[Paper Review] A symplectic Gysin sequence

Timothy Perutz|ArXiv.org|Jul 11, 2008

Geometric and Algebraic Topology21 references19 citations

TL;DR

This paper establishes a symplectic Gysin sequence in symplectic Floer homology for spherically fibred coisotropic submanifolds, using pseudo-holomorphic quilts to construct a long exact sequence analogous to the classical Gysin sequence in homology. The key result is a canonical exact triangle relating the Floer homology of a symplectomorphism μ on N to the twisted Floer homology of a Lagrangian correspondence associated with a sphere bundle, under monotonicity and Maslov index conditions.

ABSTRACT

We use the theory of pseudo-holomorphic quilts to establish a counterpart, in symplectic Floer homology, to the Gysin sequence for the homology of a sphere-bundle. In a motivating class of examples, this "symplectic Gysin sequence" is precisely analogous to an exact sequence describing the behaviour of Seiberg-Witten monopole Floer homology for 3-manifolds under connected sum.

Motivation & Objective

• To construct a symplectic counterpart to the classical Gysin sequence in homology for sphere bundles using symplectic Floer homology.
• To extend the analogy between symplectic Floer homology and Seiberg–Witten monopole Floer homology under connected sum operations.
• To provide a Floer-theoretic framework for understanding the behavior of Lagrangian correspondences in spherically fibred coisotropic submanifolds.
• To establish a long exact sequence relating the fixed-point Floer homology of a symplectomorphism μ on N to the twisted Floer homology of the Lagrangian correspondence.
• To explore the implications of this sequence in moduli spaces of flat connections and in 3-manifold topology via symplectic analogues of instanton and Heegaard Floer theories.

Proposed method

• Utilizes the theory of pseudo-holomorphic quilts to define and compute Floer homology groups for Lagrangian correspondences in symplectic manifolds.
• Constructs the Lagrangian correspondence $\widehat{V} \subset M_{-} \times N$ as the graph of the projection $\rho: V \to N$, where $V$ is a spherically fibred coisotropic submanifold of $M$.
• Imposes monotonicity and Maslov index conditions: $m_{\widehat{V}}^{\min} \geq k+2$ to ensure well-definedness and exactness of the sequence.
• Applies the universal Novikov ring $\Lambda_R$ to handle Novikov coefficients in the Floer homology groups.
• Derives the symplectic Gysin sequence as an exact triangle involving $\operatorname{HF}(\mu)$, $\operatorname{HF}(\mu)\langle e(V)\rangle$, and $\operatorname{HF}(\widehat{V}, (\mathrm{id}_{M_-} \times \mu)\widehat{V})$.
• Uses the global angular chain (e.g., a section $S$) to represent the Euler class $e(V)$, which acts via the cap product in the exact triangle.

Experimental results

Research questions

• RQ1 Can a symplectic analogue of the Gysin sequence be constructed in Floer homology for spherically fibred coisotropic submanifolds?
• RQ2 How does the Floer homology of a Lagrangian correspondence associated with a sphere bundle relate to the fixed-point Floer homology of a symplectomorphism on the base?
• RQ3 To what extent does the symplectic Gysin sequence mirror the behavior of Seiberg–Witten monopole Floer homology under connected sum operations?
• RQ4 What is the role of the Euler class in the symplectic Gysin sequence, and how is it realized geometrically in the context of moduli spaces of flat connections?
• RQ5 Can the symplectic Gysin sequence be used to model the behavior of instanton Floer homology under internal connected sums?

Key findings

• Under the hypotheses of monotonicity and $m_{\widehat{V}}^{\min} \geq k+2$, the symplectic Gysin sequence forms a canonical exact triangle involving $\operatorname{HF}(\mu)$, $\operatorname{HF}(\mu)\langle e(V)\rangle$, and $\operatorname{HF}(\widehat{V}, (\mathrm{id}_{M_-} \times \mu)\widehat{V})$.
• The sequence is isomorphic to the classical Gysin sequence in homology when the base $N$ is a closed manifold and $V \to N$ is a sphere bundle with $F \simeq S^k$.
• In the case of moduli spaces of flat SU(2) connections on punctured surfaces, the sequence yields $\operatorname{HF}_{*}(\widehat{V}, \widehat{V}) \cong \operatorname{HF}_{*}(\theta(\phi)) \oplus \operatorname{HF}_{*}(\theta(\phi))[3]$ for all $\phi \in \operatorname{Diff}^+(Σ)$, under a choice of section.
• The minimal Maslov index condition $m_{\widehat{V}}^{\min} = 4$ for $g \geq 2$ in the twisted character variety $M_g$ places the setup on the borderline for applicability, but the sequence still holds.
• The global angular chain representing $e(V)$ is realized as a section $S$ of the $S^3$-bundle $\rho: V \to M_{g-1}$, which induces the cap product action in the exact triangle.
• The result is consistent with the known behavior of Heegaard Floer homology under 1-handle attachment, suggesting a deeper analogy between symplectic and 3-manifold Floer theories.