Journal of Symplectic Geometry

Volume 21 (2023)

Number 6

Embedded contact homology of prequantization bundles

Pages: 1077 – 1189

DOI: https://dx.doi.org/10.4310/JSG.2023.v21.n6.a1

Authors

Jo Nelson (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)

Morgan Weiler (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.; and Department of Mathematics, University of California, Riverside, Calif., U.S.A.)

Abstract

The 2011 PhD thesis of Farris [Fa] demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a $\mathbb{Z}^2$-graded group to the exterior algebra of the homology of its base. We extend this result by computing the $\mathbb{Z}$-grading on the chain complex, permitting a finer understanding of this isomorphism and a stability result for ECH. We fill in a number of technical details, including the Morse–Bott direct limit argument and the classification of certain $J$-holomorphic buildings. The former requires the isomorphism between filtered Seiberg–Witten Floer cohomology and filtered ECH as established by Hutchings–Taubes [HT13]. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner–Hutchings–Zhang [CGHZ] to obtain the writhe bounds necessary to appeal to an intersection theory argument of Hutchings–Nelson [HN16].

Received 31 July 2020

Received revised 3 January 2023

Accepted 12 April 2023

Published 6 June 2024