Spectral sequences in string topology

Meier, Lennart¹

¹Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Algebraic and Geometric Topology, Tome 11 (2011) no. 5, pp. 2829-2860

Résumé

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 ΩnM → LnM → M for an h∗–oriented manifold M and the looped fiber bundle LnF → LnE → LnB of a fiber bundle F → E → 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 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.

Détail

DOI : 10.2140/agt.2011.11.2829

Keywords: string topology, free loop space, Serre spectral sequence, Gysin morphism

Bibliographie

[1] M Chas, D Sullivan, String topology

[2] D Chataur, A bordism approach to string topology, Int. Math. Res. Not. (2005) 2829

[3] R L Cohen, V Godin, A polarized view of string topology, from: "Topology, geometry and quantum field theory" (editor U Tillmann), London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 127

[4] R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773

[5] R L Cohen, J D S Jones, J Yan, The loop homology algebra of spheres and projective spaces, from: "Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001)" (editors G Arone, J Hubbuck, R Levi, M Weiss), Progr. Math. 215, Birkhäuser (2004) 77

[6] R L Cohen, A A Voronov, Notes on string topology, from: "String topology and cyclic homology", Adv. Courses Math. CRM Barcelona, Birkhäuser (2006) 1

[7] P E Conner, Differentiable periodic maps, Lecture Notes in Math. 738, Springer (1979)

[8] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monogr. 47, Amer. Math. Soc. (1997)

[9] Y Félix, S Halperin, J C Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer (2001)

[10] M Goresky, N Hingston, Loop products and closed geodesics, Duke Math. J. 150 (2009) 117

[11] M Jakob, An alternative approach to homology, from: "Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999)" (editors D Arlettaz, K Hess), Contemp. Math. 265, Amer. Math. Soc. (2000) 87

[12] S Kallel, P Salvatore, Rational maps and string topology, Geom. Topol. 10 (2006) 1579

[13] M Kreck, Differential algebraic topology: From stratifolds to exotic spheres, Graduate Studies in Math. 110, Amer. Math. Soc. (2010)

[14] J F Le Borgne, The loop-product spectral sequence, Expo. Math. 26 (2008) 25

[15] L Meier, A geometric view on string topology, Diplomarbeit, Rheinische Friedrich-Wilhelms-Universität Bonn (2009)

[16] L Meier, A Hilbert manifold model for mapping spaces, preprint (2009)

[17] J Milnor, Morse theory, Annals of Math. Studies 51, Princeton Univ. Press (1963)

[18] F Quinn, Transversal approximation on Banach manifolds, from: "Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968)", Amer. Math. Soc. (1970) 213

[19] H Tamanoi, Loop coproducts in string topology and triviality of higher genus TQFT operations, J. Pure Appl. Algebra 214 (2010) 605

[20] M Vigué-Poirrier, D Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976) 633

[21] C T C Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960) 292