Projective twists and the Hopf correspondence

Abstract

Given Lagrangian (real, complex) projective spaces $K_1, \dots , K_m$ in a Liouville manifold $(X, \omega)$ satisfying a certain cohomological condition, we show there is a Lagrangian correspondence that assigns a Lagrangian sphere $L_i \subset K$ of another Liouville manifold $(Y, \Omega)$ to any given projective Lagrangian $K_i \subset X$, $i=1, \dots m$. We use the Hopf correspondence to study \emph{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 $\mathcal{F}uk(X)$ induced by the (real, complex, quaternionic) projective twists $\tau_{K_i} \in \pi_{0}(\mathrm{Symp}_{ct}(X))$ with the corresponding autoequivalences of $\mathcal{F}uk(Y)$ induced by the Dehn twists $\tau_{L_i} \in \pi_{0}(\mathrm{Symp}_{ct}(Y))$, for $i=1, \dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a \emph{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 $T^*\mathbb{CP}$) in $\mathrm{Symp}_{ct}(T^*\mathbb{CP}^n)$ does depend on a choice of framing, for $n\geq19$. Another application of the Hopf correspondence delivers two examples of smooth homotopy complex projective spaces $K\simeq \mathbb{CP}^n$ that do not admit Lagrangian embeddings into $(T^*\mathbb{CP}^n, d\lambda_{\mathbb{CP}^n})$, for $n=4,7$.