Equivariant moduli problems, branched manifolds, and the Euler class

Abstract

The purpose of this paper is to explain the equivariant Euler class associated to an oriented G-equivariant Fredholm section S : B → E of a Hilbert space bundle over a Hilbert manifold. The key hypotheses are that the Lie group G is compact, the isotropy subgroups are finite, and the zero set of the section is compact. The present paper is motivated by our joint work with Gaio [9] on invariants of Hamiltonian group actions. In this work the Fredholm section arises from a version of the vortex equations, where the target space is a symplectic manifold with a Hamiltonian G-action [8, 19, 20]. In many interesting cases the resulting moduli spaces are compact and so the results of the present paper can be applied. Other examples of Fredholm sections with compact zero sets are the Seiberg–Witten equations over a four-manifold [25] or the harmonic map equations when the target space is a negatively curved manifold (see e.g. [14]). This is in sharp contrast to the Gromov–Witten invariants of general (compact) symplectic manifolds [11, 16, 17, 22] and to the Donaldson invariants of smooth four-manifolds [10], where the moduli spaces are noncompact and the compactifications are the source of some major difficulties of the theory. Since the unperturbed moduli space is compact, our framework is considerably simpler than the one required for the construction of the Gromov–Witten invariants. Our exposition follows closely the work of Li–Robbin–Ruan [16]. In the case G = {1l} similar results were proved in [6, 12, 21]. In [12] Fulton proved that, if B is a finite dimensional complex manifold, E → B is a holomorphic vector bundle, and S : B → E is a holomorphic section, then the zero set M := S−1(0) carries a fundamental cycle (in singular homology) which is Poincare dual to the Euler class. This was extended to the infinite ∗Supported by National Science Foundation grant DMS-0072267