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SO(3)-monopole cobordism formula (Part 2)
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Abstract:
In this series of lectures we shall describe the SO(3)-monopole cobordism approach to proving two results concerning gauge-theoretic invariants of closed, four-dimensional, smooth manifolds. First, we shall explain how the SO(3)-monopole cobordism are used to prove that all four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data. Second, we shall explain how the SO(3)-monopole cobordism and the superconformal simple type property are used to prove Witten's Conjecture (1994) relating the Donaldson and Seiberg-Witten invariants. In the first lecture, we shall give an introduction to SO(3) monopoles and overview of how the SO(3)-monopole cobordism may be used to prove the Mari{\~n}o-Moore-Peradze and Witten conjectures. In the second lecture, we shall discuss the SO(3)-monopole cobordism, its compactification, Kuranishi-style gluing models, and explain how the cobordism may be used to prove the SO(3)-monopole link-pairing formula, which gives a very general (though non-explicit) relationship between Donaldson and Seiberg-Witten invariants. In the third lecture, we shall explain how a combination of blow-up formulae for Donaldson and seiberg-Witten invariants, key examples of four-dimensional manifolds, and the SO(3)-monopole link-pairing formula can be used to prove the Marino-Moore-Peradze and Witten conjectures. Our lectures are primarily based on our articles arXiv:1408.5307 and arXiv:1408.5085 and book arXiv:math/0203047 (to appear in Memoirs of the American Mathematical Society), all joint with Thomas Leness.
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