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 greater than or equal to 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 S-m, i.e., the space of n-tuples of points (x(1),..,x(n)) where x(i) is an element of S-m and x(i) not equal x(i+1) for i = 1,...,n. (C) 2002 Elsevier Science Ltd. All rights reserved.