[Paper Review] Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds

Helmut Hofer, Kris Wysocki+1 more|arXiv (Cornell University)|Jul 11, 2014

Geometric and Algebraic Topology23 references17 citations

TL;DR

This paper establishes the foundational theory of Sc-Fredholm sections within M-polyfolds—generalized Banach manifolds with varying local dimension—by introducing a scale calculus framework and developing a robust Fredholm theory for nonlinear elliptic PDEs with compactness and transversality issues. The key contribution is a systematic approach to handling singular limits in symplectic field theory, including a new implicit function theorem and a consistent determinant bundle construction for Fredholm operators on these spaces.

ABSTRACT

The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate local symmetries. The whole package provides a functional analytic framework to deal with compactness and transversality issues as they occur in moduli problems of symplectic geometry. Applications of the theory cover Floer-type theories as they occur in symplectic geometry. M-polyfolds and the more general polyfolds are smooth spaces which can be finite-dimensional as well as infinite-dimensional. In applications of interest they in general have locally varying dimensions. Despite the fact that the spaces are much more general than Banach manifolds a nonlinear Fredholm theory with the usual features is possible (Sard-Smale type perturbation theory). This generalized Fredholm theory can be applied to classes of nonlinear elliptic problems which show bubbling-off phenomena but allow for certain kind of compactifications.

Motivation & Objective

• To develop a generalized differential and Fredholm theory for nonlinear elliptic PDEs on spaces with varying dimension, particularly in symplectic field theory (SFT).
• To address compactness and transversality problems arising from bubbling and degenerations in pseudoholomorphic curve theory.
• To introduce a new class of spaces—M-polyfolds—generalizing manifolds via retracts on decreasing Banach space sequences.
• To establish a consistent Fredholm theory with well-defined determinant bundles and orientation propagation for proper Sc-Fredholm sections.
• To provide a reusable analytical framework ('Black Boxes') for constructing polyfolds in applications, enhancing transparency and reducing error-prone proofs.

Proposed method

• Introduce sc-structures on Banach spaces as decreasing sequences of Banach spaces with compact inclusions and dense intersection, forming the basis of scale calculus.
• Define M-polyfolds as locally modeled on retracts of sc-Banach spaces, generalizing manifolds to allow varying local dimension and boundary/corners.
• Develop Sc-Fredholm sections as sections of strong bundles over M-polyfolds with finite-dimensional kernel and cokernel, satisfying a sc-Fredholm condition.
• Construct a chain of exact sequences involving kernels and cokernels to define the determinant bundle of a Fredholm operator in the sc-setting.
• Prove an implicit function theorem in partial quadrants of R^n and extend it to M-polyfolds, enabling local solution theory for nonlinear equations.
• Define orientation propagation via local determinant bundles and establish invariants for proper Sc-Fredholm sections using the determinant bundle.

Experimental results

Research questions

• RQ1 How can Fredholm theory be generalized to spaces with varying local dimension and boundary structure, such as those arising in SFT?
• RQ2 What is the correct analytical framework to handle compactness and transversality in moduli spaces of pseudoholomorphic curves with nodal or bubbling limits?
• RQ3 How can the determinant bundle of a Fredholm operator be consistently defined and transported across different levels of a scale structure?
• RQ4 Can a consistent implicit function theorem be formulated in the context of M-polyfolds, particularly in spaces with corners and partial quadrants?
• RQ5 How can orientations and invariants be systematically propagated and computed for proper Sc-Fredholm sections in this generalized setting?

Key findings

• The paper establishes a well-defined Sc-Fredholm theory on M-polyfolds, where Fredholm sections have finite-dimensional kernel and cokernel and satisfy a scale-regularity condition.
• An implicit function theorem is proven in partial quadrants of R^n and extended to M-polyfolds, enabling local solution theory for nonlinear equations with boundary and corner constraints.
• The determinant bundle of a Fredholm operator is constructed via exact sequences involving kernels and cokernels, with a canonical isomorphism between determinant bundles of related operators.
• The composition of the determinant isomorphisms γ^Q_T and γ^P_QT is shown to equal γ^P_T, proving consistency of the determinant bundle construction across operator compositions.
• The paper proves that the determinant bundle is well-defined and compatible with composition of operators, as demonstrated by the equality γ^P_QT ∘ γ^Q_T = γ^P_T on the level of determinant elements.
• The theory supports the construction of invariants for proper Sc-Fredholm sections through the determinant bundle, providing a foundation for symplectic invariants in SFT and related theories.