Floer lasagna modules for some 2-handlebodies

Abstract

Morrison, Walker and Wedrich used Khovanov homology to define the skein lasagna module for a 4-manifold with a link in its boundary. The skein lasagna module was shown to be a deep and powerful invariant; in particular, Ren and Willis proved that it detects exotic smooth structures. Still, computing the skein lasagna module outside some carefully chosen grading range remains an important open problem. Motivated by this construction, Chen defined the Floer lasagna module using Heegaard Floer link homology. In this paper, we give a complete, but partially conjectural description of Floer lasagna modules for a class of 4-manifolds obtained by adding 2-handles along plumbed L-space knots (with certain framing constraints). In particular, we prove that for $\overline{\mathbb{CP}^2}$ it is infinite-dimensional, and conjecture that for $\mathbb{CP}^2$ it is infinite-dimensional as well. We compute this module for $\overline{\mathbb{CP}^2}$ (and conjecturally for $\mathbb{CP}^2$) in all Alexander and Maslov gradings. This strengthens and extends the recent results of McCulloch.