Constructing the virtual fundamental class of a Kuranishi atlas

Abstract

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 S −1 M (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.1 See [C1, C2] for a weak form of these requirements.CONSTRUCTING THE VIRTUAL FUNDAMENTAL CLASS OF A KURANISHI ATLAS 3 axiom. Instead of gluing the chart domains together to form a topological space |K|, Pardon works with K-homotopy sheaves of (co)chain complexes defined on homotopy colimits of spaces that are obtained from the chart domains. This gives a flexible way of assembling local homological information into a global object. Though this approach may be useful in many contexts, it is hard for a nonexpert in sheaf theory to understand where the technical difficulties are, and what actually has to be checked to ensure that the method works in any particular case. This becomes an issue if one wants to extend the method to cases (such as Hamiltonian Floer theory, or symplectic field theory) in which one must deal with a family of related moduli spaces and so should work on the chain level. The current paper was prompted by the desire to develop a different approach, that would replace Pardon's sophisticated sheaf theory by more elementary arguments that yet do not require smoothness.This note only considers the simplest case, appropriate to Gromov-Witten theory, in which the aim is to construct a homology class [X] vir K ∈Ȟ d (X; Q). Working with Pardon's submersion axiom, we define a consistent thickening of the domains of the atlas charts to make them all have the same dimension. In the case with trivial isotropy, one thereby constructs an oriented topological manifold M of dimension D := d + dim E A , together with a map S M : M → E A whose zero set can be identified with X. If the isotropy is nontrivial, M is a branched manifold with a weighting function Λ and a global action of the total isotropy group Γ A , and there is a homeomorphismis the union of two circles, each of weight 1 2 , identified along a closed subarc A, so that the points x ∈ A have weight Λ(x) = 1, while the others all have weight Λ(x) = 1 2 . See also §1.4.) Here is the first main result. (See Theorem 1.3.4 for a more precise statement.)Theorem A: Let K be a d-dimensional Kuranishi atl...