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

Sean Eli, Jennifer Hom, Tye Lidman

TL;DR

The paper develops an end Floer framework to distinguish exotic $\mathbb{R}^4$s obtained by attaching a Casson handle to slice disk complements. By computing end Floer invariants $HE(X,\mathfrak{s})$ via grading-admissible exhaustions and translating cobordism information through the surgery exact triangle and mapping cone formula, it ties the knot Floer data of slice knots (via Whitehead doubles) to nontrivial end invariants. It then proves the existence of countably infinite families of pairwise nondiffeomorphic exotic $\mathbb{R}^4$s built with the simplest positive Casson handle, including branching phenomena with Casson handles carrying infinite positive and negative chains. The work also reaffirms that $Y\times\mathbb{R}$ admits infinitely many smooth structures and shows that end-summing exotic $\mathbb{R}^4$s yields infinitely many smoothings of $M\times\mathbb{R}$, all detectable via end Floer homology. Overall, the paper provides a robust, end Floer–based approach for distinguishing noncompact smooth structures in four dimensions and connects knot Floer data to global end phenomena.

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 Bižaca-Etnyre that $Y \times \mathbb{R}$, where $Y$ is any closed $3$-manifold, has infinitely many distinct smooth structures.

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

TL;DR

The paper develops an end Floer framework to distinguish exotic s obtained by attaching a Casson handle to slice disk complements. By computing end Floer invariants via grading-admissible exhaustions and translating cobordism information through the surgery exact triangle and mapping cone formula, it ties the knot Floer data of slice knots (via Whitehead doubles) to nontrivial end invariants. It then proves the existence of countably infinite families of pairwise nondiffeomorphic exotic s built with the simplest positive Casson handle, including branching phenomena with Casson handles carrying infinite positive and negative chains. The work also reaffirms that admits infinitely many smooth structures and shows that end-summing exotic s yields infinitely many smoothings of , all detectable via end Floer homology. Overall, the paper provides a robust, end Floer–based approach for distinguishing noncompact smooth structures in four dimensions and connects knot Floer data to global end phenomena.

Abstract

Attaching a Casson handle to a slice disk complement yields a smooth 4-manifold that is homeomorphic to . We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic 's made using the simplest positive Casson handle are not diffeomorphic, giving us a countably infinite family of pairwise nondiffeomorphic chiral exotic 's. Our main tool is Gadgil's end Floer homology and we use this to produce families of exotic with various phenomena. As an application, we reprove a result of Bižaca-Etnyre that , where is any closed -manifold, has infinitely many distinct smooth structures.
Paper Structure (18 sections, 20 theorems, 57 equations, 11 figures)

This paper contains 18 sections, 20 theorems, 57 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background and context.
  3. Overview of proof strategy
  4. Organization.
  5. Acknowledgements
  6. Noncompact 4-manifold background
  7. Casson handles
  8. Exhaustions of slice $\mathbb{R}^4$'s
  9. End-sums and more exotica
  10. End Floer homology
  11. Heegaard Floer preliminaries
  12. The knot Floer complex
  13. The mapping cone formula
  14. The cobordism map between $0$-surgeries
  15. A key theorem
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{R}$ be built from attaching the Casson handle $\mathit{CH}^+$ to a slice disk complement $(B^4,S^3)-\nu(D^2,K)$ for a nontrivial slice knot $K$ and removing the boundary.

Figures (11)

  • Figure 1: A Kirby diagram for an exotic $\mathbb{R}^4$ obtained by attaching the Casson handle $\mathit{CH}^+$ to a ribbon disk complement for $T_{2,3}\# -T_{2,3}$.
  • Figure 2: A Kirby diagram for $B^4\cup T^+_3$. We see three copies of $T_1^+$: the 0-framed unknots are their attaching circles, and meridians of the dotted circles are their double point loops. Continuing the pattern infinitely to the right and deleting the boundary yields $\text{int}(B^4\cup \mathit{CH}^+)$.
  • Figure 3: A Kirby diagram for $B^4$ together with the first 3 stages of $CH^*$, attached along a 0-framed unknot in $S^3$.
  • Figure 4: A cobordism from $S^3_0(K)$ to $S^3_0(\mathit{Wh}(K))$ consisting of a $1$- and $2$-handle attachment.
  • Figure 5: The knot Floer complex $\mathit{CFK}^-(4_1)$.
  • ...and 6 more figures

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 44 more