Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence

Let $Y$ be a prequantization bundle over a closed spherically monotone symplectic manifold $\Sigma$. Adapting an idea due to Diogo and Lisi, we study a split version of Rabinowitz Floer homology for $Y$ in the following two settings. First, $\Sigma$ is a symplectic hyperplane section of a closed symplectic manifold $X$ satisfying a certain monotonicity condition; in this case, $X \setminus \Sigma$ is a Liouville filling of $Y$. Second, the minimal Chern number of $\Sigma$ is greater than one, which is the case where the Rabinowitz Floer homology of the symplectization $\mathbb{R} \times Y$ is defined. In both cases, we construct a Gysin-type exact sequence connecting the Rabinowitz Floer homology of $X\setminus\Sigma$ or $\mathbb{R} \times Y$ and the quantum homology of $\Sigma$. As applications, we discuss the invertibility of a symplectic hyperplane section class in quantum homology, the isotopy problem for fibered Dehn twists, the orderability problem for prequantization bundles, and the existence of translated points. We also provide computational results based on the exact sequence that we construct.