Mathematical Research Letters
Volume 29 (2022)
On contact invariants of non-simply connected Gorenstein toric contact manifolds
Pages: 1 – 42
The first two authors showed in  how the Conley–Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley–Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underlying contact manifold. We compute these invariants for two sets of examples that satisfy these conditions: $5$‑dimensional contact manifolds that arise as unit cosphere bundles of $3$‑dimensional lens spaces, and $2n + 1$‑dimensional Gorenstein contact lens spaces. As applications, we will see that these invariants can be used to show that diffeomorphic lens spaces might not be contactomorphic and that there are homotopy classes of diffeomorphisms of some lens spaces that do not contain any contactomorphism. Following a suggestion by one referee, we will also see that this type of applications can be proved alternatively by looking at the total Chern class of these canonical contact structures on lens spaces.
Partially funded by FCT/Portugal through UID/MAT/04459/2020. MA and MM were also funded through project PTDC/MAT-GEO/1608/2014. MA and LM were also funded by CNPq/Brazil. The present work started as part of MA and LM activities within BREUDS, a research partnership between European and Brazilian research groups in dynamical systems, supported by an FP7 International Research Staff Exchange Scheme (IRSES) grant of the European Union. MM received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (ERC-2017-AdG-786580-MACI).
Received 19 March 2019
Received revised 10 January 2022
Accepted 19 January 2022
Published 6 September 2022