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Spectral sequences in string topology. (English) Zbl 1227.55007

This paper describes the behaviour of Serre spectral sequences with respect to the Chas-Sullivan product and coproduct in generalized homology theories. More precisely let \(h_*\) be a homology theory and let \(M\) be a \(d\)-dimensional \(h_*\)-oriented smooth manifold. The author proves the following two theorems.
Theorem 1. The Serre spectral sequence for the loop space fibration \(\Omega^nM\to L^nM\to M\) is a multiplicative spectral sequence converging to the Chas-Sullivan algebra \(h_*(L^nM)\). The \(E^2\)-term, \(H_*(M;h_*(\Omega^nM))\), is the intersection homology on \(M\) with coefficients in the Pontryagin algebra \(H_*(\Omega^nM)\).
Theorem 2. Let \(M\to N\to O\) be a fiber bundle of \(h_*\)-oriented manifolds. Then the Serre spectral sequence for the fibration \(L^nM\to L^nN\to L^nO\) is a multiplicative spectral sequence converging to the Chas-Sullivan algebra \(h_*(L^nN)\) whose \(E^2\)-term \(h_*(L^nO;h_*(L^nM))\) is the Chas-Sullivan algebra on \(L^nO\) with coefficients in \(h_*(L^nM)\).
The text contains similar results on coproduct structures. Indeed Theorems 1 and 2 and their applications generalize to arbitrary homology theories ideas already developed in singular homology. Theorem 1 was introduced by R. L. Cohen, J. D. S. Jones and J. Yan to compute explicit loop homology algebras [Prog. Math. 215, 77–92 (2004; Zbl 1054.55006)]. Theorem 2 has been described in singular homology by J.-F. Le Borgne [Expo. Math. 26, No. 1, 25–40 (2008; Zbl 1137.55004)].
The proofs of the results use Jakob’s bordism description of string topology as introduced by David Chataur. The results are applied to study ordinary homology of sphere bundles and generalized homologies of the free loop space of spheres and projective spaces.

MSC:

55P50 String topology
55P35 Loop spaces
55T10 Serre spectral sequences
57R19 Algebraic topology on manifolds and differential topology
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References:

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