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

TL;DR

This paper demonstrates that differences in knot Floer homology can distinguish exotic smooth structures on ^4, producing infinitely many non-diffeomorphic examples using end Floer homology.

Contribution

It introduces a method to distinguish exotic ^4's via knot Floer homology and constructs infinitely many such structures with various phenomena.

Findings

Different knot Floer homologies lead to non-diffeomorphic exotic ^4's.

Constructed countably infinite family of exotic ^4's.

Reproved that Y  for any closed 3-manifold Y has infinitely many smooth structures.

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.