Localization of Lagrangian Floer homology for ℤ_p-actions

Abstract

For an odd prime p, we prove a Smith inequality over 𝔽_p for exact Lagrangian Floer homology under a symplectic ℤ_p-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a p-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free ℤ_p-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.

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BibTeX RIS

BibTeX

@misc{bao2026localization,
  title = {{Localization of Lagrangian Floer homology for $\mathbbZ_p$-actions}},
  author = {Erkao Bao and Tyler Lawson and Cheuk Yu Mak},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14468},
  eprint = {2610.14468},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14468},
}

RIS

TY  - UNPB
TI  - Localization of Lagrangian Floer homology for ℤ_p-actions
AU  - Erkao Bao
AU  - Tyler Lawson
AU  - Cheuk Yu Mak
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.14468
UR  - https://arxiv.org/abs/2610.14468
AB  - For an odd prime p, we prove a Smith inequality over 𝔽_p for exact Lagrangian Floer homology under a symplectic ℤ_p-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a p-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free ℤ_p-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.
ER  -

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