Thin Compactifications and Virtual Fundamental Classes

We define a notion of virtual fundamental class that applies to moduli spaces in gauge theory and in symplectic Gromov-Witten theory. For universal moduli spaces over a parameter space, the virtual fundamental class specifies an element of the Čech homology of the compactification of each fiber; it is defined if the compactification is “thin” in the sense that its boundary has homological codimension at least two. The moduli spaces that occur in symplectic Gromov-Witten theory and in many gauge theories are often orbifolds that can be compactified by adding “boundary strata” of lower dimension. Often, it is straightforward to prove that each stratum is a manifold, but more difficult to prove “collar theorems” that describe the structure of neighborhoods of the boundary strata. The lack of collar theorems is an impediment to applying singular homology to the compactified moduli space, and in particular to defining its fundamental homology class. The purpose of this paper is to show that collar theorems are not needed to define a virtual fundamental class as an element of Čech homology. Indeed, existing results in the literature are enough to prove the existence of virtual fundamental classes in some cases. There are two classes of homology theories, exemplified by singular homology and by Čech homology. We will use two Čech-type theories: Čech and Steenrod homologies. These have two features that make them especially well-suited for applications to compactified moduli spaces: (1) For any closed subset A of a locally compact Hausdorff space X, the relative group Hp(X,A) is identified with Hp(X ∖A). As Massey notes [Ma2, p. vii]: . . . one does not need to consider the relative homology or cohomology groups of a pair (X,A); the homology or cohomology groups of the complementary space X−A serve that function. In many ways these “single space” theories are simpler than the usual theories involving relative homology groups of pairs. The analog of the excision property becomes a tautology, and never needs to be considered. It makes possible an intuitive and straightforward discussion of the homology and cohomology of a manifold in the top dimension, without any assumption of differentiability, triangulability, compactness, or even paracompactness! (2) Čech homology satisfies a “continuity property” ((1.9) below) that allows one to define virtual fundamental classes by a limit process. We briefly review Steenrod homology in Section 1. Then, in Section 2, we apply Property (1) to manifolds M that admit compactifications M for which the “boundary” M ∖M is “thin” in the sense that it has homological codimension at least 2. There may be many such The research of E.I. was partially supported by a Simons Foundation fellowship, and that of T.P. by the NSF grant DMS-1011793. 1 compactifications. If M is oriented and d-dimensional, every thin compactification carries a fundamental class [M] ∈ Hd(M ;Z) in Steenrod homology. This class pushes forward under maps M → Y that extend continuously over M , and many properties of the fundamental classes of manifolds continue to hold. We then enlarge the setting by considering thinly compactified families. For this we start with a Fredholm map M