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10/8-inequality is an inequality relating the signature and the second

Betti number of closed spin 4-manifolds.

This inequality was proved by M.Furuta by using a method of finite

dimensional approximation of Seiberg-Witten monopole equation as an

approach toward the 11/8-Conjecture.

The 11/8-Conjecture can be traced to the estimates of a homology

cobordism invariant "Bounding genus" for integral homology 3-spheres

introduced by Y.Matsumoto in his empirical study on the kernel of the

Rochlin invariants.

In this talk, I would like to explain briefly a proof of

10/8-inequality and give several applications. In particular, I would

like to introduce the bounding genus for rational homology spheres to

give their lower bounds in terms of Neumann-Siebenmann invariant.

10/8-inequality and plumbed rational homology 3-spheres