#### Volume 19, issue 1 (2019)

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Constructing the virtual fundamental class of a Kuranishi atlas

### Dusa McDuff

Algebraic & Geometric Topology 19 (2019) 151–238
##### Abstract

Consider a space $X\phantom{\rule{0.3em}{0ex}}$, 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 ${\mathsc{S}}_{M}:M\to E\phantom{\rule{0.3em}{0ex}}$, where $E$ is a finite-dimensional vector space and the domain $M$ is an oriented, weighted branched topological manifold. Moreover, ${\mathsc{S}}_{M}$ is equivariant under the action of the global isotropy group $\Gamma$ on $M$ and $E\phantom{\rule{0.3em}{0ex}}$. This tuple $\left(M,E,\Gamma ,{\mathsc{S}}_{M}\right)$ together with a homeomorphism from ${\mathsc{S}}_{M}^{-1}\left(0\right)∕\Gamma$ to $X$ forms a single finite-dimensional model (or chart) for $X\phantom{\rule{0.3em}{0ex}}$. 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.

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