Floerfolds and Floer functions
Urs Frauenfelder, Joa Weber
TL;DR
The article introduces Floerfolds and Floer functions to make the Hessian in Floer theory intrinsic under coordinate changes. It develops two-level and three-level strong scale differentiability, defines Floer maps and Floeromorphisms, and constructs Floer atlases culminating in Floerfolds. The main result shows that pulling back a Floer function along a Floeromorphism yields another Floer function, making the Floer-function property intrinsic on a Floerfold. As an application, the loop space of a manifold is endowed with a Floerfold structure via chartwise Floeromorphisms, illustrating the framework's alignment with loop-space Floer theories and its potential for uniform treatment of Floer-type invariants.
Abstract
In this article we introduce the notion of Floer function which has the property that the Hessian is a Fredholm operator of index zero in a scale of Hilbert spaces. Since the Hessian has a complicated transformation under chart transition, in general this is not an intrinsic condition. Therefore we introduce the concept of Floerfolds for which we show that the notion of Floer function is intrinsic.