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Mathematics, Differential Geometry, Geometric Topology
Abstract:
We prove an analogue of the Kotschick-Morgan conjecture in the context of
SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten
invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible 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 SO(3) monopoles (Feehan and Leness, PU(2) monopoles. I. Regularity, Uhlenback compactness, and transversality, 1998). In this monograph, we prove - modulo a gluing theorem which is an extension of our earlier work in PU(2) monopoles. III: Existence of gluing and obstruction maps (arXiv:math/9907107) - that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten
invariants of the four-manifold. Our proofs that the SO(3)-monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze (Superconformal invariance and the geography of four-manifolds, 1999; Four-manifold geography and superconformal symmetry, 1999) and Witten's Conjecture (Monopoles and four-manifolds, 1994) in full generality for all closed, oriented, smooth four-manifolds with and odd appear in Feehan and Leness, Superconformal simple type and Witten's conjecture (arXiv:1408.5085) and monopole cobordism and superconformal simple type (arXiv:1408.5307).
