Twisted generating functions and the nearby Lagrangian conjecture

We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is nullhomotopic for all spheres. The key tool that we introduce in order to prove this is the notion of twisted generating function, and we show that every closed exact Lagrangian can be described using such an object by extending a doubling argument developed in the setting of sheaf theory. Floer theory and sheaf theory constrain the type of twisted generating functions that can appear to a class which is closely related to Waldhausen’s tube space, and our main result follows by a theorem of Bökstedt that computes the rational homotopy type of the tube space.