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The loop-product spectral sequence. (English) Zbl 1137.55004

For a smooth embedding \(f:X\to Y\) of finite codimension \(k\) between Hilbert connected smooth manifolds without boundaries, it is known that there is a well defined \(H_*(Y)\)-comodule homomorphism \(f_{!}:H_*(Y)\to H_*(X)\) called the homology shriek map.
In this paper the author proves that the shriek map associated to a sub-fiberwise embedding behaves properly in the associated Serre spectral sequences and that it induces a homomorphism of spectral sequences between them. In particular, he applies this result to the Chas-Sullivan loop product of the total space of a fibration and computes the loop homology of sphere bundles.

MSC:

55P35 Loop spaces
55N33 Intersection homology and cohomology in algebraic topology
55T10 Serre spectral sequences
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References:

[1] G.E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer, Berlin, 1993, pp. 223-228.; G.E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer, Berlin, 1993, pp. 223-228.
[2] M. Chas, D. Sullivan, String topology, Preprint math.GT/9911159.; M. Chas, D. Sullivan, String topology, Preprint math.GT/9911159. · Zbl 1185.55013
[3] R.L. Cohen, J.D.S. Jones, Jun Yan, The Loop-homology Algebra of Spheres and Projective Spaces, Categorical Decomposition Techniques in Algebraic Topology, Progress in Mathematics, vol. 215, BirkhaÄuser-Verlag, 2004.; R.L. Cohen, J.D.S. Jones, Jun Yan, The Loop-homology Algebra of Spheres and Projective Spaces, Categorical Decomposition Techniques in Algebraic Topology, Progress in Mathematics, vol. 215, BirkhaÄuser-Verlag, 2004.
[4] Crabb, M.; James, I., Fiberwise Homotopy Theory, Springer Monographs in Mathematics (1998), Springer: Springer Berlin
[5] Fadell, E. R.; Husseini, S. Y., Geometry and Topology of Configuration Spaces, Springer Monograph in Mathematics (2001), Springer: Springer Berlin, Heidelberg
[6] K. Gruher, P. Salvatore, Generalized string topology operations, Preprint Arxiv:math.AT/0602210v1, 10 February 2006.; K. Gruher, P. Salvatore, Generalized string topology operations, Preprint Arxiv:math.AT/0602210v1, 10 February 2006.
[7] S. Lang, Differential and Riemannian manifolds, third ed., Graduate Text in Mathematics, vol. 160, Springer, New York, 1995, pp. 108-113.; S. Lang, Differential and Riemannian manifolds, third ed., Graduate Text in Mathematics, vol. 160, Springer, New York, 1995, pp. 108-113.
[8] J. Mac Cleary, Users Guide to Spectral Sequences, second ed., Cambridge Studies in Advanced Mathematics, vol. 58, 2001.; J. Mac Cleary, Users Guide to Spectral Sequences, second ed., Cambridge Studies in Advanced Mathematics, vol. 58, 2001.
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