#### Volume 11, issue 4 (2011)

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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
##### Abstract

Let $X$ be a Poincaré duality space, $Y$ a space and $f:X\to Y$ 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 ${L}_{f}Y$ 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 ${H}_{\ast }\left({L}_{f}Y;ℚ\right)$ defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.

##### Keywords
string topology, Hochschild (co)homology, mapping space, free loop space, rational homotopy theory
##### Mathematical Subject Classification 2010
Primary: 55P35, 55P50
Secondary: 55P62