Symplectic homology and the Eilenberg-Steenrod axioms

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology. In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.Comment: v3: corrected Lemma 7.11. Various other minor modifications and reformatting. Final version to be published in Algebraic and Geometric Topolog

Symplectic hypersurfaces and transversality in Gromov-Witten theory

We use Donaldson hypersurfaces to construct pseudo-cycles which define Gromov-Witten invariants for any symplectic manifold which agree with the invariants in the cases where transversality could be achieved by perturbing the almost complex structure.Comment: 53 pages, final versio

The topology of rationally and polynomially convex domains

We give in this article necessary and sufficient conditions on the topology of rationally and polynomially convex domains.Comment: 23 pages, no figures, final version to appear in Invent. Mat

A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions

In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a K\"ahler-Einstein manifold. An immediate consequence is that in K\"ahler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.Comment: J. Symplectic Geom. vol 2 (2004), issue 2, 261-26