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Symplectic Floer homology and the mapping class group. (English) Zbl 1061.53065
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 $$\langle a,[l]\rangle=1.$$

MSC:
 53D40 Symplectic aspects of Floer homology and cohomology 53D35 Global theory of symplectic and contact manifolds
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