[论文解读] Holonomy perturbations in a cylinder, and regularity for traceless SU(2) character varieties of tangles

Christopher M. Herald, Paul Kirk|arXiv (Cornell University)|Nov 1, 2015

Geometric and Algebraic Topology参考文献 20被引用 6

一句话总结

该论文通过规范扰动，建立了同调3-球面中辫子的迹为零的SU(2)特征簇的正则性与辛结构。通过在圆柱体中沿曲线构造规范扰动，诱导戈德曼的哈密顿扭转变换流，作者证明了扰动后的特征簇是分层流形，其光滑最高层维数为$2n-3$，且其在边界2-球面上的限制映射给出到亏格-$n$曲面特征簇的辛约化中的拉格朗日子浸入，这是构建弗洛尔理论辫子不变量的关键一步。

ABSTRACT

The traceless $SU(2)$ character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the $SU(2)$ character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimension $4n-6$. For generic holonomy perturbations $π$, the traceless $SU(2)$ character variety $R_π(Y,L)$ of an $n$-stranded tangle $L$ in a homology 3-ball $Y$ is stratified with a finite singular stratum and top stratum a smooth manifold. The restriction to $R(S^2, 2n)$ is a Lagrangian immersion which preserves the cone neighborhood structure near the singular stratum. For generic holonomy perturbations $π$, the variant $R_π^ atural(Y,L)$, obtained by taking the connected sum of $L$ with a Hopf link and considering $SO(3)$ representations with $w_2$ supported near the extra component, is a smooth compact manifold without boundary of dimension $2n-3$, which Lagrangian immerses into the smooth stratum of $R(S^2,\{a_i,b_i\}_{i=1}^n)$. The proofs of these assertions consist of stratified transversality arguments to eliminate non-generic strata in the character variety and to insure that the restriction map to the boundary character variety is also generic. The main tool introduced to establish abundance of holonomy perturbations is the use of holonomy perturbations along curves $C$ in a cylinder $F imes I$, where $F$ is a closed surface. When $C$ is obtained by pushing an embedded curve on $F$ into the cylinder, we prove that the corresponding holonomy perturbation induces one of Goldman's generalized Hamiltonian twist flows on the $SU(2)$ character variety $\mathcal{M}(F)$ associated to the curve $C$.

研究动机与目标

• 通过规范扰动，建立同调3-球面中$n$-股辫的迹为零SU(2)特征簇的正则性。
• 证明扰动后的特征簇$R_{\tilde{\nu}}(Y,L)$是分层结构，其光滑最高层维数为$2n-3$，且奇异层为有限集。
• 证明边界2-球面上的限制映射给出到亏格-$n$曲面特征簇的辛约化中的拉格朗日子浸入。
• 定义变体$R_{\tilde{\nu}}^{\natural}(Y,L)$，其为光滑紧致流形，且拉格朗日子浸入边界特征簇的光滑层中。
• 为通过拉格朗日-弗洛尔同调建立明确定义的辫子不变量奠定基础，其猜想同构于Kronheimer-Mrowka的$I^{\natural}$。

提出的方法

• 在圆柱体$F \times I$中的嵌入曲线$C$上使用规范扰动，变形Chern-Simons泛函，以消除特征簇中的非典型分层。
• 证明此类规范扰动在闭曲面$F$的$SU(2)$特征簇$\mathcal{M}(F)$上诱导出戈德曼的广义哈密顿扭转变换流。
• 应用分层横截性论证，确保从$R_{\tilde{\nu}}(Y,L)$到$R(S^2, \{a_i,b_i\})$的限制映射为横截且典型。
• 通过与Hopf链的连通和构造变体$R_{\tilde{\nu}}^{\natural}(Y,L)$，以保证$w_2$-支撑的$SO(3)$表示和光滑性。
• 利用规范扰动的丰富性，证明向辛约化中拉格朗日子浸入是明确定义的，且在小扰动下稳定。
• 利用$R(S^2, \{a_i,b_i\})$作为$\mathcal{M}(F)$在哈密顿$n$-环面作用下的商的辛约化结构。

实验结果

研究问题

• RQ1 能否使用规范扰动来正则化同调3-球面中辫子的迹为零$SU(2)$特征簇？
• RQ2 扰动后特征簇在边界2-球面上的限制是否给出到曲面特征簇辛约化中的拉格朗日子浸入？
• RQ3 变体$R_{\tilde{\nu}}^{\natural}(Y,L)$是否为维数$2n-3$的光滑紧致流形？
• RQ4 能否定义$R_{\tilde{\nu}}^{\natural}(Y_1,L_1)$与$R_{\tilde{\nu}}(Y_2,L_2)$的拉格朗日-弗洛尔同调，作为辫子$(Y,L)$的明确定义不变量？
• RQ5 是否存在该拉格朗日-弗洛尔同调与Kronheimer-Mrowka的$I^{\natural}(Y,L)$之间的猜想同构？

主要发现

• 对于一般规范扰动$\pi$，特征簇$R_{\pi}(Y,L)$为分层结构，其光滑最高层维数为$2n-3$，奇异层为有限集。
• 奇异层中每一点具有同胚于$\mathbb{C}P^{n-2}$的锥邻域，表明其为复射影型的孤立奇点。
• 限制映射$R_{\pi}(Y,L) \to R(S^2, \{a_i,b_i\})$将光滑最高层拉格朗日子浸入到维数为$4n-6$的辛流形$R(S^2, \{a_i,b_i\})^{\mathbb{Z}/2}$中。
• 变体$R_{\pi}^{\natural}(Y,L)$为维数$2n-3$的光滑紧致流形，且拉格朗日子浸入边界特征簇的光滑层中。
• 当横截时，$R_{\pi}^{\natural}(Y_1,L_1)$与$R_{\pi}(Y_2,L_2)$的交集恰好是扰动Chern-Simons函数的临界集，其同胚于$R_{\pi}^{\natural}(Y,L)$。
• 本文提供了强有力的证据，表明拉格朗日-弗洛尔同调$FH(R_{\pi}^{\natural}(Y_1,L_1), R_{\pi}(Y_2,L_2))$是明确定义的辫子不变量，其猜想同构于$I^{\natural}(Y,L)$。