Dehn surgery along torus knots
By using the Seiberg-Witten gauge theory, \textit{M. Furuta} [Monopole equation and the 11/8 conjecture, Preprint] proved the 10/8-conjecture, i.e. if \(Q\) is the intersection form of a closed smooth spin 4-manifold, then \(8\cdot \text{rank} (Q)>10\cdot |\text{sign}(Q) |\) (see for example [\textit{S. Akbulut}, Turk. J. Math. 20, No. 1, 95-118 (1996; Zbl 0871.57015)]). Let \(\Sigma= \Sigma(p,q, pqm\pm 1)\) be the Seifert fibered homology sphere obtained by \((-1/m)\)-Dehn surgery along a \((p,q)\)-torus knot, where \(p\), \(q\geq 2\). The author constructs for each odd \(m>0\) a smooth simply-connected 4-manifold with boundary \(\Sigma\) and even intersection form \(Q\) for which the inequality above fails. This will imply that neither \(\Sigma\) nor any of its multiples can bound a smooth acyclic 4-manifold. As a consequence, any Rokhlin invariant one homology 3-sphere obtained by Dehn surgery on a torus knot has infinite order in the homology cobordism group of oriented homology 3-spheres.
scientific article; zbMATH DE number 1308037