On the mapping space homotopy groups and the free loop space homology groups

Naito, Takahito¹

¹Interdisciplinary Graduate School of Science and Technology, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan

Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 2369-2390

Résumé

Let X be a Poincaré duality space, Y a space and f : X → Y a based map. We show that the rational homotopy group of the connected component of the space of maps from X to Y containing f is contained in the rational homology group of a space LfY which is the pullback of f and the evaluation map from the free loop space LY to the space Y . As an application of the result, when X is a closed oriented manifold, we give a condition of a noncommutativity for the rational loop homology algebra H∗(LfY ; ℚ) defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.

Détail

DOI : 10.2140/agt.2011.11.2369

Classification : 55P35, 55P50, 55P62
Keywords: string topology, Hochschild (co)homology, mapping space, free loop space, rational homotopy theory

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