2014/09/24 by Dominic Joyce, Joyce, Dominic
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG
paper · pdf · doi:10.48550/arxiv.1409.6908
Preliminary version of book, comments welcome. (v3) 193 pages. Revised and corrected, new material added
arxiv created 2015/10/27 · arxiv updated 2015/10/28
'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of J-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a satisfactory definition of Kuranishi space has been the subject of recent debate (see e.g. arXiv:1208.1340, arXiv:1209.4410, arXiv:1510.06849). We propose three new definitions of Kuranishi space: a simple 'manifold' version, 'μ-Kuranishi spaces', which form an ordinary category \boldsymbolμ\bf Kur; a more complicated 'manifold' version, 'm-Kuranishi spaces', which form a weak 2-category \bf mKur; and an 'orbifold' version, 'Kuranishi spaces', which form a weak 2-category \bf Kur. These are related by an equivalence of categories \boldsymbolμ\bf Kur≃\rm Ho(\bf mKur), where \rm Ho(\bf mKur) is the homotopy category of \bf mKur, and by a full and faithful embedding \bf mKur\hookrightarrow\bf Kur. We also define (μ-, m-)Kuranishi spaces with boundary, and with corners. We hope our definitions will become accepted as final, replacing previous definitions. Any Fukaya-Oh-Ohta-Ono Kuranishi space \bf X can be made into a compact Kuranishi space \bf X' uniquely up to equivalence in \bf Kur (that is, up to isomorphism in \rm Ho(\bf Kur)). The same holds for topological spaces with Fukaya-Oh-Ohta-Ono 'good coordinate systems', and for McDuff and Wehrheim's 'Kuranishi atlases' in arXiv:1508.01556. A compact topological space \bf X with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder can be made into a Kuranishi space \bf X uniquely up to equivalence in \bf Kur. This book is surveyed in arXiv:1510.07444.