An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants

The authors prove an analogue of the Kotschick-Morgan Conjecture in the context of \mathrm{SO(3)} monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the \mathrm{SO(3)}-monopole cobordism. The main technical difficulty in the \mathrm{SO(3)}-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible \mathrm{SO(3)} monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of \mathrm{SO(3)} monopoles. In this monograph, the authors prove--modulo a gluing theorem which is an extension of their earlier work--that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. Their proofs that the \mathrm{SO(3)}-monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze and Witten's Conjecture in full generality for all closed, oriented, smooth four-manifolds with b_1=0 and odd b^+\ge 3 appear in earlier works.