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

✍ Scribed by Pascal Lambrechts

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 142 KB
• Volume: 161
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(00)00098-0

✦ Synopsis

Let X be a simply connected ÿnite CW-complex such that dim * (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 Hn(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 ÿnite-dimensional cohomology.