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 Birkhoff in the 1920s. Our proof is based on topological methods of calculus of variations — equivariant Morse and Lusternik–Schnirelman 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≠xi+1 for i=1,…,n.