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Monopole equation and the \(\frac{11}{8}\)-conjecture. (English) Zbl 0984.57011
If \(M\) is a smooth oriented closed spin 4-dimensional manifold, a well-known open conjecture is that \(M\) satisfies \(b_2(M)\geq {{11}\over{8}} |\sigma(M)|\), where \(b_2(M)\) and \(\sigma(M)\) are the second Betti number and signature of \(M\), respectively. (This is known as the 11/8 conjecture.) In this paper, the author proves the weaker inequality \(b_2(M)\geq {{5}\over{4}} |\sigma(M)|+2\). This result, first announced in 1995, is proven using Seiberg-Witten theory. A central idea in the proof is to make use of a finite-dimensional approximation to the usual Seiberg-Witten equations. The author analyzes the \(\text{Pin}_2\) symmetry of these equations, and uses equivariant K-theory to derive the above inequality.

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
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