Loop homology of spheres and complex projective spaces

In his Inventiones paper, Ziller (1977) computed the integral homology as a graded abelian group of the free loop space of compact, globally symmetric spaces of rank 1. Chas and Sullivan (1999) showed that the homology of the free loop space of a compact closed orientable manifold can be equipped with a loop product and a BV-operator making it a Batalin-Vilkovisky algebra. Cohen, Jones and Yan (2004) developed a spectral sequence which converges to the loop homology as a spectral sequence of algebras. They computed the algebra structure of the loop homology of spheres and complex projective spaces by using Ziller's results and the method of Brown-Shih (1959, 1962). In this note we compute the loop homology algebra by using only spectral sequences and the technique of universal examples. We therefore not only obtain Ziller's and Brown-Shih's results in an elementary way, but we also replace the roundabout computations of Cohen, Jones and Yan (2004) making them independent of Ziller's and Brown-Shih's work. Moreover we offer an elementary technique which we expect can easily be generalized and applied to a wider family of spaces, not only the globally symmetric ones.