Symplectic Homology

Symplectic homology extends Floer homology to non-compact symplectic manifolds with contact-type boundary. It detects subtle features of symplectic topology and is a key invariant for distinguishing exotic symplectic structures.

Definition

Definition8.4Symplectic Homology

For a Liouville domain $(W, \omega = d\lambda)$ with contact boundary $(\partial W, \alpha = \lambda|_{\partial W})$ , the symplectic homology $SH_*(W)$ is defined as a direct limit $SH_*(W) = \varinjlim_{H} HF_*(H)$ over Hamiltonians $H$ that are linear with increasing slope on the cylindrical end $\partial W \times [1, \infty)$ .

Definition8.5Wrapped Floer Cohomology

For two exact Lagrangians $L_0, L_1$ in a Liouville domain $W$ , the wrapped Floer cohomology $HW^*(L_0, L_1)$ is a direct limit of Lagrangian Floer cohomology over Hamiltonians with increasing slope at infinity. The wrapped Fukaya category $\mathcal{W}(W)$ has objects: exact Lagrangians, and morphisms: wrapped Floer cochain complexes.

Example$SH_*(T^*Q) \cong H_*(\mathcal{L}Q)$

For the cotangent bundle $T^*Q$ of a closed manifold $Q$ , Viterbo and Abbondandolo-Schwarz proved $SH_*(T^*Q) \cong H_*(\mathcal{L}Q; \mathbb{Z}_2)$ , where $\mathcal{L}Q$ is the free loop space of $Q$ . This relates symplectic invariants to classical algebraic topology of loop spaces.

Applications

RemarkWeinstein Conjecture

Symplectic homology provides a tool for proving the Weinstein conjecture: every Reeb vector field on a contact manifold has a periodic orbit. If $SH_*(W) \neq 0$ , the contact boundary $\partial W$ carries a periodic Reeb orbit. Taubes proved the Weinstein conjecture in dimension 3 using Seiberg-Witten theory, and Floer-theoretic proofs work in many higher-dimensional cases.