NCSGS Higher homotopy groups of spaces of eventually linear Hamiltonians

Stanford meeting of Northern California Symplectic Geometry Seminar

Abstract: The community hasn't done too much to explore the *continuous* functoriality of wrapped Fukaya categories{for example, are Fukaya-categorical invariants sensitive to the topology of the space of isotopies? We show that the answer is yes, and that this leads to some concrete applications: By judiciously applying infinity-categorical machinery (but only standard Floer-theoretic tools) we prove that for any Liouville sector, the group of eventually-linear Hamiltonian automorphisms acts on the wrapped Fukaya category. In particular, higher homotopy groups of the eventually-linear Hamiltonian group map to the higher Hochschild cohomology groups of the wrapped Fukaya category. In the case of cotangent bundles, this allows us to prove the existence of some non-trivial higher homotopy groups of the eventually-linear Hamiltonian automorphism group through some rational homotopy theory computations. This is joint work with Yong-Geun Oh.