Algebraic & Geometric Topology

Spectral sequences in string topology

Lennart Meier

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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 ΩnMLnMM for an h–oriented manifold M and the looped fiber bundle LnFLnELnB of a fiber bundle FEB 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 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.

Article information

Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2829-2860.

Received: 8 June 2010
Revised: 12 September 2011
Accepted: 14 September 2011
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P35: Loop spaces 55T10: Serre spectral sequences
Secondary: 57R19: Algebraic topology on manifolds

string topology free loop space Serre spectral sequence Gysin morphism


Meier, Lennart. Spectral sequences in string topology. Algebr. Geom. Topol. 11 (2011), no. 5, 2829--2860. doi:10.2140/agt.2011.11.2829.

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