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Cobordism theory. --- Four-manifolds (Topology) --- Seiberg-Witten invariants. --- Cobordismes, Théorie des. --- Seiberg-Witten, Invariants de. --- Théorie des cobordismes --- Variétés topologiques à 4 dimensions --- Invariants de Seiberg-Witten --- Cobordism theory --- Seiberg-Witten invariants --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- Invariants --- Differential topology
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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.
Cobordism theory. --- Four-manifolds (Topology) --- Seiberg-Witten invariants.
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"We prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions generalize that of the pure Yang-Mills energy function due to the first author (Feehan, 2014) for base manifolds of arbitrary dimension and due to R"ade (1992, Proposition 7.2) for dimensions two and three"--
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