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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.

##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R57 Applications of global analysis to structures on manifolds
##### Keywords:
Seiberg-Witten theory; 4-manifold
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