Commuting symplectomorphisms and Dehn twists in divisors

Tonkonog, Dmitry

doi:10.2140/gt.2015.19.3345

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Commuting symplectomorphisms and Dehn twists in divisors

Dmitry Tonkonog

Geometry & Topology 19 (2015) 3345–3403

DOI: 10.2140/gt.2015.19.3345

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 ${HF}^{*} (f)$ 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

References

Publication

Received: 31 May 2014

Revised: 16 March 2015

Accepted: 26 April 2015

Published: 6 January 2016

Proposed: Yasha Eliashberg

Seconded: Leonid Polterovich, Ciprian Manolescu

Authors

Dmitry Tonkonog
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB UK

Volume 19, issue 6 (2015)