A new definition of Kuranishi space

Abstract

'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, '$\mu$-Kuranishi spaces', which form an ordinary category $\mathbf{\mu }\mathbf{Kur}$; a more complicated 'manifold' version, 'm-Kuranishi spaces', which form a weak 2-category $\mathbf{mKur}$; and an 'orbifold' version, 'Kuranishi spaces', which form a weak 2-category $\mathbf{Kur}$. These are related by an equivalence of categories $\mathbf{\mu }\mathbf{Kur}\simeq \mathrm{Ho}(\mathbf{mKur})$, where $\mathrm{Ho}(\mathbf{mKur})$ is the homotopy category of $\mathbf{mKur}$, and by a full and faithful embedding $\mathbf{mKur}\hookrightarrow \mathbf{Kur}$. We also define ($\mu$-, 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 $\mathbf{X}$ can be made into a compact Kuranishi space ${\mathbf{X}}^{\prime }$ uniquely up to equivalence in $\mathbf{Kur}$ (that is, up to isomorphism in $\mathrm{Ho}(\mathbf{Kur})$). 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 $\mathbf{X}$ with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder can be made into a Kuranishi space $\mathbf{X}$ uniquely up to equivalence in $\mathbf{Kur}$. This book is surveyed in arXiv:1510.07444.

Preliminary version of book, comments welcome. (v3) 193 pages. Revised and corrected, new material added

References in corpus (12)

• Topological and Smooth Stacks
• Set theory for category theory
• Technical details on Kuranishi structure and virtual fundamental chain
• Smooth Kuranishi atlases with isotropy
• Kuranishi homology and Kuranishi cohomology
• Lagrangian Floer theory and mirror symmetry on compact toric manifolds
• Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds
• Integration Theory for Zero Sets of Polyfold Fredholm Sections
• Log differentiable spaces and manifolds with corners
• The polyfold--Kuranishi correspondence I: A choice-independent theory of Kuranishi structures
• Kuranishi homology and Kuranishi cohomology: a User's Guide
• Virtual harmony

Cited by in corpus (8)

• Contact homology and virtual fundamental cycles
• Conjectures on counting associative 3-folds in $G_2$-manifolds
• Derived Differentiable Manifolds
• Some new homology and cohomology theories of manifolds and orbifolds
• Manifolds with analytic corners
• The inverse function theorem for curved L-infinity spaces
• Casson invariants via virtual counting
• Homotopy L-infinity spaces and Kuranishi manifolds, I: categorical structures