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Floer theory for Lagrangian cobordisms

Baptiste Chantraine, Georgios Dimitroglou Rizell, Paolo Ghiggini, Roman Golovko

TL;DR

The paper develops a Floer-theoretic framework, the Cthulhu complex, for exact Lagrangian cobordisms between Legendrian ends in the contactisation of a Liouville manifold, leveraging augmentations of Chekanov–Eliashberg algebras. It proves the complex is acyclic and yields long exact sequences linking Morse homology of cobordisms with bilinearised Legendrian contact homology of the ends, enabling topological restrictions on cobordisms. By introducing twisted and $L^2$-coefficients, it extends these results to fundamental-group data and noncommutative augmentations, establishing functoriality of the fundamental class and rigidity phenomena such as $L^2$-rigidity and homology-cylinder conclusions for endocobordisms. The work culminates in applications and explicit constructions that demonstrate obstructions to concordances, restrictions on endomorphisms, and concrete examples of non-invertible cobordisms, highlighting the interplay between symplectic topology and Legendrian invariants. The framework provides a powerful toolkit for understanding rigidity versus flexibility of Lagrangian cobordisms through exact sequences, dualities, and high-level algebraic structures.

Abstract

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.

Floer theory for Lagrangian cobordisms

TL;DR

The paper develops a Floer-theoretic framework, the Cthulhu complex, for exact Lagrangian cobordisms between Legendrian ends in the contactisation of a Liouville manifold, leveraging augmentations of Chekanov–Eliashberg algebras. It proves the complex is acyclic and yields long exact sequences linking Morse homology of cobordisms with bilinearised Legendrian contact homology of the ends, enabling topological restrictions on cobordisms. By introducing twisted and -coefficients, it extends these results to fundamental-group data and noncommutative augmentations, establishing functoriality of the fundamental class and rigidity phenomena such as -rigidity and homology-cylinder conclusions for endocobordisms. The work culminates in applications and explicit constructions that demonstrate obstructions to concordances, restrictions on endomorphisms, and concrete examples of non-invertible cobordisms, highlighting the interplay between symplectic topology and Legendrian invariants. The framework provides a powerful toolkit for understanding rigidity versus flexibility of Lagrangian cobordisms through exact sequences, dualities, and high-level algebraic structures.

Abstract

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.

Paper Structure

This paper contains 76 sections, 44 theorems, 164 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Main results.
  3. Remarks about grading and orientation.
  4. Long exact sequences for LCH induced by a Lagrangian cobordism.
  5. A generalisation of the long exact sequence of a pair.
  6. A generalisation of the duality long exact sequence and fundamental class
  7. A generalisation of the Mayer-Vietoris long exact sequence
  8. Topological restrictions on Lagrangian cobordisms
  9. The homology of an exact Lagrangian cobordism from a Legendrian submanifold to itself
  10. Restrictions on the fundamental group of certain exact Lagrangian fillings and cobordisms
  11. Obstructions to the existence of a Lagrangian concordance
  12. Remarks about the hypotheses.
  13. Restrictions on the ambient manifolds.
  14. Restrictions on the Lagrangian submanifolds.
  15. Preliminary definitions
  16. ...and 61 more sections

Key Result

Theorem 1.1

Let $\Sigma$ be a graded exact Lagrangian cobordism from $\Lambda^-$ to $\Lambda^+$ and let $\varepsilon^-_0$ and $\varepsilon^-_1$ be two augmentations of $\mathcal{A}(\Lambda^-)$ inducing augmentations $\varepsilon_0^+$, $\varepsilon_1^+$ of $\mathcal{A}(\Lambda^+)$. There is a long exact sequence where the map $\Phi^{\varepsilon^-_0,\varepsilon^-_1}_\Sigma \colon LCH^{k}_{\varepsilon^-_0,\varep

Figures (15)

  • Figure 1: Two Lagrangian cobordisms inside a symplectisation $\mathdj{R} \times Y$, where the vertical axis corresponds to the $\mathdj{R}$-coordinate.
  • Figure 2: A mixed cobordism LCH curve.
  • Figure 3: A punctured Floer strip.
  • Figure 4: A pseudoholomorphic cthulhu.
  • Figure 5: A pseudoholomorphic cultist
  • ...and 10 more figures

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4: Caps
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 88 more