  We construct and discuss new numerical homotopy invariants of topological
spaces that are suitable for the study of functions on loop and sphere spaces.
These invariants resemble the Lusternik-Schnirelmann category and provide lower
bounds for the numbers of critical orbits of SO(n)-invariant functions on
spaces of n-spheres in a manifold. Lower bounds on these invariants are derived
using weights of cohomology classes. As an application, we prove new existence
results for closed geodesics on Finsler manifolds of positive flag curvature
satisfying a pinching condition.
