Microlocal sheaf categories and the J-homomorphism

2020/04/29 by Xin Jin, Jin, Xin · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2004.14270

openalex publication_date 2020/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a smooth manifold and k be a commutative (or at least 𝔼₂) 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 k-spectra Mod(k). We show that the classifying map for the local system of categories factors through the stable Gauss map L→ U/O and the delooping of the J-homomorphism U/O→ BPic(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→ U/O→ BPic(S), when L is in addition compact.

Citations

• Quantization of conic Lagrangian submanifolds of cotangent bundles
• Sheaf quantization in Weinstein symplectic manifolds

Cited by

• C⁰-rigidity of Legendrians and coisotropics via sheaf quantization
• Twisted generating functions and the nearby Lagrangian conjecture

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