Instantons, indefinite 4-manifolds, and Dehn surgery

Abstract: We prove that there exist hyperbolic integer homology spheres with arbitrarily large Dehn surgery number. Previously, no integer homology sphere was known to have a surgery number larger than $2$. Our approach uses Froyshov's invariant $q_3$ of integer homology spheres, which is defined in terms of mod 2 instanton homology. We show that if $W: Y \to Y&#39;$ is a cobordism between integer homology spheres with no $2$-torsion in its first homology, then $-b^+(W) \le q_3(Y&#39;) - q_3(Y) \le b<sup>-(W)$. We also extend both $q_3$ and the inequality to rational homology spheres.