A Polyfold Proof of the Arnold Conjecture

TL;DR

This paper provides a detailed polyfold-based proof of the homological Arnold conjecture for nondegenerate Hamiltonians on closed symplectic manifolds, utilizing advanced moduli space regularization techniques.

Contribution

It introduces a comprehensive polyfold framework for moduli spaces of pseudoholomorphic curves, extending the proof of the Arnold conjecture to general closed symplectic manifolds.

Findings

Successful construction of a coherent polyfold description for moduli spaces

Development of a regularization theorem for moduli spaces with evaluation maps

Complete proof of the homological Arnold conjecture in this setting

Abstract

We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds $M$ via a direct Piunikhin-Salamon-Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $\mathbb{C}\mathbb{P}^1\times M$ to $\mathbb{C}^+ \times M$ and $\mathbb{C}^-\times M$, as developed by Fish-Hofer-Wysocki-Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.