Constructing the virtual fundamental class of a Kuranishi atlas

TL;DR

This paper presents a method to define the virtual fundamental class of a space represented by a Kuranishi atlas using a finite dimensional model, accommodating topological and smooth structures.

Contribution

It introduces a construction of the virtual fundamental class via a finite dimensional model that works under topological conditions and extends to sc-Fredholm operators with sc-smooth partitions.

Findings

Defines the virtual fundamental class using a finite dimensional model.

Provides a construction applicable under topological index conditions.

Outlines a direct construction for sc-Fredholm zero sets using sc-smooth partitions.

Abstract

Consider a space $X$, such as a compact space of $J$-holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of $X$ by representing $X$ via the zero set of a map $S_M: M\to E$, where $E$ is a finite dimensional vector space and the domain $M$ is an oriented, weighted branched topological manifold. Moreover, $S_M$ is equivariant under the action of the global isotropy group $\Gamma$ on $M$ and $E$. This tuple $(M,E, \Gamma, S_M)$ together with a homeomorphism $S_M^{-1}(0)/\Gamma \to X$ forms a single finite dimensional model (or chart) for $X$. The construction assumes only that the atlas satisfies a topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. However if $X$ is presented as the zero set of an sc-Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold $M$ that uses an sc-smooth partition of unity.