Xin Jin: Microlocal sheaf categories and the J-homomorphism

Xin Jin (Boston College): Microlocal sheaf categories and the J-homomorphism (7/10/20 7pm GMT)

The theory of microlocal sheaves, developed by Kashiwara--Schapira, has found many applications in the study of symplectic topology. For a smooth Lagrangian L in a cotangent bundle of a smooth manifold and a commutative ring spectrum k, one can associate a sheaf of microlocal categories, which is locally constant with fiber equivalent to Mod(k). It admits a classifying map L---) BPic(k). We will show that the classifying map factors through the Gauss map L---) U/O and the delooping of the J-homomorphism U/O---) BPic(S), where S is the sphere spectrum. As an application, combining with previous results of Guillermou, we show that if L is a compact smooth exact Lagrangian, then the classifying map is homotopically trivial, recovering a result of Abouzaid--Kragh.

00:18:39 Oleg Lazarev: Can you distinguish coisotropics (that are higher dimensional, not Lagrangian) this way?