Let M1 and M2 be two simply connected closed manifolds of the same dimension. It is proved that
(1) if k is a coefficient field such that neither M1 nor M2 has the same cohomology as a sphere, then
the sequence (bk)k[ges ]1 of Betti numbers of the free loop space on M1 #M2 is unbounded;
(2) if, moreover, the cohomology H*(M1;k) is not generated as algebra by only one element, then the
sequence (bk)k[ges ]1 has an exponential growth.
Thanks to theorems of Gromoll and Meyer and of Gromov, this implies, in case 1, that there exist
infinitely many closed geodesics on M1#M2 for each Riemannian metric, and, in case 2, that for a generic
metric, the number of closed geodesics of length [les ]t grows exponentially with t.