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Equivariant Floer homology is isomorphic to reduced Floer homology

Julio Sampietro Christ

TL;DR

This work constructs a quantum, Novikov-ring refinement of equivariant Lagrangian Floer homology for a symplectic manifold with a Hamiltonian $G$-action and $G$-invariant Lagrangians in the zero-level set, proving Cazassus' conjecture: when the $G$-action on $\mu^{-1}(0)$ is free, the equivariant Floer homology is isomorphic to the Floer homology of the Marsden–Weinstein reduced Lagrangians. The strategy combines a quantum-model approach (inspired by Biran–Cornea and Schmäschke) with a spectral-sequence-type argument made precise via energy filtrations and quilted cascades, organized through a telescope construction over symplectic Borel spaces. A central technical advance is the demonstration of uniform energy gaps for differentials and quilt-increment maps, ensuring the filtration is preserved and enabling a controlled comparison to reduced Floer theory. The results sharpen the theory of equivariant Lagrangian Floer homology, supply a robust bridge to reduction theory, and yield an invariant framework adaptable to monotone and non-clean-intersection settings, with potential applications to probing singular reductions and orbifold-type phenomena in symplectic topology.

Abstract

Given a symplectic manifold equipped with a Hamiltonian $G$-action and two $G$-invariant Lagrangians, we lift the construction of equivariant Lagrangian Floer homology of G.\@~Cazassus to the Novikov ring by constructing a ``quantum'' model in the vein of Biran and Cornea and Schmäschke. Using this refinement, we prove Cazassus' conjecture that, if the action on the zero-level of the moment map is free, then equivariant Floer homology agrees with the Floer homology of the Marsden-Weinstein reduction.

Equivariant Floer homology is isomorphic to reduced Floer homology

TL;DR

This work constructs a quantum, Novikov-ring refinement of equivariant Lagrangian Floer homology for a symplectic manifold with a Hamiltonian -action and -invariant Lagrangians in the zero-level set, proving Cazassus' conjecture: when the -action on is free, the equivariant Floer homology is isomorphic to the Floer homology of the Marsden–Weinstein reduced Lagrangians. The strategy combines a quantum-model approach (inspired by Biran–Cornea and Schmäschke) with a spectral-sequence-type argument made precise via energy filtrations and quilted cascades, organized through a telescope construction over symplectic Borel spaces. A central technical advance is the demonstration of uniform energy gaps for differentials and quilt-increment maps, ensuring the filtration is preserved and enabling a controlled comparison to reduced Floer theory. The results sharpen the theory of equivariant Lagrangian Floer homology, supply a robust bridge to reduction theory, and yield an invariant framework adaptable to monotone and non-clean-intersection settings, with potential applications to probing singular reductions and orbifold-type phenomena in symplectic topology.

Abstract

Given a symplectic manifold equipped with a Hamiltonian -action and two -invariant Lagrangians, we lift the construction of equivariant Lagrangian Floer homology of G.\@~Cazassus to the Novikov ring by constructing a ``quantum'' model in the vein of Biran and Cornea and Schmäschke. Using this refinement, we prove Cazassus' conjecture that, if the action on the zero-level of the moment map is free, then equivariant Floer homology agrees with the Floer homology of the Marsden-Weinstein reduction.
Paper Structure (34 sections, 31 theorems, 137 equations, 16 figures)

This paper contains 34 sections, 31 theorems, 137 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Strategy of proof
  4. Organization of the paper
  5. Acknowledgements
  6. Preliminaries
  7. Algebra
  8. Novikov rings and gapped modules
  9. Telescopes
  10. Symplectic geometry
  11. Hamiltonian actions
  12. Floer homology of Lagrangians in clean intersection
  13. Pseudoholomorphic quilts
  14. Equivariant Morse homology
  15. Classical equivariant homology
  16. ...and 19 more sections

Key Result

Theorem 1

If the action of $G$ on the zero-level of the moment map is free, then there is an isomorphism Where $\Lambda_0$ denotes the Novikov ring over $\mathbb{Z}_2$ and $\overline{L_j}=L_j\slash G$.

Figures (16)

  • Figure 1: A cascade from $p$ to $q$ with three jumps.
  • Figure 2: Case 1: A sequence of cascades may develop an interior bubble $(a)$ or a disc bubble $(b)$.
  • Figure 3: Case 2: In a sequence of cascades, one or more $u_k$ may break.
  • Figure 4: Case 3: In a sequence of cascades, a gradient line may shrink to a point.
  • Figure 5: Case 4: One of the gradient lines could break.
  • ...and 11 more figures

Theorems & Definitions (59)

  • Theorem 1
  • Remark 2
  • Proposition 3
  • Proposition 4
  • proof
  • Theorem 5: Marsden-Weinstein reduction
  • Example 6
  • Remark 7
  • Remark 8: Marsden-Weinstein correspondence
  • Proposition 9
  • ...and 49 more