Spectral sequences in string topology

Abstract

In this paper, we investigate the behavior of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specifically with the Chas–Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fiber bundles: the fiber bundle ${\Omega }^{n}M\to L^{n}M\to M$ for an $h_{\ast }$–oriented manifold $M$ and the looped fiber bundle $L^{n}F\to L^{n}E\to L^{n}B$ of a fiber bundle $F\to E\to B$ of $h_{\ast }$–oriented manifolds. Our method lies in the construction of Gysin morphisms of spectral sequences. We apply these results to study the ordinary homology of the free loop spaces of sphere bundles and some generalized homologies of the free loop spaces of spheres and projective spaces. For the latter purpose, we construct explicit manifold generators for the homology of these spaces.