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For every compact almost Kahler manifold and an integral second homology class , we describe a natural closed subspace of the moduli space of stable –holomorphic genus-one maps such that contains all stable maps with smooth domains. If is the standard complex projective space, is an irreducible component of . We also show that if an almost complex structure on is sufficiently close to , the structure of the space is similar to that of . This paper’s compactness and structure theorems lead to new invariants for some symplectic manifolds, which are generalized to arbitrary symplectic manifolds in a separate paper. Relatedly, the smaller moduli space is useful for computing the genus-one Gromov–Witten invariants, which arise from the larger moduli space .
We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multitwists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary.
We introduce a new Floer theory associated to a pair consisting of a Cartan hypercontact –manifold and a hyperkähler manifold . The theory is a based on the gradient flow of the hypersymplectic action functional on the space of maps from to . The gradient flow lines satisfy a nonlinear analogue of the Dirac equation. We work out the details of the analysis and compute the Floer homology groups in the case where is flat. As a corollary we derive an existence theorem for the –dimensional perturbed nonlinear Dirac equation.
The symplectic Floer homology of a symplectomorphism encodes data about the fixed points of using counts of holomorphic cylinders in , where is the mapping torus of . We give an algorithm to compute for a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel’s for any orientation-preserving mapping class.
Let be a Gorenstein orbifold with projective coarse moduli space and let be a crepant resolution of . We state a conjecture relating the genus-zero Gromov–Witten invariants of to those of , which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan–Graber, and prove our conjecture when and . As a consequence, we see that the original form of the Bryan–Graber Conjecture holds for but is probably false for . Our methods are based on mirror symmetry for toric orbifolds.
The paper focuses on the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits.
We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action-index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of with exactly periodic orbits a stronger result holds. Namely, for such a Hamiltonian diffeomorphism, the difference of the action and the mean index on a periodic orbit is independent of the orbit, provided that the symplectic structure on is normalized to be in the same cohomology class as the first Chern class.
For all systolic groups we construct boundaries which are –structures. This implies the Novikov conjecture for torsion-free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in spaces.
This paper explores the topology of monotone Lagrangian submanifolds inside a symplectic manifold by exploiting the relationships between the quantum homology of and various quantum structures associated to the Lagrangian .
We prove that, for a link in a rational homology 3–sphere, the link Floer homology detects the Thurston norm of its complement. This result has been proved by Ozsváth and Szabó for links in . As an ingredient of the proof, we show that knot Floer homology detects the genus of null-homologous links in rational homology spheres, which is a generalization of an earlier result of the author. Our argument uses the techniques due to Ozsváth and Szabó, Hedden and the author.
Let be any locally compact unimodular metrizable group. The main result of this paper, roughly stated, is that if is any finitely generated free group and any lattice, then up to a small perturbation and passing to a finite index subgroup, is a subgroup of . If is noncompact then we require additional hypotheses that include .