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SO(3) monopoles and relations between Donaldson and Seiberg-Witten invariants

Banff International Research Station for Mathematical Innovation and Discovery

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. Our presentation is 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.

Mathematics; Manifolds and cell complexes; Global analysis, analysis on manifolds; Low dimensional topology

video/mp4

Author affiliation: Rutgers, The State University of New Jersey

2018-02-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64564

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/