Polyfolds: A First and Second Look (1210.6670v3)

Abstract: Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory.

• The paper introduces scale differentiability to refine classical calculus for infinite-dimensional reparametrization settings.
• The paper details sc-retractions and splicing cores as local models that regularize singular solutions in moduli spaces.
• The paper establishes an M-polyfold regularization theorem that ensures smooth compactness and rigorous definition of symplectic invariants.

Overview of "Polyfolds: A First and Second Look" by Fabert, Fish, Golovko, and Wehrheim

The paper "Polyfolds: A First and Second Look" by Oliver Fabert, Joel W. Fish, Roman Golovko, and Katrin Wehrheim presents a comprehensive exploration of polyfold theory, an innovative framework developed by Hofer, Wysocki, and Zehnder to address analytical challenges in symplectic topology. The paper provides a detailed exposition of the core concepts and technical machinery of polyfold theory, including definitions, applications, and insights into its broader implications for several geometric elliptic partial differential equations (PDEs) that arise in the paper of moduli spaces of pseudoholomorphic curves.

Polyfold Theory and Its Application

Polyfold theory is an abstract analytical framework designed to systematically address the issues of "compactification" and "transversality" that are commonly encountered in symplectic topology. The theory introduces a novel notion of smoothness on Banach spaces and employs local models for differential geometry, alongside a nonlinear Fredholm theory in this new context. The primary objective of the polyfold approach is to provide a unified framework that simplifies and generalizes the transversality problem for moduli spaces, which, in turn, facilitates rigorous proofs and computations of invariants in symplectic topology.

Main Contributions

1. Scale Calculus: The authors introduce the concept of scale differentiability, which refines the classical notions of differentiability tailored to settings involving infinite-dimensional reparametrization groups. Scale calculus supports the pertinent structure necessary for defining Fredholm sections in polyfold bundles, emphasizing the regularizing properties of retractions on Banach manifolds.
2. Sc-Retractions and Splicing Cores: The paper details the fundamental idea of sc-retractions, which serve as local models for polyfolds. These constructs permit the accommodation of broken or nodal curves within the ambient space, effectively transforming traditionally singular solutions into regular ones. Splicing cores facilitate the seamless integration of pregluing maps, further bridging gaps between distinct solutions.
3. M-Polyfolds and Fredholm Theory: The authors establish the structure of M-polyfolds, discussing their boundary and corner stratifications essential for analytical and topological investigations. The Fredholm framework is expanded to encompass nonlinear PDEs, incorporating the distinct scale-smoothness properties of polyfolds to handle the intricate behavior of solutions under reparametrization.
4. Polyfold Regularization Theorem: The central theoretical result is the polyfold regularization theorem, which asserts that for any proper Fredholm section of a strong polyfold bundle, there exists a class of perturbations yielding a smooth, compact manifold. This is achieved while ensuring transversality and suffices to define Gromov-Witten invariants rigorously.

Practical and Theoretical Implications

The significance of polyfold theory extends beyond simplifying the transversality problem. It offers a robust methodology for analyzing infinite-dimensional moduli spaces, providing a scalable and unified approach for future research in symplectic geometry and related fields. The implications of this work are profound, not only for foundational research in topology and geometry but also potentially influencing computational approaches to complex geometric problems.

Future Directions

The future development of polyfold theory is anticipated to address complex moduli problems such as those arising in gauge theory and higher-dimensional symplectic manifolds. Enhancements to scale calculus and further exploration of splicing and retraction notions may offer deeper insights into broader applications, including dynamically evolving systems and mathematical physics.

In conclusion, the paper lays a robust groundwork for polyfold theory, providing a detailed exposition suitable for experienced researchers in symplectic topology and offering pivotal insights that promise to significantly advance the analysis of moduli spaces in geometric research.