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

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 $H_{\ast }\left(L_{f}Y;\mathbb{Q}\right)$ defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.