We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the openclosed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid's generation criterion follows.
We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from "counting" pseudo-holomorphic curves.We introduce the notion of an implicit atlas on a moduli space, which is (roughly) a convenient system of local finite-dimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudoholomorphic curves. The main technical step in applying this strategy in any particular setting is to prove appropriate gluing theorems. We require only topological gluing theorems, that is, smoothness of the transition maps between gluing charts need not be addressed. Our approach to virtual fundamental cycles is algebraic rather than geometric (in particular, we do not use perturbation). Sheaf-theoretic tools play an important role in setting up our functorial algebraic "VFC package".We illustrate the methods we introduce by giving definitions of Gromov-Witten invariants and Hamiltonian Floer homology over Q for general symplectic manifolds. Our framework generalizes to the S 1 -equivariant setting, and we use S 1 -localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, Hofer-Salamon, Ono, Liu-Tian, Ruan, and Fukaya-Ono) is a well-known corollary of this calculation.
We give a construction of contact homology in the sense of Eliashberg-Givental-Hofer. Specifically, we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo-holomorphic curves.The aim of this work is to provide a rigorous construction of contact homology, an invariant of contact manifolds and symplectic cobordisms due to Eliashberg-Givental-Hofer [Eli98, EGH00]. The contact homology of a contact manifold (Y, ξ) is defined by counting pseudo-holomorphic curves in the sense of Gromov [Gro85] in its symplectization R × Y . The main problem we solve in this paper is simply to give a rigorous definition of these curve counts, the essential difficulty being that the moduli spaces of such curves are usually not cut out transversally. It is therefore necessary to construct the virtual fundamental cycles of these moduli spaces (which play the same enumerative role that the ordinary fundamental cycles do for transversally cut out moduli spaces). For this construction, we use the framework developed in [Par16]. Our methods are quite general, and apply equally well to many other moduli spaces of interest.We use a compactification of the relevant moduli spaces which is smaller than the compactification considered in [EGH00, BEHWZ03]. Roughly speaking, for curves in symplectizations R × Y , we do not keep track of the relative vertical positions of different components (in particular, no trivial cylinders appear). Our compactification is more convenient for proving the master equations of contact homology: the codimension one boundary strata in our compactification correspond bijectively with the desired terms in the "master equations", whereas the compactification from [EGH00, BEHWZ03] contains additional codimension one boundary strata. If we were to use the compactification from [EGH00, BEHWZ03], we would need to additionally argue that the contribution of each such extra codimension one boundary stratum vanishes.Remark 0.1 (Historical discussion). The theory of pseudo-holomorphic curves in closed symplectic manifolds was founded by Gromov [Gro85]. Hofer's breakthrough work on the threedimensional Weinstein conjecture [Hof93] introduced pseudo-holomorphic curves in symplectizations and their relationship with Reeb dynamics. The analytic theory of such curves was then further developed by Hofer-Wysocki-Zehnder [HWZ96, HWZ98, HWZ95, HWZ99, *
Our main result is a nontrivial lower bound for the distortion of some specific knots. In particular, we show that the distortion of the torus knot $T_{p,q}$ satisfies $\delta(T_{p,q})>\frac 1{160}\min(p,q)$. This answers a 1983 question of Gromov
We show that every Stein or Weinstein domain may be presented (up to deformation) as a Lefschetz fibration over the disk. The proof is an application of Donaldson's quantitative transversality techniques.Comment: 28 pages. Final version to appear in Gometry and Topolog
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