On the Betti numbers of the free loop space of a coformal space

Abstract

Let X be a simply connected finite CW-complex such that dim π ∗(X)⊗ Q =∞ . We prove that if X is coformal (this is an hypothesis coming from the rational homotopy theory) then the sequence of rational Betti numbers of its free loop space, ( dim H n(X S 1 ; Q )) n≥1 , has an exponential growth. Since the Betti numbers of the free loop space on a simply connected closed Riemannian manifold bound below the number of closed geodesics, we deduce from the inequality above that on hyperbolic coformal manifolds, the number of closed geodesics of length ≤ t grows exponentially with t. Our methods permit also to prove a dichotomy theorem for the growth of Hochschild homology of graded Lie algebras of finite-dimensional cohomology.