I will give a gentle introduction to the study of smooth geometric spaces with finite isotropy - namely, orbifolds and polyfolds.
The key to actually proving many results in polyfold Gromov-Witten theory is a well-defined intersection theory for orbifolds.
For this, I will discuss the Steenrod problem for orbifolds, and prove that the rational homology groups of an orbifold have a basis consisting of suborbifolds.
This enables us to define the polyfold Gromov-Witten invariants as an intersection number against a basis of representing suborbifolds.
This interpretation is essential for the Gromov-Witten axioms, and a proof that the classical pseudocycle Gromov-Witten invariants are a strict subset of the polyfold Gromov-Witten invariants.