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Instantons, indefinite 4-manifolds, and Dehn surgery

  • Aliakbar Daemi
  • Xingpei Liu
  • Mike Miller Eismeier
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Abstract

We prove that there exist hyperbolic integer homology spheres with arbitrarily large Dehn surgery number. Previously, no integer homology sphere was known to have a surgery number larger than $2$. Our approach uses Froyshov's invariant q3 of integer homology spheres, which is defined in terms of mod 2 instanton homology. We show that if W: Y → Y' is a cobordism between integer homology spheres with no $2$-torsion in its first homology, then −b+(W) ≤ q3(Y') − q3(Y) ≤ b-(W). We also extend both q3 and the inequality to rational homology spheres.

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Cite

BibTeXRIS
BibTeX
@misc{daemi2026instantons,
  title = {{Instantons, indefinite 4-manifolds, and Dehn surgery}},
  author = {Aliakbar Daemi and Xingpei Liu and Mike Miller Eismeier},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27904},
  eprint = {2608.27904},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27904},
}
RIS
TY  - UNPB
TI  - Instantons, indefinite 4-manifolds, and Dehn surgery
AU  - Aliakbar Daemi
AU  - Xingpei Liu
AU  - Mike Miller Eismeier
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27904
UR  - https://arxiv.org/abs/2608.27904
AB  - We prove that there exist hyperbolic integer homology spheres with arbitrarily large Dehn surgery number. Previously, no integer homology sphere was known to have a surgery number larger than $2$. Our approach uses Froyshov's invariant q_3 of integer homology spheres, which is defined in terms of mod 2 instanton homology. We show that if W: Y → Y' is a cobordism between integer homology spheres with no $2$-torsion in its first homology, then −b^+(W) ≤ q_3(Y') − q_3(Y) ≤ b^-(W). We also extend both q_3 and the inequality to rational homology spheres.
ER  -