Loop homology of spheres and complex projective spaces

TL;DR

This paper provides an elementary, spectral sequence-based method to compute the loop homology algebra of spheres and complex projective spaces, simplifying previous complex computations and enabling broader applications.

Contribution

It introduces a new elementary technique using spectral sequences and universal examples for computing loop homology, independent of prior complex methods.

Findings

Computed loop homology algebra of spheres and projective spaces using spectral sequences.

Simplified previous computations, making them more accessible.

Proposed a method adaptable to a wider class of spaces.

Abstract

In his Inventiones paper, Ziller (Invent. Math: 1-22, 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 (String Topology, 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 (The loop homology algebra of spheres and projective spaces, 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 (Ann. of Math. 69:223-246, 1959, Publ. Math. Inst. Hautes \'Etudes Sci. 3: 93-176, 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 Zillers' and Brown-Shihs' results in an elementary way, we also replace the roundabout computations of Cohen, Jones and Yan (The loop homology algebra of spheres and projective spaces, 2004) making them independent of Ziller's and Brown-Shihs' 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.