We extend the definition of Khovanov-Lee homology to links in connected sums of S 1 × S 2 's, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in S 1 ×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 four-manifolds: B 2 × S 2 , S 1 × B 3 , 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 non-standard.
Contents1. Introduction 2. Lee's deformation in # r (S 1 × S 2 ) 2.1. Notation and conventions 2.2. The deformed complex and Lee homology 2.3. Behavior under diffeomorphisms 2.4. Lee generators in KC Lee (T ∞ n ) and KC Lee (D) 3. A generalization of Rasmussen's invariant 3.1.