An index inequality for embedded pseudoholomorphic curves in symplectizations

Abstract

Abstract.Let be a surface with a symplectic form, let φ be a symplectomorphism of , and let Y be the mapping torus of φ. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in R × Y , with cylindrical ends asymptotic to periodic orbits of φ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces.This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of Y in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact topology.

Symplectic Floer homology of area-preserving surface diffeomorphisms

“…In this case, by Lemma 3.3, we must have˛0 either nullhomologous or boundary parallel. 18 If it's nullhomologous, we may argue as in Lemma 3.2 and conclude that ! .A / D 0.…”

“…Symplectic Floer homology for surface symplectomorphisms is the d D 1 part of periodic Floer homology 5 (PFH) of (see Hutchings and Sullivan [21,Section 2] and Hutchings [18]), a Floer homology theory whose chain complex is generated by certain multisets of periodic orbits of and whose differential counts certain embedded pseudoholomorphic curves in R M . This theory was created to be a candidate for a 3-dimensional version of Taubes' "SW D Gr" result [46].…”

Embedded contact homology and Seiberg–Witten Floer cohomology III

“…The group C 1 .M I U.1// acts on the space of solutions to (1)(2)(3)(4)(5)(6)(7)(8) as follows: Suppose that u is a map from M to U.1/ and c D .A; / is a solution to (1)(2)(3)(4)(5)(6)(7)(8). Then uc D .A u 1 du; u / is also a solution to (1)(2)(3)(4)(5)(6)(7)(8).…”

“…H 2 .M I Z/ denotes the Poincaré duality isomorphism. Hutchings [6] explains how the elements in Z can be given a relative Z=pZ grading. There are six steps involved.…”

“…It also comes with a natural, complete Kahler metric; but this is not the flat Kahler metric unless m D 1. In any event, this metric defines a symplectic form and thus the Hamiltonian dynamical system that is defined using the time dependent Hamiltonian function (1)(2)(3)(4)(5)(6) h D The spectrum of this operator is a discrete subset of R with finite multiplicities and no accumulation points. What is denoted here by C‚ consists of the elements in C‚ of the form fc g .…”