Symplectic Floer homology and the mapping class group. (Q1858320)

Let \(M\) be a closed connected oriented surface of genus \(\geq 2.\) If \(\omega \) is an everywhere positive two-form on \(M\) then a theorem of Moser says that each \(g\in \pi _0(\text{Diff}^+(M))\) admits a representative \(\phi \) which preserves \(\omega\), the symplectic structure \((M,\omega ) \) on \(M.\) The author proves that the quantum cap action of \(H^2(M,\mathbb{Z}/2),\) through the quantum cap product \(*: H^*(M,\mathbb{Z}/2)\otimes HF_*(\phi )\to HF_*(\phi ),\) on the Floer homology \(HF_*(g)\) is zero for all nontrivial elements of the group \(\Gamma =\pi _0(\text{Diff}^+(M))\). He also proves that if \(a\in H^1(M;\mathbb{Z}/2)\) is a class whose quantum cap action on \(HF_*(g)\) is nonzero, then there is a loop \(l:S^1\to M\) with \(g(l)\cong l\) and \(\langle a,[l]\rangle=1.\)