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 $W:Y\to Y^{\prime }$ is a cobordism between integer homology spheres with no $2$-torsion in its first homology, then $-b^{+}(W)\le q_3(Y^{\prime })-q_3(Y)\le b^{-}(W)$. We also extend both $q_3$ and the inequality to rational homology spheres.

67 pages, 2 figures, comments welcome