A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds

Abstract

We extend the definition of Khovanov–Lee homology to links in connected sums of $S^1\times S^2$’s and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1\times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following 4-manifolds: $B^2\times S^2$, $S^1\times B^3$, ${\mathbb{CP}}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen’s invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are nonstandard.