An SO(3)-monopole cobordism formula relating donaldson and seiberg-witten invariants
Article
Feehan, PMN, Leness, TG. (2018). An SO(3)-monopole cobordism formula relating donaldson and seiberg-witten invariants
. MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, 256(1226), 1-254. 10.1090/MEMO/1226
Feehan, PMN, Leness, TG. (2018). An SO(3)-monopole cobordism formula relating donaldson and seiberg-witten invariants
. MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, 256(1226), 1-254. 10.1090/MEMO/1226
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, Uhlenbeck 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 b1 = 0 and odd b+ ≥ 3 appear in Feehan and Leness, Superconformal simple type and Witten's conjecture (arXiv:1408.5085) and SO(3) monopole cobordism and superconformal simple type (arXiv:1408.5307).
