Hilbert manifold structures on path spaces

TL;DR

This paper establishes conditions under which the space of paths on a two-level manifold can be given a Hilbert manifold structure, facilitating Floer theory applications.

Contribution

It introduces the concept of tame two-level manifolds and demonstrates that their path spaces form Hilbert manifolds, providing a new framework for Floer homology constructions.

Findings

Path spaces on tame two-level manifolds are Hilbert manifolds.

Tame maps are closed under composition.

A new approach to defining charts on path spaces is proposed.

Abstract

In Floer theory one has to deal with two-level manifolds like for instance the space of $W^{2,2}$ loops and the space of $W^{1,2}$ loops. Gradient flow lines in Floer theory are then trajectories in a two-level manifold. Inspired by our endeavor to find a general setup to construct Floer homology we therefore address in this paper the question if the space of paths on a two-level manifold has itself the structure of a Hilbert manifold. In view of the two topologies on a two-level manifold it is unclear how to define the exponential map on a general two-level manifold. We therefore study a different approach how to define charts on path spaces of two-level manifolds. To make this approach work we need an additional structure on a two-level manifold which we refer to as tameness. We introduce the notion of tame maps and show that the composition of tame is tame again. Therefore it makes sense to introduce the notion of a tame two-level manifold. The main result of this paper shows that the path spaces on tame two-level manifolds have the structure of a Hilbert manifold.