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

TL;DR

This paper generalizes Rasmussen's invariant to links in certain 4-manifolds, providing genus bounds for surfaces and exploring implications for homotopy 4-balls, but showing limitations in detecting non-standard smooth structures.

Contribution

It extends Khovanov-Lee homology and Rasmussen's invariant to links in connected sums of S^1 x S^2, and applies these to genus bounds in various 4-manifolds.

Findings

Computed the invariant for certain links in S^1 x S^2 using Hochschild homology.

Proved inequalities relating the invariant to surface genus in specific 4-manifolds.

Showed the invariant cannot distinguish certain exotic smooth structures on 4-balls.

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 four-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 non-standard.