Generic equivariant metrics yielding cleanly cut moduli spaces

Determine whether, for any finite group action on a manifold and any fixed equivariant Morse function, a generic equivariant Riemannian metric produces moduli spaces of Morse trajectories that are cleanly cut out in the sense that, near every trajectory, the section defining the gradient flow equation intersects the zero section cleanly after a finite-dimensional reduction.

Background

The paper relaxes the usual Morse–Smale transversality condition to a clean intersection condition, enabling computable obstruction-bundle gluing and semi-global Kuranishi structures. It establishes clean intersection in several settings, such as for index 0 moduli spaces in regular 1-parameter metric families and for manifolds with boundary where Morse–Smale may be unattainable.

Motivated by these examples, the authors consider the equivariant setting, where transversality is often obstructed due to symmetry constraints. They ask whether an analogous genericity result holds for equivariant metrics—namely, whether the moduli spaces of equivariant Morse trajectories can be made cleanly cut out by a generic choice of metric that respects the group action.