Morse-Bott Split Symplectic Homology

TL;DR

This paper develops a Morse-Bott approach to symplectic homology for Liouville domains with prequantization boundary, establishing transversality and counting results for cascades of Floer solutions.

Contribution

It introduces a new chain complex framework for symplectic homology using Morse-Bott techniques and proves transversality and finiteness results for cascade moduli spaces.

Findings

Constructed a chain complex for symplectic homology using Morse-Bott cascades.

Proved transversality for moduli spaces of cascades and holomorphic spheres.

Showed that the differential counts only finitely many cascade types under monotonicity.

Abstract

We introduce a chain complex associated to a Liouville domain $(\overline{W}, d\lambda)$ whose boundary $Y$ admits a Boothby--Wang contact form (i.e. is a prequantization space). The differential counts cascades of Floer solutions in the completion $W$ of $\overline{W}$, in the spirit of Morse--Bott homology (as in work of Bourgeois, Frauenfelder arXiv:math/0309373 and Bourgeois-Oancea arXiv:0704.1039). The homology of this complex is the symplectic homology of the completion $W$. We identify a class of simple cascades and show that their moduli spaces are cut out transversely for generic choice of auxiliary data. If $X$ is obtained by collapsing the boundary along Reeb orbits and $\Sigma$ is the quotient of $Y$ by the $S^1$-action induced by the Reeb flow, we also establish transversality for certain moduli spaces of holomorphic spheres in $X$ and in $\Sigma$. Finally, under monotonicity assumptions on $X$ and $\Sigma$, we show that for generic data, the differential in our chain complex counts elements of moduli spaces that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential.