Projective twists and the Hopf correspondence

Torricelli, Brunella Charlotte¹

¹ Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom

Algebraic and Geometric Topology, Tome 24 (2024) no. 8, pp. 4139-4200

Résumé

Given Lagrangian (real, complex) projective spaces K1,… ⁡,Km in a Liouville manifold (X,ω) satisfying a certain cohomological condition, we show there is a Lagrangian correspondence (in the sense of Wehrheim and Woodward (2012)) that assigns a Lagrangian sphere Li ⊂ K of another Liouville manifold (Y,Ω) to any given projective Lagrangian Ki ⊂ X for i = 1,… ⁡,m.

We use the Hopf correspondence to study projective twists, a class of symplectomorphisms akin to Dehn twists, but defined starting from Lagrangian projective spaces. When this correspondence can be established, we show that it intertwines the autoequivalences of the compact Fukaya category ℱuk ⁡ (X) induced by the projective twists τKi ∈ π0(Symp ⁡ ct ⁡ (X)) with the autoequivalences of ℱuk ⁡ (Y ) induced by the Dehn twists τLi ∈ π0(Symp ⁡ ct ⁡ (Y )) for i = 1,… ⁡,m.

Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and various results about products of positive powers of Dehn/projective twists in Liouville manifolds.

The same techniques are also used to show that the Hamiltonian isotopy class of the projective twist (along the zero section in T∗ℂℙn) in Symp ⁡ ct ⁡ (T∗ℂℙn) does depend on a choice of framing for n ≥ 19. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces K ≃ ℂℙn that do not admit Lagrangian embeddings into (T∗ℂℙn,dλℂℙn) for n = 4,7.

Détail

DOI : 10.2140/agt.2024.24.4139

Keywords: symplectic topology, Dehn twists, symplectic mapping class group, product of twists, framing of twists, nearby Lagrangian conjecture