微局部层范畴与$J$ -同态
Microlocal sheaf categories and the $J$-homomorphism
摘要：
设$X$是光滑流形，$\mathbf{k}$是交换（或至少是$\mathbb{E}_2$）环谱.给定一个光滑的精确拉格朗日$L\hookrightarrow T^*X$，微局部层理论（以下Kashiwara-Schapira）自然分配一个局部恒定层的范畴$L$与纤维等价的范畴的$\mathbf{k}$ -谱$\mathrm{Mod}(\mathbf{k})$。本文通过稳定高斯映射$L\rightarrow U/O$和$J$ -同态$U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$的去环证明了局部范畴系统的分类映射因子。作为应用，结合Guillermou [Gui]以前的结果，我们恢复了Abouzaid-Kragh [AbKr]关于$L\rightarrow U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$的平凡性的一个结果，当$L$是加紧的时.
英文摘要：
Let $X$ be a smooth manifold and $\mathbf{k}$ be a commutative (or at least $\mathbb{E}_2$) ring spectrum. Given a smooth exact Lagrangian $L\hookrightarrow T^*X$, the microlocal sheaf theory (following Kashiwara--Schapira) naturally assigns a locally constant sheaf of categories on $L$ with fiber equivalent to the category of $\mathbf{k}$-spectra $\mathrm{Mod}(\mathbf{k})$. We show that the classifying map for the local system of categories factors through the stable Gauss map $L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$. As an application, combining with previous results of Guillermou [Gui], we recover a result of Abouzaid--Kragh [AbKr] on the triviality of the composition $L\rightarrow U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$, when $L$ is in addition compact.
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