with respect to its natural identification topology. In Chapter 3, we consider the moduli space Bk of all connections of charge k and focus on its C-sequences, namely, sequences of gauge equivalence classes of connections with bounded Yang-Mills energy and functional gradient tending to zero. We employ Taubes’ results concerning the limiting behaviour of C-sequences and also certain properties of a general gluing construction, in order to construct a ‘limit space’ for the C-sequences of Bk• In Chapter 4, we outline the construction of the //-map in gauge theory and use the construction of determinant line bundles over Aik associated to certain families of Dirac operators over X, to show that the map // actually extends over the compactified space. Moreover, we see that the restriction of this extended map to the links of certain low-dimensional strata yields the corresponding //-map and a symmetric product of the Poincare-dual of a reference homology class. In Chapter 5, we study the restriction of certain products of //-type cohomology classes to lower strata of the ideal moduli space TAik- The formulae emerged from the computation of the associated Kronecker pairings consist of Donaldson polynomials of certain charge and symmetric functions which are defined in terms of the intersection form of the 4-manifold X