Constructing the virtual fundamental class of a Kuranishi atlas

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 ${\mathcal{S}}_M:M\to E$, where $E$ is a finite-dimensional vector space and the domain $M$ is an oriented, weighted branched topological manifold. Moreover, ${\mathcal{S}}_M$ is equivariant under the action of the global isotropy group $\mathrm{\Gamma }$ on $M$ and $E$. This tuple $\left(M,E,\mathrm{\Gamma },{\mathcal{S}}_M\right)$ together with a homeomorphism from ${\mathcal{S}}_M^{-1}\left(0\right)∕\mathrm{\Gamma }$ 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.