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Homological Lagrangian monodromy and Wang exact sequence

Yoshihiro Sugimoto

TL;DR

This work studies homological Lagrangian monodromy for a closed Lagrangian submanifold $L$ in a symplectic manifold by comparing the Hofer energy of Hamiltonian isotopies to the minimal energy of $J$-holomorphic spheres and discs. The authors introduce a Seidel-type construction on a Hamiltonian fibration and employ a Morse-theoretic description of the Wang exact sequence to reduce the monodromy problem to an injectivity criterion for the inclusion of the fiber homology. By defining an $\Omega$-coefficient Seidel-type map and establishing chain-homotopies with the inverse isotopy, they prove that if the energy bound $||[\{\phi_t\}]||<\sigma(M,L,[\{\phi_t\}])$ holds, the induced action on $H_*(L;\mathbb{Z}_2)$ is trivial. The results extend to rational Lagrangians and provide a framework that connects symplectic fibrations, Morse theory, and Floer-type counts to detect monodromy, with potential implications for understanding rigidity phenomena in Lagrangian topology.

Abstract

In this paper, we study homological monodromy of a Lagrangian submanifold. We prove that homological Lagrangian monodromy is trivial if Hofer energy of a Hamiltonian isotopy is smaller than the minimum energy of J-holomorphic spheres and discs.

Homological Lagrangian monodromy and Wang exact sequence

TL;DR

This work studies homological Lagrangian monodromy for a closed Lagrangian submanifold in a symplectic manifold by comparing the Hofer energy of Hamiltonian isotopies to the minimal energy of -holomorphic spheres and discs. The authors introduce a Seidel-type construction on a Hamiltonian fibration and employ a Morse-theoretic description of the Wang exact sequence to reduce the monodromy problem to an injectivity criterion for the inclusion of the fiber homology. By defining an -coefficient Seidel-type map and establishing chain-homotopies with the inverse isotopy, they prove that if the energy bound holds, the induced action on is trivial. The results extend to rational Lagrangians and provide a framework that connects symplectic fibrations, Morse theory, and Floer-type counts to detect monodromy, with potential implications for understanding rigidity phenomena in Lagrangian topology.

Abstract

In this paper, we study homological monodromy of a Lagrangian submanifold. We prove that homological Lagrangian monodromy is trivial if Hofer energy of a Hamiltonian isotopy is smaller than the minimum energy of J-holomorphic spheres and discs.

Paper Structure

This paper contains 3 sections, 7 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Symplectic fibration over $D^2$
  3. Seidel-type map and proof of Theorem 1

Key Result

Theorem 1

Let ${L\subset M}$ be a closed Lagrangian submanifold. Let ${\{\phi_t\}\subset \mathrm{Ham}(M,\omega)}$ be a Hamiltonian isotopy such that ${\phi_0=\mathrm{Id}}$ and ${\phi_1\in \mathrm{Ham}_L(M,\omega)}$ hold. If holds, then is equal to the identity.

Theorems & Definitions (9)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Lemma 3
  • Lemma 4
  • Corollary 2