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

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We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in ᵐ⁺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 Lusternik - Schirelman 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=1,2,..., n.