论文标题
投影曲折和HOPF通信
Projective twists and the Hopf correspondence
论文作者
论文摘要
给定Liouville歧管$（x，ω）中的Lagrangian（真实，复杂）的投影空间$ k_1，\ dots，k_m $满足某种同胞条件，我们表明有一个lagrangian的对应关系，可以分配一个lagrangian Sphere $ l_i $ l_i \ l_i \ li \ subset k $ supseian $ k $ y， x $，$ i = 1，\ dots m $。 我们使用HOPF通信来研究\ Emph {投影曲折}，这是类似于Dehn Twists的一类符号切除型，但从Lagrangian投射空间开始定义。当可以建立这种对象时，我们表明它与紧凑型福卡亚类别的自动等量相结合了$ \ Mathcal {f} uk（x）$，由（真实，复杂，quaternionic）投射曲折$τ_{k_i} \ inπ_{0}（0} $} { $ \ MATHCAL {F} uk（y）$的相应自动等同于dehn Twist $τ_{l_i} \inπ_{0}（\ Mathrm {symp} _ {ct} _ {ct}（y）（y Mathrm {symp} _ {ct}（y））$，对于$ i = 1，\ dots m $。 使用HOPF通信，我们为投影空间的\ emph {Clean Plumbing}中的投射曲折获得了自由世代的结果，以及有关Dehn/Projective Twists在Liouville歧管中的阳性产物的各种结果。 还使用相同的技术来表明投影旋转的哈密顿同位素类别（沿$ t^*\ mathbb {cp} $中的零部分沿$ \ mathrm {symp} _ {ct} _ {ct}（ct}（t^*\ mathbbb {cp}^n）$ n of framing $ n $ n ope $ n。 HOPF对应关系的另一个应用程序提供了两个平滑同质复杂的射射线空间的示例$ k \ simeq \ simeq \ Mathbb {cp}^n $，这些空间不接收lagrangian嵌入$（t^*\ mathbb {cp}^n，dλ_
Given Lagrangian (real, complex) projective spaces $K_1, \dots , K_m$ in a Liouville manifold $(X, ω)$ 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 $(Y, Ω)$ 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 $τ_{K_i} \in π_{0}(\mathrm{Symp}_{ct}(X))$ with the corresponding autoequivalences of $\mathcal{F}uk(Y)$ induced by the Dehn twists $τ_{L_i} \in π_{0}(\mathrm{Symp}_{ct}(Y))$, 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 $(T^*\mathbb{CP}^n, dλ_{\mathbb{CP}^n})$, for $n=4,7$.
