Let be a Poincaré duality space, a space and a based map. We show that the rational homotopy group of the connected component of the space of maps from to containing is contained in the rational homology group of a space which is the pullback of and the evaluation map from the free loop space to the space . As an application of the result, when is a closed oriented manifold, we give a condition of a noncommutativity for the rational loop homology algebra defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.
"On the mapping space homotopy groups and the free loop space homology groups." Algebr. Geom. Topol. 11 (4) 2369 - 2390, 2011. https://doi.org/10.2140/agt.2011.11.2369