Mathematical Research Letters 10, 435–445 (2003) ON THE MORSE INEQUALITIES FOR GEODESICS ON LORENTZIAN MANIFOLDS A. Abbondandolo, V. Benci, D. Fortunato, and A. Masiello Abstract. We extend the classical Morse inequalities in Riemannian Geometry to the geodesics joining two nonconjugate points on a Lorentzian manifold. The Morse inequalities are obtained developing a Morse Theory for a class of strongly indefinite functionals. 1. Introduction Morse Theory for Riemannian geodesics relates the set of the geodesics joining two nonconjugate points on a complete Riemannian manifold to the topologi- cal structure of the manifold. In particular the Morse Inequalities give a lower bound on the number of such geodesics by the Betti numbers of the based loop space. Let (M,g) be a smooth, connected and complete Riemannian manifold and let Ω(M) be the based loop space of the manifold M, equipped with the compact–open topology. Moreover, let H k (Ω(M); K) be the k–th singular ho- mology group of the space Ω(M) with respect to the field K. The Betti numbers β k (Ω(M); K), k ∈ N, are defined as the dimension of H k (Ω(M); K). Let p and q two nonconjugate points for the Riemannian metric g on M and denote by G(p,q) the set of the geodesics joining p and q. The Morse Relations state that there exists a formal series Q(λ)= ∑ ∞ k=0 a k λ k , with a k ∈ N ∪{+∞}, such that  x∈G(p,q) λ m(x) = ∞  k=0 β k (M; K)λ k +(1+ λ)Q(λ) The series ∑ ∞ k=0 β k (M; K)λ k is often called Poincar´ e polynomial of the manifold M with coefficients in the field K. For any k ∈ N let G k (p,q) be the number of geodesics z in G(p,q) having Morse index m(z ) equal to k. From the Morse Relations one can deduce the Morse Inequalities for the geodesics joining p and q, which state that for any k ∈ N, (1.1) G k (p,q) ≥ β k (Ω(M); K). TheproofoftheMorseRelationsisobtainedapplyingtheabstractMorseTheory to the action integral (1.2) E(x)=  1 0 g(x(s))[˙ x(s), ˙ x(s)]ds Received October 1, 2002. 435