2020/08/06 by Brunella Charlotte Torricelli, Torricelli, Brunella Charlotte
Mathematics · #53D12 #53D35 #53D37 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2008.02758
openalex publication_date 2020/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We use Picard-Lefschetz theory to introduce a new local model for the planar projective twists τ_\mathbbAℙ2 ∈ Sympct(T^*\mathbbAℙ2), \mathbbA ∈ \ ℝ, ℂ \. In each case, we construct an exact Lefschetz fibration π\colon T^*\mathbbAℙ2→ ℂ with three singular fibres, and define a compactly supported symplectomorphism φ∈ Sympct(T^*\mathbbAℙ2) on the total space. Given two disjoint Lefschetz thimbles Δα,Δβ ⊂ T^*\mathbbAℙ2, we compute the Floer cohomology groups HF(φk(Δα), Δβ; ℤ/2ℤ) and verify (partially for ℂℙ2) that φ is indeed isotopic to (a power of) the projective twist in its local model. The constructions we present are governed by generalised lantern relations, which provide an isotopy between the total monodromy of a Lefschetz fibration and a fibred twist along an S1-fibred coisotropic submanifold of the smooth fibre. We also use these relations to generate non-exact fillings for the contact manifolds (ST^*ℂℙ2, ξstd), (ST^*ℝℙ3,ξstd), and to study two classes of monotone Lagrangian submanifolds of (T^*ℂℙ2, dλℂℙ2).