Localization for involutions in Floer cohomology
If \(M\) is a compact smooth manifold with a smooth action of the group \(G=\mathbb Z/2\) on it, and \(M^{\text{inv}}\) is its fixed point set, then the classical Smith inequality states that \(\dim H^*(M;\mathbb Z_2)\geq \dim H^*(M^{\text{inv}};\mathbb Z_2)\). To prove this inequality, one may use the equivariant cohomology \(H^*_G(M;\mathbb Z_2)\) and the localization theorem, which says that the restriction map \(H^*_G(M;\mathbb Z_2)\to H^*(M^{\text{inv}};\mathbb Z_2)\otimes_{\mathbb Z_2}H^*(BG;\mathbb Z_2)\) becomes an isomorphism once one inverts the generator \(q\) of \(H^*(BG;\mathbb Z_2)=\mathbb Z_2q\). In this paper, the authors develop an analogue of this inequality for Lagrangian Floer cohomology, with a view to applications to the link invariant. Let \(M\) be an exact and convex at infinity symplectic manifold which carries a symplectic involution \(\imath\). Let \(L_0\) and \(L_1\) be exact Lagrangian submanifolds of \(M\) invariant under \(\imath\), which are either compact or convex at infinity. Let \(M^{\text{inv}}\) be the fixed point set. The fixed parts \(L_k^{\text{inv}}=L_k\cap M^{\text{inv}}\) are again automatically Lagrangian submanifolds. There are three kinds of Floer cohomology groups involved, the ordinary Floer cohomology \(HF(L_0^{\text{inv}},L_1^{\text{inv}})\) in \(M^{\text{inv}}\), the corresponding group \(HF(L_0,L_1)\) in \(M\), and the \(G\)-equivariant analogue of the latter. The first two are taken with \(\mathbb Z_2\) coefficients, and the last one is a module over \(\mathbb Z_2q\). Under the assumption of the existence of a stable normal trivialization near the fixed point set, the authors prove the following Smith inequality: \(\dim HF(L_0,L_1)\geq \dim HF(L_0^{\text{inv}},L_1^{\text{inv}})\). Also, they include two applications to symplectic Khovanov cohomology.
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