论文标题
微局部分层类别和$ j $ - 肌形态
Microlocal sheaf categories and the $J$-homomorphism
论文作者
论文摘要
令$ x $为平滑的多种流形，$ \ mathbf {k} $为换向（或至少$ \ mathbb {e} _2 $）环频谱。给定一个流畅的Lagrangian $ l \ hookrightarrow t^*X $，微局部捆捆理论（遵循kashiwara-schapira）自然会在$ l $上分配$ l $的本地恒定类别，其纤维等效于$ \ mathbf {k} $ \ spectra $ \ mathrm {mod mathrm {mathrm {mathrm {mathrm {mathrm {mathrm {mathbf}（\ mathrm {mathrm {mathrm {math}（我们表明，通过稳定的高斯地图$ l \ rightarrow u/o $和$ j $ -J $ -HOMOMORPHISM $ u/o \ rightarrow b \ Mathrm {pic}（\ mathbf {s}）$ delooping的分类映射和$ J $ -HOMORPHISM $ u/o \ rightarrow b \ rightArrow b \ rightArrow b \ rightArrow b \ rightarrow b \ rightarrow b \ rightArrow $ u/o $的分类地图（\ mathbf {s}）$。作为一个应用程序，与Guillermou [Gui]的先前结果相结合，我们恢复了Abouzaid-Kragh [Abkr]的结果，涉及组成的琐碎性$ l \ rightArrow U/O \ rightArrow u/o \ rightArrow b \ Mathrm {pic}（pic}（pic}）
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.
