definition 0.0.1
Let \(Q\) be a manifold. The free loop space of \(Q\), denote by \(\mathcal L Q\), consists of all continuous maps \[\gamma:S^1\to Q.\] The projection to base point is the map \begin{align*} \ev_0: \mathcal L Q\to& Q\\ \gamma\mapsto &\gamma(0). \end{align*}theorem 0.0.2 [Vit99]
Let \((Q, g)\) be a compact Riemannian manifold. Let \(X=B^*Q\) be the unit cotangent ball. Then \[\SH(X)= H_\bullet(\mathcal L Q).\]application 0.0.3
We now look at an application of Viterbo's theorem. Let \(X\) be a Liouville domain, and suppose that there exists \(L\subset X\) an exact Lagrangian submanifold. Then a Weinstein neighborhood \(B^*L\subset X\) provides an example of a Liouville subdomain. We therefore have a unital ring homomorphism \(\SH(X)\to \SH(B^*L)\). Since the latter is isomorphic (as a vector space) to \(H_\bullet(\mathcal L)\), it is non-vanishing. Since a unital ring homomorphism to a non-trivial target cannot have trivial domain, we conclude that \(\SH(X)\) is non-vanishing. This application is more striking in the reverse direction. Let \(X\) be a Liouville domain with vanishing symplectic cohomology (for instance, a subcritical Stein domain). Then \(X\) contains no exact Lagrangian submanifolds.References
[Vit99] | Claude Viterbo. Functors and computations in Floer homology with applications, i. Geometric & Functional Analysis GAFA, 9(5):985--1033, 1999. |