Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology

Abstract

Attaching a Casson handle to a slice disk complement yields a smooth 4-manifold that is homeomorphic to $\mathbb{R}^4$. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic $\mathbb{R}^4$'s made using the simplest positive Casson handle are not diffeomorphic, giving us a countably infinite family of pairwise nondiffeomorphic chiral exotic $\mathbb{R}^4$'s. Our main tool is Gadgil's end Floer homology and we use this to produce families of exotic $\mathbb{R}^4$ with various phenomena. As an application, we reprove a result of Bi\v{z}aca-Etnyre that $Y \times \mathbb{R}$, where $Y$ is any closed $3$-manifold, has infinitely many distinct smooth structures.