C0-rigidity in contact topology via microlocal sheaf theory

Contact homeomorphisms are points in the closure of the (compactly supported) contactomorphism group in the homeomorphism group under the C0-topology. Recently, Dimitroglou Rizell and Sullivan showed that the images of closed Legendrians under contact homeomorphisms are still Legendrian if they are smooth. One can ask further questions: (1) whether images of coisotropics are still coisotropic if they are smooth, and (2) whether Maslov data and Floer theory invariants of closed Legendrians are preserved. We provide an approach of understanding such questions in cosphere bundles (with the standard contact structures) using microlocal sheaf theory. This is joint work with Tomohiro Asano, Yuichi Ike and Christopher Kuo.