Embedded contact homology is a topological invariant of three-manifolds defined by choosing a contact structure on the manifold and studying pseudoholomorphic curves in the four-dimensional symplectization. We compute this invariant for circle bundles over Riemann surfaces (prequantization spaces).
David gave a brief introduction to ECH and then sketched the proof of his thesis result. He computed the ECH invariant for non-trivial circle bundles over surfaces by taking the prequantization contact form. Note that since ECH is a smooth invariant, one is free to take any contact structure on this.
The main result is that, with $Y$ denoting the circle bundle and $\Sigma$ the base,
\[
ECH(Y) = \oplus_{d \ge 1} Sym^d( H_*(\Sigma) )
\]
Furthermore, recall that ECH has a filtration by $H_1(Y)$, representing the homology class of the generators in question. We note that $H_1(Y) = \mathbb{Z}/|e| \oplus \mathbb{Z}^{2g+2}$, where $g$ is the genus of $\Sigma$. The torsion factor is the only relevant one. Then,
\[
ECH(Y, l) = ???see notes??
\]
The key difficulty is that for an $S^1$-invariant complex structure, one can show that the index $1$ moduli spaces are empty... however, these $S^1$-invariant acs are not generic. David solves this problem by first showing the moduli spaces are empty for a domain dependent perturbation. He is able to carry out the Cieliebak-Mohnke/Fabert methods, even though he has genus, essentially because the formation of a ghost bubble with genus is codimension at least 2. (This uses the fact we are in dimension 4 quite crucially). Then, the next tricky point is to show that the count of embedded curves remains $0$ as he deforms the $S^1$-invariant domain-dependent almost complex structure to a generic domain-independent complex structure. This relies on a theorem of Hutchings and Taubes about bifurcations, since David can rule out connectors in the compactness argument.
In thinking about this, I wonder if there is a way of making sense of this as some statement about moduli spaces with an $S^1$ action. This is an idea Jian He explained to me, and is the heart of the geometric picture behind his work on descendants. The general principle (stolen from localization in algebraic geometry) is that if you have a non-generic $J$ and a moduli space of which a connected component has a free $S^1$ action, then after perturbation, this connected component still can be arranged to have an $S^1$ action, in particular then, not contribute to the count of isolated solutions. If one could understand this principle in dimension 4 for embedded curves, it might then be possible to extend David's result to the case of a trivial circle bundle. (Ironically, that case seems harder.)
