2012/08/07 by Dusa McDuff, McDuff, Dusa, Katrin Wehrheim +1 · 4 citations
Mathematics · #53D45 #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.SG #msc:53D45
paper · pdf · doi:10.48550/arxiv.1208.1340
Version 2 added material in 2.6 and the appendix. Versions 3 and 4 clarify comments on Siebert's work and changed the title and name of the main object to Kuranishi atlases. Version 5 made further small corrections to citations. Version 6 split off some parts. Versions 7 reverted to Version 5. Version 8 added arxiv numbers in abstract
openalex publication_date 2012/08/07 · arxiv created 2015/08/11 · arxiv updated 2015/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Kuranishi structures were introduced to symplectic topology by Fukaya and Ono and recently refined by Joyce, in order to extract homological data from compactified moduli spaces of holomorphic maps in cases where geometric regularization approaches such as perturbations of the almost complex structure do not yield a smooth structure on the moduli space. We give a general survey of regularization techniques in symplectic topology, pointing to some general analytic issues, and discussing some specific topological issues of the Kuranishi approach. In the main body of the paper we provide an abstract framework of Kuranishi atlases which separates the analytic and topological issues. Throughout, we focus on the most fundamental issues, which are already present in applying virtual transversality techniques to moduli spaces of holomorphic spheres without nodes or nontrivial isotropy. This is the reinstated 2013 version of this survey and sample construction. A generalized version of the topological theory is now available under 'The topology of Kuranishi atlases' arxiv:1508.01844, with the survey parts and VMC construction updated in 'The fundamental class of smooth Kuranishi atlases with trivial isotropy' arxiv:1508.01560.