[Paper Review] The Gysin exact sequence for $S^1$-equivariant symplectic homology

Frédéric Bourgeois, Alexandru Oancea|arXiv (Cornell University)|Sep 24, 2009

Geometric and Algebraic TopologyMathematics27 references28 citations

TL;DR

This paper establishes a Gysin exact sequence for $S^1$-equivariant symplectic homology of symplectically aspherical manifolds with contact-type boundary, using a parametrized Floer homology construction as a bridge between non-equivariant and equivariant theories. The key result is a long exact sequence relating $SH^a_k(W)$, $SH^{a,S^1}_k(W)$, and $SH^{a,S^1}_{k-2}(W)$, which is compatible with the tautological exact sequences of symplectic homology and provides a geometric realization of the Gysin sequence in symplectic topology.

ABSTRACT

We define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.

Motivation & Objective

• To provide a complete geometric construction of $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact-type boundary, following Viterbo's Floer-type approach.
• To define a parametrized version of symplectic homology as a foundational tool to interpolate between non-equivariant and equivariant theories.
• To establish a Gysin exact sequence relating $SH^a_k(W)$, $SH^{a,S^1}_k(W)$, and $SH^{a,S^1}_{k-2}(W)$, proving its compatibility with existing tautological exact sequences.
• To demonstrate the isomorphism of the Gysin sequence with the classical Gysin sequence on the free loop space $\Lambda L$ for cotangent bundles.
• To clarify the relationship between the Strong Algebraic Weinstein Conjecture (SAWC) and the Strong Equivariant Algebraic Weinstein Conjecture (EWC), showing SAWC implies EWC.

Proposed method

• Constructs $S^1$-equivariant symplectic homology via a Floer-type chain complex using Hamiltonian perturbations and $S^1$-action on the space of loops.
• Introduces a parametrized symplectic homology theory, where Hamiltonians are families indexed by a finite-dimensional smooth parameter space, to mediate the non-equivariant-to-equivariant transition.
• Applies Morse-Bott techniques to the parametrized theory, using moduli spaces of gradient and Floer trajectories with breaking and gluing structures.
• Defines a chain homotopy $K = K_1 + K_2$ using counts of 1-dimensional moduli spaces to prove the homotopy equivalence of continuation maps.
• Uses the cone construction and filtered continuation maps to relate different parametrized chain complexes and establish spectral sequence convergence.
• Applies the $S^1$-equivariant homology of the pair $(W, \partial W)$ with trivial $S^1$-action to prove compatibility of the Gysin sequence with tautological sequences.

Experimental results

Research questions

• RQ1 How can $S^1$-equivariant symplectic homology be rigorously defined for symplectically aspherical manifolds with contact-type boundary using Floer-theoretic methods?
• RQ2 What is the precise relationship between non-equivariant symplectic homology and its $S^1$-equivariant counterpart, and can this be captured by a long exact sequence?
• RQ3 How does the parametrized symplectic homology construction serve as a bridge between non-equivariant and equivariant theories in this context?
• RQ4 Is the Gysin exact sequence for $S^1$-equivariant symplectic homology compatible with the tautological exact sequences of symplectic homology?
• RQ5 Does the Strong Algebraic Weinstein Conjecture imply the Strong Equivariant Algebraic Weinstein Conjecture in the $S^1$-equivariant setting?

Key findings

• The $S^1$-equivariant symplectic homology $SH^{a,S^1}_*(W)$ fits into a long exact sequence: $\cdots \to SH^a_k(W) \to SH^{a,S^1}_k(W) \xrightarrow{D} SH^{a,S^1}_{k-2}(W) \to SH^a_{k-1}(W) \to \cdots$, establishing a Gysin-type exact sequence.
• The Gysin differential $D$ is compatible with the tautological exact sequences: a commutative diagram is constructed showing compatibility between the Gysin and tautological sequences for both non-equivariant and equivariant theories.
• For the cotangent bundle $W = DT^*L$, the Gysin sequence (1.4) is isomorphic to the classical Gysin sequence on the free loop space $\Lambda L$, i.e., $\cdots \to H_*(\Lambda L) \to HS^1_*(\Lambda L) \xrightarrow{D} HS^1_{*-2}(\Lambda L) \to H_{*-1}(\Lambda L) \to \cdots$
• For subcritical Stein manifolds with $c_1(W) = 0$, the $S^1$-equivariant symplectic homology vanishes: $SH^{S^1}_*(W) = 0$, and the Gysin sequence becomes an isomorphism of exact sequences with the $S^1$-homology of the pair $(W, \partial W)$.
• The Strong Algebraic Weinstein Conjecture (SAWC) implies the Strong Equivariant Algebraic Weinstein Conjecture (EWC), as shown by the commutative diagram in Theorem 1.2 and the vanishing of $SH^*(W)$ under SAWC.
• The isomorphism type of $S^1$-equivariant symplectic homology is independent of the choice of homotopy data in the parametrized construction, as shown by spectral sequence convergence and homotopy invariance of the continuation maps.