A General Fredholm Theory III: Fredholm Functors and Polyfolds

TL;DR

This paper develops a comprehensive Fredholm theory for polyfolds, a new class of ambient spaces with locally varying dimensions, enabling advanced analysis in symplectic geometry and related fields.

Contribution

It introduces a general nonlinear Fredholm framework for polyfolds, facilitating the study of complex geometric problems with variable local dimensions.

Findings

Established a Fredholm theory applicable to polyfolds with varying local dimensions.

Provided a functional analytic foundation for nonlinear problems in symplectic geometry.

Enabled rigorous analysis of Gromov-Witten, Floer, and Symplectic Field Theory problems.

Abstract

We describe a very general (nonlinear) Fredholm theory for a new class of ambient spaces, called polyfolds. The basic feature of these new spaces is that in general they may have locally varying dimensions. These new spaces are needed for a functional analytic treatment of nonlinear problems involving analytic limiting behavior. This theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory.