Algebraic & Geometric Topology

Spectral sequences in string topology

Lennart Meier

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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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715307

Digital Object Identifier
doi:10.2140/agt.2011.11.2829

Mathematical Reviews number (MathSciNet)
MR2846913

Zentralblatt MATH identifier
1227.55007

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

Keywords
string topology free loop space Serre spectral sequence Gysin morphism

Citation

Meier, Lennart. Spectral sequences in string topology. Algebr. Geom. Topol. 11 (2011), no. 5, 2829--2860. doi:10.2140/agt.2011.11.2829. https://projecteuclid.org/euclid.agt/1513715307


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