2024/05/09 by Joonghyun Bae, Jungsoo Kang, Sungho Kim
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Cellular homology #Floer homology #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematics #Morse homology #Pure mathematics #Sequence (biology) #Symplectic geometry
paper · pdf · doi:10.1007/s00208-024-02878-w
openalex publication_date 2024/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Abstract Let Y be a prequantization bundle over a closed spherically monotone symplectic manifold Σ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> . 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, Σ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> is a symplectic hyperplane section of a closed symplectic manifold X satisfying a certain monotonicity condition; in this case, X ∖ Σ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mo></mml:mo><mml:mi>Σ</mml:mi></mml:mrow></mml:math> is a Liouville filling of Y . Second, the minimal Chern number of Σ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> is greater than one, which is the case where the Rabinowitz Floer homology of the symplectization \mathbb R × Y <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>×</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math> is defined. In both cases, we construct a Gysin-type exact sequence connecting the Rabinowitz Floer homology of X∖ Σ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mo></mml:mo><mml:mi>Σ</mml:mi></mml:mrow></mml:math> or \mathbb R × Y <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>×</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math> and the quantum homology of Σ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> . 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.