Spectral Sequences in String Topology

TL;DR

This paper explores how the Serre spectral sequence interacts with string topology structures in generalized homology theories, establishing compatibility for specific fiber bundles and applying results to free loop spaces of spheres and projective spaces.

Contribution

It introduces Gysin morphisms of spectral sequences to analyze the compatibility of algebraic structures in string topology within generalized homology theories.

Findings

Compatibility of spectral sequences with string topology operations established

Gysin morphisms constructed for spectral sequences in this context

Results applied to homology of free loop spaces of spheres and projective spaces

Abstract

In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundles: the fibre bundle $\Omega^n M \to L^n M \to M$ for an $h_*$-oriented manifold $M$ and the looped fibre bundle $L^n F \to L^n E \to L^n B$ of a fibre bunde $F \to E \to B$ of $h_*$-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 generalized homologies of the free loop spaces of spheres and projective spaces.