Consider a space , such as a compact space of –holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of by representing via the zero set of a map , where is a finite-dimensional vector space and the domain is an oriented, weighted branched topological manifold. Moreover, is equivariant under the action of the global isotropy group on and . This tuple together with a homeomorphism from to forms a single finite-dimensional model (or chart) for . 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 is presented as the zero set of an –Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold that uses an –smooth partition of unity.
"Constructing the virtual fundamental class of a Kuranishi atlas." Algebr. Geom. Topol. 19 (1) 151 - 238, 2019. https://doi.org/10.2140/agt.2019.19.151