Polyfold fundamental classes and globally structured multivalued perturbations

TL;DR

This paper introduces a new method for constructing globally structured multivalued perturbations of polyfold Fredholm sections, simplifying the extraction of topological invariants like the fundamental class in symplectic geometry.

Contribution

It develops a finite dimensional, explicit construction of multivalued perturbations that are globally structured and regularizing, improving the analytic and topological analysis of moduli spaces.

Findings

Perturbations are globally structured and explicitly described.

Almost all perturbations are regularizing.

Allows transparent definition of the fundamental class in certain cases.

Abstract

Work of Hofer--Wysocki--Zehnder has shown that many spaces of pseudoholomorphic curves that arise when studying symplectic manifolds may be described as the zero set of a polyfold Fredholm section. This framework has many analytic advantages. However the methods they develop to extract useful topological information from it are rather cumbersome. This paper develops a general construction of a finite dimensional space of multivalued perturbations of a polyfold Fredholm section such that almost all elements are regularizing. These perturbation are globally structured and explicitly described, and, in cases where the moduli space has no formal boundary, permit a transparent definition of its (rational Cech) fundamental class.