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Constructing the virtual fundamental class of a Kuranishi atlas

Dusa McDuff

Algebraic & Geometric Topology 19 (2019) 151–238

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 SM: M E, where E is a finite-dimensional vector space and the domain M is an oriented, weighted branched topological manifold. Moreover, SM is equivariant under the action of the global isotropy group Γ on M and E. This tuple (M,E,Γ,SM) together with a homeomorphism from SM1(0)Γ 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.

virtual fundamental cycle, virtual fundamental class, pseudoholomorphic curve, Kuranishi structure, weighted branched manifold, polyfold
Mathematical Subject Classification 2010
Primary: 18B30, 53D35, 53D45, 57R17, 57R95
Received: 16 October 2017
Revised: 5 September 2018
Accepted: 15 September 2018
Published: 6 February 2019
Dusa McDuff
Mathematics Department
Barnard College
Columbia University
New York, NY
United States