Volume 19, issue 6 (2015)

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Commuting symplectomorphisms and Dehn twists in divisors

Dmitry Tonkonog

Geometry & Topology 19 (2015) 3345–3403
Abstract

Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism $f$ commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of ${HF}^{\ast }\left(f\right)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in the symplectic mapping class group.

Keywords
Floer cohomology, elliptic relation, symplectic involution, Dehn twist
Mathematical Subject Classification 2010
Primary: 53D40
Secondary: 14F35, 14D05