Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908761716686848 |
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| author | Eli, Sean Hom, Jennifer Lidman, Tye |
| author_facet | Eli, Sean Hom, Jennifer Lidman, Tye |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08767 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology Eli, Sean Hom, Jennifer Lidman, Tye Geometric Topology 57K18, 57K31, 57K40, 57K41 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. |
| title | Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology |
| topic | Geometric Topology 57K18, 57K31, 57K40, 57K41 |
| url | https://arxiv.org/abs/2601.08767 |