Embedded contact homology of prequantization bundles

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].