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

Takahito Naito

Algebraic & Geometric Topology 11 (2011) 2369–2390

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 ; ) defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.

string topology, Hochschild (co)homology, mapping space, free loop space, rational homotopy theory
Mathematical Subject Classification 2010
Primary: 55P35, 55P50
Secondary: 55P62
Received: 26 January 2011
Revised: 10 May 2011
Accepted: 10 July 2011
Published: 5 September 2011
Takahito Naito
Interdisciplinary Graduate School of Science and Technology
Shinshu University
3-1-1 Asahi
Matsumoto, Nagano 390-8621