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2019 Constructing the virtual fundamental class of a Kuranishi atlas
Dusa McDuff
Algebr. Geom. Topol. 19(1): 151-238 (2019). DOI: 10.2140/agt.2019.19.151


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 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 Γ on M and E . This tuple ( M , E , Γ , S M ) together with a homeomorphism from S M 1 ( 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.


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Dusa McDuff. "Constructing the virtual fundamental class of a Kuranishi atlas." Algebr. Geom. Topol. 19 (1) 151 - 238, 2019.


Received: 16 October 2017; Revised: 5 September 2018; Accepted: 15 September 2018; Published: 2019
First available in Project Euclid: 12 February 2019

zbMATH: 07053573
MathSciNet: MR3910580
Digital Object Identifier: 10.2140/agt.2019.19.151

Primary: 18B30, 53D35, 53D45, 57R17, 57R95

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.19 • No. 1 • 2019
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