Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in Rm+1 for m ≥ 3. For plane billiards (when m = 1) such bounds were obtained by G. Birkhoff in the 1920’s. Our proof is based on topological methods of calculus of variations – equivariant Morse and LusternikSchnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere Sm, i.e., the space of n-tuples of points (x1, . . . , xn), where xi ∈ Sm and xi 6= xi+1 for i = 1, . . . , n.