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 $\tau AP2\in \mathrm{Symp}ct(T\ast AP2), A\in \{R,C\}$. In each case, we construct an exact Lefschetz fibration $\pi :T\ast AP2\to C$ with three singular fibres, and define a compactly supported symplectomorphism $\phi \in \mathrm{Symp}ct(T\ast AP2)$ on the total space.

Given two disjoint Lefschetz thimbles $\Delta \alpha ,\Delta \beta \subset T\ast AP2$, we compute the Floer cohomology groups $\mathrm{HF}(\phi k(\Delta \alpha ),\Delta \beta ;Z/2Z)$ and verify (partially for $CP2$) that $\phi$ 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 $(T\ast CP2,d\lambda T\ast CP2)$.

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 $L1,\dots ,Lm$ of a Liouville manifold $(Y,\Omega )$ to given Lagrangian (real, complex) projective spaces $K1,\dots ,Km$ of a Liouville manifold $(W,\omega )$. When this correspondence can be established, it intertwines the (real, complex) projective twists $\tau Ki\in \pi 0(\mathrm{Symp}ct(W))$ (and the induced autoequivalences of the compact Fukaya category $F\mathrm{uk}(W)$) with the Dehn twists $\tau Li\in \pi 0(\mathrm{Symp}ct(Y))$ (and the corresponding autoequivalences of $F\mathrm{uk}(Y)$), 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 $\mathrm{Symp}ct(T\ast CPn)$ 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 $K\simeq CPn$, that do not admit Lagrangian embeddings into $(T\ast CPn,d\lambda T\ast CPn)$, for $n=4,7$.