Journal of Symplectic Geometry

Volume 15 (2017)

Number 4

The positive equivariant symplectic homology as an invariant for some contact manifolds

Pages: 1019 – 1069

DOI: https://dx.doi.org/10.4310/JSG.2017.v15.n4.a3

Author

Jean Gutt (Department of Mathematics, University of California at Berkeley; and Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)

Abstract

We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley–Zehnder indices of all good periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the positive $S^1$-equivariant symplectic homology can be computed; it is generated by those orbits. We prove a “Viterbo functoriality” property: when one Liouville domain is embedded into an other one, there is a morphism (reversing arrows) between their positive $S^1$-equivariant symplectic homologies and morphisms compose nicely.

These properties allow us to give a proof of Ustilovsky’s result on the number of non isomorphic contact structures on the spheres $S^{4m+1}$. They also give a new proof of a Theorem by Ekeland and Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in $\mathbb{R}^{2n}$. We extend this result to some hypersurfaces in some negative line bundles.

Received 8 March 2015

Accepted 27 May 2016

Published 28 November 2017