Spectral Sequences in String Topology

Details Spectral Sequences in String Topology Spectral Sequences in String Topology

2010-01-27

arxiv.org/abs/1001.4906v2

Mathematics

Algebraic Topology

22 pages

Scientific paper

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 the 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.

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