The Steenrod problem for orbifolds and polyfold invariants as intersection numbers

Abstract

The Steenrod problem for closed orientable manifolds was solved completely by Thom. Following this approach, we solve the Steenrod problem for closed orientable orbifolds, proving that the rational homology groups of a closed orientable orbifold have a basis consisting of classes represented by suborbifolds whose normal bundles have fiberwise trivial isotropy action. Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, has yielded a well-defined Gromov-Witten invariant via the regularization of moduli spaces. As an application, we demonstrate that the polyfold Gromov-Witten invariants, originally defined via branched integrals, may equivalently be defined as intersection numbers against a basis of representing suborbifolds.