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A sharp lower bound on fixed points of surface symplectomorphisms in each mapping class

Andrew Cotton-Clay

Geometry & Topology 27 (2023) 1657–1690
Abstract

Given a compact, oriented surface Σ, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (ie area-preserving map) in the given mapping class, both with and without nondegeneracy assumptions on the fixed points. This generalizes the Poincaré–Birkhoff fixed point theorem to arbitrary surfaces and mapping classes. These bounds often exceed those for non-area-preserving maps. We give a fixed point bound on symplectic mapping classes for monotone symplectic manifolds in terms of the rank of a twisted-coefficient Floer homology group, with computations in the surface case. For the case of possibly degenerate fixed points, we use quantum-cup-length-type arguments for certain cohomology operations we define on summands of the Floer homology.

Keywords
symplectomorphisms, symplectic Floer homology, fixed point bounds, Nielsen theory, Poincaré–Birkhoff, mapping classes, Thurston classification, degenerate fixed points, quantum cup length, Novikov ring
Mathematical Subject Classification 2010
Primary: 37E30, 37J10, 53D40
Secondary: 37C25
References
Publication
Received: 25 February 2014
Revised: 7 November 2021
Accepted: 8 December 2021
Published: 27 July 2023
Proposed: Peter Teichner
Seconded: András I Stipsicz, Ciprian Manolescu
Authors
Andrew Cotton-Clay
Fullpower Technologies
Santa Cruz, CA
United States

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