We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, as well as a rigorous definition of cylindrical/linearized contact homology based on an $S^1$-equivariant construction.

S1-Equivariant Symplectic Homology and Linearized Contact Homology

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.

https://msp.org/agt/2018/18-4/agt-v18-n4-p03-p.pdf

Symplectic homology and the Eilenberg–Steenrod axioms

We use positive S1–equivariant symplectic homology to define a sequence of symplectic capacities ck for star-shaped domains in ℝ2n. These capacities are conjecturally equal to the Ekeland–Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities ck of any “convex toric domain” or “concave toric domain”. As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities ck to functions of Liouville domains which are almost but not quite symplectic capacities.

https://msp.org/agt/2018/18-6/agt-v18-n6-p11-s.pdf

Symplectic capacities from positive $S^1$–equivariant symplectic homology

We develop a variant of Lusternik–Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. As an application, we prove new multiplicity results for simple closed Reeb orbits on the standard contact sphere, the unit cotangent bundle to the sphere and some other contact manifolds. We also show that the lower Conley–Zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of Long and Zhu. This is the second, combinatorial ingredient in the proof of the multiplicity results.

Lusternik–Schnirelmann theory and closed Reeb orbits

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology; in certain cases, we prove that all flexible fillings are symplectomorphic. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Similar methods are used to construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. We also prove relative analogs of our results, which we apply to Lagrangians in cotangent bundles.

Contact manifolds with flexible fillings

The goal of this note is to make two corrections to the original article. The first correction concerns the transversality assumptions for the definition of linearized contact homology. The second correction concerns the description of the connecting map D in terms of the contact complex. These corrections are described, respectively, in Sects. 1 and 2 below. The equations in the original article are numbered by 1, 2, 3, . . .. The equations in the current note are numbered by x .1, x .2, x .3, . . . where x is the number of the section.

/pdf/erratum-to-an-exact-sequence-for-contact-and-symplectic-42v1x9v43m.pdf

Alexandru Oancea

Papers

S1-Equivariant Symplectic Homology and Linearized Contact Homology

Erratum to: An exact sequence for contact- and symplectic homology