Instanton 2-torsion and Dehn surgeries
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918115353296896 |
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| author | Li, Zhenkun Ye, Fan |
| author_facet | Li, Zhenkun Ye, Fan |
| contents | In our earlier work on $2$-torsion in instanton Floer homology, we considered only integral surgeries on a knot $K\subset S^3$ and showed that the absence of $2$-torsion forces $K$ to be fibered. The present paper extends the result to all rational surgeries. We prove that if the framed instanton homology $I^{\sharp}(S^3_r(K);\mathbb{Z})$ is $2$-torsion-free for some $r\in \mathbb{Q}_+$, then $K$ is an instanton L-space knot and $r>2g(K)-1$. Leveraging this $2$-torsion perspective, we also obtain new small-surgery obstructions: If either $S^{3}_{5}(K)$ or $S^{3}_{11/2}(K)$ is $SU(2)$-abelian, then $K$ must be the unknot or the right-handed trefoil. This result sharpens the small-$SU(2)$-abelian surgery theorems of Kronheimer--Mrowka, Baldwin--Sivek, and Baldwin--Li--Sivek--Ye. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03394 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Instanton 2-torsion and Dehn surgeries Li, Zhenkun Ye, Fan Geometric Topology In our earlier work on $2$-torsion in instanton Floer homology, we considered only integral surgeries on a knot $K\subset S^3$ and showed that the absence of $2$-torsion forces $K$ to be fibered. The present paper extends the result to all rational surgeries. We prove that if the framed instanton homology $I^{\sharp}(S^3_r(K);\mathbb{Z})$ is $2$-torsion-free for some $r\in \mathbb{Q}_+$, then $K$ is an instanton L-space knot and $r>2g(K)-1$. Leveraging this $2$-torsion perspective, we also obtain new small-surgery obstructions: If either $S^{3}_{5}(K)$ or $S^{3}_{11/2}(K)$ is $SU(2)$-abelian, then $K$ must be the unknot or the right-handed trefoil. This result sharpens the small-$SU(2)$-abelian surgery theorems of Kronheimer--Mrowka, Baldwin--Sivek, and Baldwin--Li--Sivek--Ye. |
| title | Instanton 2-torsion and Dehn surgeries |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2508.03394 |