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 $S_M:M\to E$, 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.

71 pages, 5 figures; v 3 is revised after a referee report

References in corpus (5)

• Smooth Kuranishi atlases with isotropy
• The polyfold--Kuranishi correspondence I: A choice-independent theory of Kuranishi structures
• The topology of Kuranishi atlases
• Genus zero Gromov-Witten axioms via Kuranishi atlases
• Smoothness of Kuranishi atlases on Gromov-Witten moduli spaces

Cited by in corpus (2)

• Contact homology and virtual fundamental cycles
• Bounds on spectral norms and barcodes