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

Manolescu, Ciprian and Marengon, Marco and Sarkar, Sucharit and Willis, Michael (2023) A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds. DUKE MATHEMATICAL JOURNAL, 172 (2). pp. 231-311. ISSN 0012-7094

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Abstract

We extend the definition of Khovanov-Lee homology to links in connected sums of S1 × S2’s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in S1 ×S2, 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: B2 × S2, S1 × B3, CP2, 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 B4 by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.

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