Projective twists and the Hopf correspondence

This dissertation is the fruit of a research project on a class of symplectic automorphisms called $projective$ $twists$. In the first part of the thesis (Chapters 3,4) we use Picard$-$Lefschetz theory to introduce a new local model for the planar projective twists \tau_{\mathbb{A}\mathbb{P}²} \in Symp_ct(T^*\mathbb{A}\mathbb{P}²), \mathbb{A} \in \{ \mathbb{R}, \mathbb{C} \}. In each case, we construct an exact Lefschetz fibration π\colon T^*\mathbb{A}\mathbb{P}²\to \mathbb{C} with three singular fibres, and define a compactly supported symplectomorphism \varphi \in Symp_ct(T^*\mathbb{A}\mathbb{P}²) on the total space. Given two disjoint Lefschetz thimbles \Delta_α,\Delta_β \subset T^*\mathbb{A}\mathbb{P}², we compute the Floer cohomology groups \operatorname{HF}(\varphi^k(\Delta_α), \Delta_β;\mathbb{Z}/2\mathbb{Z}) and verify (partially for \mathbb{C}\mathbb{P}²) that $\varphi$ is indeed isotopic to (a power of) the standard local projective twist. The constructions we present are governed by $generalised$ $lantern$ $relations$, which provide an isotopy between the global 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 study two classes of monotone Lagrangian submanifolds of (T^*\mathbb{C}\mathbb{P}², d\lambda_{T^*\mathbb{C}\mathbb{P}²}). In the second part of the thesis, starting from Chapter 5, we investigate the properties of projective twists within the symplectic mapping class group of Liouville/Stein manifolds. We define the $Hopf$ $correspondence$, a Lagrangian correspondence (in the sense of Wehrheim$-$Woodward) aimed at assigning Lagrangian spheres L₁, \dots , L_m of a Liouville manifold $(Y, \Omega)$ to given Lagrangian (real, complex) projective spaces K₁, \dots , K_m of a Liouville manifold (W, ω). When this correspondence can be established, it intertwines the (real, complex) projective twists \tau_Ki \in π₀(Symp_ct(W)) (and the induced autoequivalences of the compact Fukaya category $\operatorname{\mathcal{F}uk}(W)$) with the Dehn twists \tau_Li \in π₀(Symp_ct(Y)) (and the corresponding autoequivalences of $\operatorname{\mathcal{F}uk}(Y)$), for $i=1, \dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and a result about products of positive powers of real projective twists in Liouville manifolds. The same techniques are also used to show that in infinitely many dimensions $n$, the Hamiltonian class of the local projective twist in Symp_ct(T^*\mathbb{C}\mathbb{P}ⁿ) does depend on a choice of framing, i.e a choice of smooth parametrisation of the Lagrangian projective space used to define the twist. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces K\simeq \mathbb{C}\mathbb{P}ⁿ, that do not admit Lagrangian embeddings into (T^*\mathbb{C}\mathbb{P}ⁿ, d\lambda_{T^*\mathbb{C}\mathbb{P}ⁿ}), for $n=4,7$.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2022

Author dc:creator

• Torricelli, Brunella Charlotte

Advisor dc:contributor.advisor

• Smith, Ivan

• symplectic topology
• Dehn twist
• Floer cohomology

Licence

• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.87065

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/339647